Time-varying Uncertainties Research Articles

An efficient uncertainty propagation method for nonlinear dynamics with distribution-free P-box processes

The distribution-free P-box process serves as an effective quantification model for time-varying uncertainties in dynamical systems when only imprecise probabilistic information is available. However, its application to nonlinear systems remains limited due to excessive computation. This work develops an efficient method for propagating distribution-free P-box processes in nonlinear dynamics. First, using the Covariance Analysis Describing Equation Technique (CADET), the dynamic problems with P-box processes are transformed into interval Ordinary Differential Equations (ODEs). These equations provide the Mean-and-Covariance (MAC) bounds of the system responses in relation to the MAC bounds of P-box-process excitations. They also separate the previously coupled P-box analysis and nonlinear-dynamic simulations into two sequential steps, including the MAC bound analysis of excitations and the MAC bounds calculation of responses by solving the interval ODEs. Afterward, a Gaussian assumption of the CADET is extended to the P-box form, i.e., the responses are approximate parametric Gaussian P-box processes. As a result, the probability bounds of the responses are approximated by using the solutions of the interval ODEs. Moreover, the Chebyshev method is introduced and modified to efficiently solve the interval ODEs. The proposed method is validated based on test cases, including a duffing oscillator, a vehicle ride, and an engineering black-box problem of launch vehicle trajectory. Compared to the reference solutions based on the Monte Carlo method, with relative errors of less than 3%, the proposed method requires less than 0.2% calculation time. The proposed method also possesses the ability to handle complex black-box problems.

Adaptive backstepping controller based on a novel framework for dynamic solution of an ankle rehabilitation spherical parallel robot

AbstractThis research offers an adaptive model-based methodology for autonomous control of 3-RRR spherical parallel manipulator (RSPM) based on a novel modeling framework. RSPM is an overconstrained parallel mechanism that has a variety of applications in medical procedures such as ankle rehabilitation because of its precision and accuracy. However, obtaining a complete explicit dynamic model of these mechanisms for tracking purposes has been a problematic challenge due to their inherent singularities, coupling effects of the limbs, and redundant constraints imposed by the intermediate joints. This paper presents a novel algorithm to obtain the analytical kinematic solutions of RSPMs based on the closed-loop vector method, which includes constraint analysis. By incorporating constrained kinematics into the dynamic model, a comprehensive explicit dynamic solution of the non-overconstrained version 3-RCC of RSPM is developed in task space, based on screw theory and the linear homogeneous property of algebraic equations on the manipulator twist. Based on the proposed computational framework, a robust self-tuning backstepping control (STBC) strategy is applied to the robot to overcome the effect of external disturbances and time-varying uncertainties. Furthermore, an observer-based compensation (OBC) method is presented for dealing with the nonlinear hysteresis loops of the ankle during trajectory tracking purposes. The closed-loop stability of the whole system including STBC and OBC is theoretically performed by Lyapunov methods. The proposed methodologies are validated by realistic co-simulations in different scenarios. For instant, in the presence of external disturbances, the maximum tracking error norm of STBC is 37.5% less than the sliding mode approach.

Automatic berthing of unmanned surface vessels with predetermined performance

To solve the problem of ship automatic berthing control due to unknown time-varying disturbance and dynamic uncertainty of model parameters, an automatic berthing control law based on predefined performance time function is proposed. First, a predefined performance time function is designed and coupled with tracking error to achieve the predetermined performance of tracking error. Secondly, radial basis function neural network is used to approach the dynamic uncertainty of ship model parameters, and the complex uncertainty of model parameters and unknown time-varying disturbance is represented by linearized parameter form with single virtual parameter, which makes the calculation simple and easy to implement in engineering. On this basis, the reverse step control law is designed. Thirdly, the stability of the system is proved based on Lyapunov stability theory. Finally, the simulation results show that the control law can make the ship reach the desired position and heading angle, and realize the automatic berthing of the ship. The control law and berthing controller designed in this paper have good applicability and robustness, which provides a theoretical basis for the subsequent control research of surface intelligent ships.

Adaptive robust control for fuzzy underactuated mechanical systems: A Stackelberg game-theoretic optimization approach

This article proposes a Stackelberg game-theoretic optimization approach for adaptive robust controller design of fuzzy underactuated mechanical systems (UMSs). As a commonly encountered challenge, several factors render it difficult to further improve the control performance of these UMSs, such as the coupling effects and nonlinearities resulting from the underactuated structure. Meanwhile, various sources of unexpected time-varying uncertainties can further add to the difficulty in the controller design procedure pipeline. To overcome the above-mentioned issues and further enhance the system performance, a dual-parameter optimization framework is presented here for adaptive robust controller design, leveraging fuzzy set theory and Stackelberg game. In this development, the time-varying uncertainty is characterized via suitable membership functions; and thus the appropriate fuzzy UMS is established. Then, an adaptive robust controller with a leakage-type adaptation law is presented to tackle the uncertainty appropriately. To further improve the system performance considering the trade-off among the transient performance; the steady-state performance; and the control cost; a Stackelberg game-based dual-parameter optimization method is proposed to determine the Stackelberg strategy. It is rigorously proved in the developments here that the proposed controller design and parameters optimization method guarantees the desired deterministic system performance, i.e., uniform boundedness (UB) and uniform ultimate boundedness (UUB). Moreover, a series of numerical simulations based on a flywheel inverted pendulum (FIP) are performed, which illustrate the effectiveness and superiority of the proposed method.

Distributed Constrained Consensus of Multiagent Systems With Uncertainties and Disturbances Under Switching Directed Graphs

This paper provides a distributed leaderless consensus control framework for nonlinear multi-agent systems with time-varying asymmetric state constraints, uncertainties, and disturbances under switching directed graphs. In such a framework, original constrained states of agents are first transformed into free states in a transformed state space. To deal with switching directed graphs, we drive agents towards consensus in the transformed space by leveraging a model reference control scheme, and it is sufficient that the original states reach consensus strictly subject to the time-varying constraints under mild assumptions. A single-layer neural network with weights adapted online is leveraged to approximate the uncertainties in agent dynamics. For external disturbances and reconstruction errors in the approximation, we introduce a robust term with an adaptive gain for compensation. Distributed consensus algorithms are proposed, respectively, for multi-agent systems with first- or second-order dynamics. We prove convergence to consensus via Lyapunov analysis and study the proposed algorithms' performance using numerical simulations.

New evidence on US monetary policy activism and the Taylor rule

Abstract We provide new evidence about US monetary policy using a model that: (i) estimates time-varying monetary policy weights without relying on stylized theoretical assumptions; (ii) allows for endogenous breakdowns in the relationship between interest rates, inflation, and output; and (iii) generates a unique measure of monetary policy activism that accounts for economic instability. The joint incorporation of endogenous time-varying uncertainty about the monetary policy parameters and the stability of the relationship between interest rates, inflation, and output materially reduces the probability of determinate monetary policy. The average probability of determinacy over the period post-1982 to 1997 is below 60% (hence well below seminal estimates of determinacy probabilities that are close to unity). Post-1990, the average probability of determinacy is 75%, falling to approximately 60% when we allow for typical levels of trend inflation.

Observer-based preview control for T-S fuzzy systems

PurposeConsidering the unmeasurable states of the systems and the previewed reference signal, a novel fuzzy observer-based preview controller, which is a mixed controller of the fuzzy observer-based controller, fuzzy integrator and preview controller, is considered to address the tracking control problem.Design/methodology/approachThe authors employ an augmentation technique to construct an augmented error system for uncertain T-S fuzzy discrete-time systems with time-varying uncertainties. Additionally, the authors obtain the corresponding linear matrix inequality (LMI) conditions for designing the preview controller.FindingsThis paper discusses the preview tracking problem for nonlinear systems. First, considering the unmeasurable states of the systems and the previewed reference signal, a novel fuzzy observer-based preview controller, which is a mixed controller of the fuzzy observer-based controller, fuzzy integrator, and preview controller, is considered to address the tracking control problem. Then, using the fuzzy Lyapunov functional with the linear matrix inequality (LMI) technique, new sufficient conditions for the asymptotic stability of the augmented system are derived by applying the LMI technique. The preview controller and fuzzy observer can be designed in one step. Finally, a numerical example is used to illustrate the effectiveness of the results.Originality/valueAn augmented error system is successfully constructed by the state augmentation approach. A novel preview controller is designed to address the tracking control problem. The preview controller and fuzzy observer can be designed in one step.

Self-tuning PID feedback control method for magnetic suspension active vibration isolation system with parameters uncertainty

A self-tuning PID (Proportion-Integral-Derivative) control method based on the adaptive mechanism was proposed for the magnetic suspension active vibration isolation system with the characteristics of nonlinear, time-varying, and model parameter uncertainty. The controller identified the controlled object parameters online and solved the controller parameters in real time. To test the effectiveness of the proposed method, a six degree-of-freedom (DOF) magnetic suspension active vibration isolation system was designed and built. The six DOF dynamic model of the vibration isolation system with uncertain model parameters was derived for control. Experiments were carried out and compared with the control effects of the tradition PID control and cascade PID control algorithm. The experimental results show that the proposed self-tuning PID controller has more outstanding practicality and effectiveness in solving the model parameter uncertainty problem compared with the existing studies at home and abroad, and it also proves that the control method has a good isolation effect on the wide band disturbances.

Adaptive Control of Unmanned Aerial Vehicles with Varying Payload and Full Parametric Uncertainties

This article investigates an adaptive tracking control problem for a six degrees of freedom (6-DOF) nonlinear quadrotor unmanned aerial vehicle (UAV) with a variable payload mass. The changing payload introduces time-varying parametric uncertainties into the dynamical model, rendering a static control strategy no longer effective. To handle this issue, two adaptive schemes are developed to maintain the uncertainties in the translational and rotational dynamics. Initially, a virtual proportional derivative (PD) is designed to stabilize the horizontal position; however, due to an unknown and time-varying mass, an adaptive controller is proposed to generate the total thrust of the UAV. Furthermore, an adaptive controller is designed for the rotational dynamics, to handle parametric uncertainties, such as inertia and external disturbance parameters. In both schemes, a standard adaptive scheme using the certainty equivalence principle is extended and designed. A stability analysis was conducted with rigorous analytical proofs to show the performance of our proposed controllers, and simulations were implemented to assess the performance against other existing methods. Tracking fitness and total control efforts were calculated and compared with closed-loop adaptive tracking control (CLATC) and adaptive sliding mode control (ASMC). The results indicated that the proposed design better maintained UAV stability.

Robust cascade observer for a disturbance unmanned aerial vehicle carrying a load under multiple time-varying delays and uncertainties

In this article, a novel approach of a robust flight observer for a disturbance quadrotor transporting a payload in the presence of multiple time-varying delays is proposed. The developed algorithm is not only pertinent to a vehicle output signal with multiple small time-varying delays but also to long time-varying delays that are common for a quadrotor transporting a hanging load because a cascade structure is also employed to achieve outstanding performance, and the delay can be removed. The multiple time-varying delays and disturbances are compensated in the system with the proposed observer. The stability of the proposed methodology is proved using the Lyapunov–Krasovskii functional associated with an H ∞ criterion. Numerical simulations are shown to assess and validate the observer's performance and robustness under multiple time-varying delays and disturbances.

Reinforcement Learning-Based 3D Trajectory Tracking Control of Hypersonic Gliding Vehicles With Time-Varying Uncertainties

Reinforcement Learning-Based 3D Trajectory Tracking Control of Hypersonic Gliding Vehicles With Time-Varying Uncertainties

Fixed-Time Composite Neural Learning Control of Flexible Telerobotic Systems.

This article is devoted to the fixed-time synchronous control for a class of uncertain flexible telerobotic systems. The presence of unknown joint flexible coupling, time-varying system uncertainties, and external disturbances makes the system different from those in the related works. First, the lumped system dynamics uncertainties and external disturbances are estimated successfully by designing a new composite adaptive neural networks (CANNs) learning law skillfully. Moreover, the fast-transient, satisfactory robustness, and high-precision position/force synchronization are also realized by design of fixed-time impedance control strategies. Furthermore, the "complexity explosion" issue triggered by traditional backstepping technology is averted efficiently via a novel fixed-time command filter and filter compensation signals. And then, sufficient conditions of system controller parameters and fixed-time stability are theoretically given by establishing the Lyapunov stability theorem. Besides, the absolute stability of the two-port networked system under complex transmission time delays is rigorously proved. Finally, simulations are performed with 2-link flexible telerobotic systems under two cases, results are presented to realistically verify the proposed control algorithm available.

Precise Tracking Control for Articulating Crane: Prescribed Performance, Adaptation, and Fuzzy Optimality by Nash Game.

Articulating crane (AC) is used in various industrial activities. The articulated multisection arm exacerbates nonlinearities and uncertainties, making the precise tracking control challenging. This study proposes an adaptive prescribed performance tracking control (APPTC) for AC to robustly fulfill the task of precise tracking control, with adaptation to resist time-variant uncertainties, whose bounds are unknown but lie in prescribed fuzzy sets. Particularly, a state transformation is applied to simultaneously track the desired trajectory and satisfy the prescribed performance. Adopting the fuzzy set theory to describe uncertainties, APPTC does not invoke any IF-THEN fuzzy rules. There is no linearizations, or nonlinear cancelation for APPTC, thus making it approximation free. The performance of the controlled AC is twofold. First, deterministic performance in fulfilling the control task is ensured by the Lyapunov analysis using uniform boundedness and uniform ultimate boundedness. Second, fuzzy-based performance is further improved by an optimal design, which seeks the optima of control parameters by formulating a two-player Nash game. The existence of Nash equilibrium is theoretically proved, and its acquisition process is given. The simulation results are provided for validations. This is the first endeavor that explores the precise tracking control for fuzzy AC.

Cross-coupling indirect iterative learning control method for batch processes with time-varying uncertainties

For batch time-varying processeswith non-repetitive disturbances, a cross-coupling indirect iterative learning control (CC-iILC) is proposed. The set trajectory of the system on the single axis is accurately tracked by an indirect iterative learning control strategy with a PI controller for its time direction and a closed-loop feedback control strategy for the batch direction. Under the asymptotically stable condition of the two-dimensional (2D) dynamic model, the optimal control law is obtained by optimizing the H-infinity control function. To avoid the contour error during coupling, the cross-coupling technique is used to distribute the contour error to each axis for compensation. The stability condition of the indirect iterative learning controller is analyzed by a two-dimensional Fornasini-Marchesini (FM) batch dynamics model and two-dimensional robust H-infinity control theory. The feasibility and superiority of the proposed control method are verified by numerical simulation and experimental tests.

A novel hybrid time-variant reliability analysis method through approximating bound-most-probable point trajectory

In the engineering field, time-variant reliability analysis (TRA) is used to measure the safety level of structures under time-variant uncertainties. Lacking in information or data, some uncertainties cannot be directly quantified as stochastic models, which results in the simultaneous existence of aleatory and epistemic uncertainties in most of problems. In general, stochastic and interval models are respectively used to describe aleatory and epistemic uncertainties. For the hybrid TRA (HTRA) problem considering the two kinds of uncertainties, the existing method needs to excessively evaluate the original time-variant limit-state function, which is too expensive for engineering problems. To address this issue, we propose the concept of bound-most-probable point trajectory (BMPPT) which can be used to construct the approximation of the limit-state hyper-surface. Moreover, we develop a HTRA method based on approximating BMPPT which can further improve the computational efficiency. First, based on time discretization, we transform the HTRA problem into a time-independent series-system reliability problem which can be solved by searching the bound-most-probable point (BMPP) at all discrete time instants. Then, with the BMPPT, the lower and upper bounds of the time-variant limit-state function are linearized into two Gaussian processes. Finally, the expansion optimal linear estimation and Monte Carlo simulation are performed to estimate the time-variant reliability. To avoid excessive BMPP searches, the active learning Kriging is used to approximate the BMPPT. Two numerical examples including a cantilever beam, and a 10-bar truss, and two engineering applications of the solid rocket engine shell and the rocket inter-stage structure are investigated, and the results reveal that the proposed method can solve the HTRA problems with high accuracy and efficiency.

Time-varying uncertainty and disturbance estimator without velocity measurements: Design and application

The uncertainty and disturbance estimator (UDE) technique has been proven to be simple but effective in reconstructing unknown disturbances. However, classic UDE with a constant gain cannot deal with more complicated issues, such as the trade-off between steady-state and transient performance of the estimator, simultaneous attenuation of input disturbances and measurement noises, etc. Motivated by this observation, the design of UDE technique is extended to the time-varying situation and an implementable form of time-varying UDE (TV-UDE) is proposed. In particular, velocity measurements are assumed to be unavailable in the design. A time-varying ordinary differential equation is constructed to build the estimation relationship. A formula for the integration by parts is thereafter introduced to derive an implementable form of the TV-UDE. The classic UDE is shown to be a special case of the TV-UDE. The UDE is combined with a nominal dynamic output-feedback controller to yield a robust control solution to a class of uncertain second-order attitude control systems without velocity measurements. The design of a TV-UDE is discussed with a low initial gain and a higher final gain to avoid transient performance issues. Finally, numerical simulations and physical experiments are carried out on a 2-DOF AERO attitude helicopter platform to demonstrate the control performance of the solution. It is illustrated that the proposed design can effectively reduce the steady-state errors in both the trajectory tracking and target pointing tasks. Meanwhile, the input saturation and peaking phenomena associated with a high-gain estimation are avoided by applying the TV-UDE.

Robust adaptive output feedback for the formation control of heterogeneous ships based on a nonlinear extended state observer

This study considers the formation control problem of heterogeneous ships with unknown time-varying disturbances, dynamic uncertainties, and unmeasurable speeds, a robust adaptive output feedback formation control scheme for the heterogeneous ships based on nonlinear extended state observer is developed. In order to solve the heterogeneous ships formation control problem with leader-following structure, transforming the formation control problem into a class of ship trajectory tracking problem by improving the design of the virtual ship. Firstly, a nonlinear extended state observer is designed for estimating ship speeds due to technical limitations and external disturbances. Then, combining the adaptive neural networks with disturbances adaptive law to deal with the unknown time-varying disturbances and dynamic uncertainties. Meanwhile, the second-order tracking differentiator is designed to calculate the derivative of the virtual control law, which has higher precision comparing to the first-order filter. Stability theory is proved that all the signal of closed-loop control system are uniformly ultimately bounded, and the tracking errors can converge into arbitrary small neighborhoods around zero. Finally, comparative simulations were performed to demonstrate the effectiveness of the proposed controller.

Repetitive process based indirect-type iterative learning control for batch processes with model uncertainty and input delay

This paper develops an indirect iterative learning control scheme for batch processes with time-varying uncertainties, input delay, and disturbances. In this paper, a predictor based on a state observer is designed to estimate the future state and to compensate for the input delay. Then a feedback controller based on the estimated state and the set-point error is used to track the specified reference trajectory, where, of the options available, a robust H∞ controller is designed in the presence of time-varying uncertainties and load disturbances. Then a proportional plus derivative type iterative learning control law is designed. An injection molding process model demonstrates the new method’s effectiveness, and a comparison with a direct-type design is given.

Reinforcement learning-based saturated adaptive robust output-feedback funnel control of surface vessels in different weather conditions

This article tackles the problem of saturated adaptive robust actor–critic learning PID tracking control design guaranteeing a prescribed funnel performance for ships with multiple dead bands in which the ship’s velocities/accelerations and dynamic parameters are unknown and time-variant subject to the availability of external disturbances. The controller is proposed by integrating an observer, adaptive actor–critic multi-layer neural networks, generalized saturation functions, the prescribed performance funnel control, an adaptive robust controller and multiple dead bands. The observer calculates the estimation of ship’s velocities/accelerations, in which the peaking phenomenon is prevented by limiting the observer signals. Within the structure of adaptive actor–critic design, adaptive multi-layer neural networks are exerted, which counteracts against ship’s time-variant uncertainties and the actuator saturation nonlinearity. In order to reduce the likelihood of actuator saturation well, generalized saturation functions are taken into account. To realize prespecified transient and steady-state performance of the tracking errors, the prescribed performance control is applied. With the purpose of avoiding the corner cutting to increase the trajectory tracking accuracy, curvature of the circular trajectories is calculated and the angle of arc is compensated. An adaptive robust controller is utilized to resist against external disturbances, and a scheme of multiple dead bands is devised in this paper to protect the actuator system in heavy seas.

Applying real options with reinforcement learning to assess commercial CCU deployment

Carbon capture and utilization (CCU), which emerged as a means to reduce anthropogenic carbon emissions, has been highlighted to close the carbon cycle and combat climate change. CCU involves utilizing or converting captured CO2 to create value-added products that can replace or supplement fossil fuel-derived products. In order to meet climate goals, commercial-scale CCU facilities need to be built and their capacities increased, but barriers to large-scale CCU deployment still exist, primarily caused by large uncertainties in product/energy prices, markets, technologies, and policies. Conventional techno-economic analysis (TEA) methods cannot appropriately assess viability of CCU deployment projects which include dynamic uncertainties and deployment strategies. More flexible methods that can account for both time-varying uncertainties and dynamic capacity building are needed. We propose a systematic framework for the evaluation of commercial CCU deployment using the theory of real options and reinforcement learning (RL). RL is needed as considering options of adding capacities or delaying/abandoning the project at multiple time points under dynamic uncertainties lead to a large-scale stochastic optimal control problem which cannot be solved using conventional optimization methods like stochastic programming. The framework consists of three steps: surrogate modeling, uncertainty modeling, and assessment modeling. The framework is dependent on the type of CCU technology, uncertainties, real options and RL algorithm. We demonstrate the application of the framework through a case study of CO2 hydrogenation-to-methanol process in Europe, which is a late-stage, i.e., high technology readiness level (TRL), technology.