Set-membership Framework Research Articles

An Adaptive-Noise-Bound-Based Set-Membership Method for Process Identification of Industrial Control Loops

Modeling of key variable data needs to consider the complex characteristics of systems in the catalytic cracking unit (CCU) of petroleum refining process, such as slow time-varying behavior, complex dynamic properties, distributed traits, and unknown stochastic noise. To fully capture the dynamics of a linear ordinary dynamic process without introducing incremental components, an adaptive-noise-bound-based set-membership method (RSMI) is proposed in this paper. Under the set-membership framework, the output set is typically represented as an ellipsoid based on the assumed conditions. Firstly, a CARMA model is considered; longer-duration historical data are selected to capture the intricate dynamic characteristics of industrial control loops. Secondly, RSMI introduces am approach to determine allowance factor, optimizing the noise bound for better suitability in real-world noise environments. The adaptive noise bound is achieved by designing an optimization algorithm that seeks the optimal parameters within the optimization framework. The stability of the RSMI algorithm is demonstrated through the application of the Lyapunov method. Next, the RSMI algorithm has been applied in engineering practice and designed for offline and online training stages of control processes. Finally, simulation experiments are performed to model and predict real-time data of flow, pressure, and liquid-level control loops within a catalytic cracking unit. Furthermore, the effectiveness of the RSMI algorithm is validated through two general examples, and frequency domain analysis is performed.

Multitrajectory Model Predictive Control for Safe UAV Navigation in an Unknown Environment

The problem of navigating an unmanned aerial vehicle (UAV) in an unknown environment is addressed with a novel model predictive control (MPC) formulation, named multitrajectory MPC (mt-MPC). The objective is to safely drive the vehicle to the desired target location by relying only on the partial description of the surroundings provided by an exteroceptive sensor. This information results in time-varying constraints during the navigation among obstacles. The proposed mt-MPC generates a sequence of position set points that are fed to control loops at lower hierarchical levels. To do so, the mt-MPC predicts two different state trajectories, a safe one and an exploiting one, in the same finite horizon optimal control problem (FHOCP). This formulation, particularly suitable for problems with uncertain time-varying constraints, allows one to partially decouple constraint satisfaction (safety) from cost function minimization (exploitation). Uncertainty due to modeling errors and sensors noise is taken into account as well, in a set membership (SM) framework. Theoretical guarantees of persistent obstacle avoidance are derived under suitable assumptions, and the approach is demonstrated experimentally out-of-the-laboratory on a prototype built with off-the-shelf components.

Virtual reference feedback tuning for linear discrete-time systems with robust stability guarantees based on set membership

In this paper we propose a novel methodology that allows us to design, in a purely data-based fashion and for linear single-input and single-output systems, both robustly stable and performing control systems for tracking piecewise constant reference signals. The approach uses both (i) virtual reference feedback tuning for enforcing suitable performance and (ii) the set membership framework for providing a-priori robust stability guarantees. Indeed, an uncertainty set for the system parameters is obtained through set membership identification, where an algorithm based on the scenario approach is proposed to estimate the inflation parameter in a probabilistic way. Based on this set, robust stability conditions are enforced as linear matrix inequality constraints within an optimization problem whose linear cost function relies on virtual reference feedback tuning. To show the generality and effectiveness of our approach, we apply it to two of the most widely used yet simple control schemes, i.e., where tracking is achieved thanks to (i) a static feedforward action and (ii) an integrator in closed-loop.The proposed method is not direct due to the set membership identification. However, the uncertainty set is used with the objective of providing robust stability guarantees for the closed-loop system and it is not used for the definition of the cost function, which instead is based on data. The effectiveness of the method is shown with reference to three simulation examples. A comparison with other data-driven methods is carried out.

Self-Tuning Nonlinear Iterative Learning for a Precision Testing Stage: A Set-Membership Approach

Iterative learning control (ILC) is an appealing method in motion control applications that can achieve the performance limit of feedforward compensation in repeating tasks. Compensating for the repetitive errors rapidly (G1) and at the same time reducing the accumulation of the non-repetitive errors effectively (G2) are two major goals of ILC. However, in the simultaneous presence of model uncertainties and noises, most ILC methods that have been used in application cannot fully achieve these two goals. To tackle this problem, this paper develops a self-tuning nonlinear ILC method (STNILC). Firstly, a new nonlinear ILC method (NILC) is designed in the set-membership framework based on the boundedness noise-uncertainty assumption. Then, a self-tuning algorithm is developed to determine the unknown noise-uncertainty bound that is required by NILC. Theoretical results confirm that desirable transient and steady-state convergence performances can be achieved by the proposed STNILC method.Comparative experimental results on a short-stroke stage for testing the lithographic projection lens verify the superiorities of STNILC in sufficiently achieving G1 and G2.

A Set Membership approach to black-box optimization for time-varying problems

A novel method to tackle black-box optimization for time-varying problems is proposed. Using a Set Membership (SM) framework, the approach directly adjusts the uncertainty associated with old data points as new samples are introduced. Uninformative old samples are discarded, and the adjusted model guides the exploitation and exploration routines as characteristic of black-box optimization. With the proposed method, there is no need to estimate the time-related rate of change of the hidden function, as required in previous literature. We provide results of a benchmark test, comparing the performance of the proposed method to other approaches to time-varying black-box optimization, with promising results.

Zonotopic multi-sensor fusion estimation with mixed delays under try-once-discard protocol: A set-membership framework

In this paper, the fusion estimation problem is investigated for a class of multi-sensor networked systems with unknown-but-bounded noises and mixed time-delays under try-once-discard protocols (TODPs). The measurements of multiple sensor nodes are sent to the remote fusion center through individual network channels, where the TODP is implemented in each channel to schedule the signal transmissions for the purpose of avoiding data collisions. To fuse the information collected from all channels, a sequential estimator is proposed whose estimator parameters are designed such that the estimation error (after each measurement update) is restrained into a zonotope with minimum F-radius at each moment. First, by using the properties of zonotopes, desired zonotopes are obtained that contain the estimation errors, and the F-radii of these zonotopes are subsequently minimized by appropriately designing the estimator parameters. It is shown that, under the proposed zonotopes-based fusion rule, the designed sequential estimator leads to the same estimation accuracy as that of the widely utilized parallel estimator. Finally, an illustrative example is provided to verify the validity of the developed fusion estimation method.

An Interpolatory Algorithm for Distributed Set Membership Estimation in Asynchronous Networks

This article addresses distributed estimation problems over asynchronous networks in a set membership framework. The agents in the network asynchronously collect and process measurements, communicate over a possibly time-varying and unbalanced directed graph and may have nonnegligible computation times. Measurements are affected by bounded errors so that they define feasible sets containing the unknown parameters to be estimated. The proposed algorithm requires each agent to compute a weighted average of its estimate and those of its neighbors and to project it onto a local feasible set. By assuming convexity of the measurement sets, the local estimates are shown to converge to a common point belonging to the global feasible set.

Zonotopic observer designs for uncertain Takagi–Sugeno fuzzy systems

This paper addresses the zonotopic observer design for nonlinear systems affected by uncertainties, i.e., state disturbances and measurement noises using Takagi–Sugeno (TS) fuzzy technique. The system uncertainties are considered as unknown but bounded, which are handled via a set-membership framework. For state estimation purposes, we develop an algorithm to recursively compute the zonotope containing the mismatching nonlinear term caused by unmeasured nonlinearities. Then, two methods are proposed to design the zonotopic observer gains. The first method is based on the minimization of the F−radius of zonotopes, for which the membership-function-dependent observer gain must be completely computed online. For the second method, an ℋ∞ approach is used together with a nonquadratic Lyapunov function to determine the observer gain. Then, the zonotopic observer design is reformulated a convex optimization problem under linear matrix inequalities (LMIs), which can be effectively solved with numerical solvers. An autonomous vehicle application is provided to demonstrate and analyze the effectiveness of both proposed methods.

On interval observer design for active Fault Tolerant Control of Linear Parameter-Varying systems

This paper proposes an active Fault Tolerant Control (FTC) scheme for polytopic uncertain Linear Parameter-Varying (LPV) systems subject to uncertainties and actuator faults. First, a fault estimation interval observer is designed to estimate the system state and the actuator fault. A novel approach is developed using the L ∞ norm to attenuate the effects of the uncertainties and to improve the accuracy of the proposed observer. Then, based on the fault estimation information, the FTC strategy is designed using a linear state feedback control law and H ∞ technique to compensate actuator faults and maintain system performance and stability, even under faulty conditions. Finally, the effectiveness of the proposed method is demonstrated by its application to a vehicle lateral dynamic nonlinear model. • Fault Tolerant Control using interval state estimation and robust stabilization. • Set-membership framework for the design of Linear Parameter Varying systems. • Fault estimation observer design for descriptor system. • L ∞ norm-based design method to attenuate the effect of fault and uncertainties and to improve the estimation performance.

SMGO-[formula omitted]: Balancing caution and reward in global optimization with black-box constraints

In numerous applications across all science and engineering areas, there are optimization problems where both the objective function and the constraints have no closed-form expression or are too complex to be managed analytically, so that they can only be evaluated through experiments. To address such issues, we design a global optimization technique for problems with black-box objective and constraints. Assuming Lipschitz continuity of the cost and constraint functions, a Set Membership framework is adopted to build a surrogate model of the optimization program, that is used for exploitation and exploration routines. The resulting algorithm, named Set Membership Global Optimization with black-box constraints (SMGO-Δ), features one tunable risk parameter, which the user can intuitively adjust to trade-off safety, exploitation, and exploration. The theoretical properties of the algorithm are derived, and the optimization performance is compared with representative techniques from the literature in several benchmarks. An extension to uncertain cost/constraint function outcomes is presented, too, as well as computational aspects. Lastly, the approach is tested and compared with constrained Bayesian optimization in a case study pertaining to model predictive control tuning for a servomechanism with disturbances and plant uncertainties, addressing practically-motivated task-level constraints.

SMGO: A set membership approach to data-driven global optimization

Many science and engineering applications feature non-convex optimization problems where the objective function cannot be handled analytically, i.e. it is a black box. Examples include design optimization via experiments, or via costly finite elements simulations. To solve these problems, global optimization routines are used. These iterative techniques must trade-off exploitation close to the current best point with exploration of unseen regions of the search space. In this respect, a new global optimization strategy based on a Set Membership (SM) framework is proposed. Assuming Lipschitz continuity of the cost function, the approach employs SM concepts to decide whether to switch from an exploitation mode to an exploration one, and vice-versa. The resulting algorithm, named SMGO (Set Membership Global Optimization) is presented. Theoretical properties regarding convergence and computational complexity are derived, and implementation aspects are discussed. Finally, the SMGO performance is evaluated on a set of benchmark non-convex problems and compared with those of other global optimization approaches.

Robust Zonotopic Observer Design: Avoiding Unmeasured Premise Variables for Takagi-Sugeno Fuzzy Systems

Abstract This paper addresses the design of a robust set-based interval observer for a nonlinear discrete-time system affected by system uncertainty (state disturbance and measurement noise) using a Takagi-Sugeno (TS) fuzzy model including an unmeasurable premise variable. The effect of unmeasurable premise variable and uncertainties is considered as of unknown but bounded nature, i.e., in the set-membership framework. A zonotopic representation of a set towards reducing set operations to simple matrix calculations is used to bound the state estimation provided by the interval observer-based approach. Furthermore, the criterion-based approach and H∞ performance technique are considered in order to compute the observer gain to achieve robustness. Finally, an example is employed to both illustrate and discuss the effectiveness of the proposed approach.

NSM Converges to a k-NN Regressor Under Loose Lipschitz Estimates

Although it is known that having accurate Lipschitz estimates is essential for certain models to deliver good predictive performance, refining this constant in practice can be a difficult task especially when the input dimension is high. In this letter, we shed light on the consequences of employing loose Lipschitz bounds in the Nonlinear Set Membership (NSM) framework, showing that the model converges to a nearest neighbor regressor (k-NN with $k=1$ ). This convergence process is moreover not uniform, and is monotonic in the univariate case. An intuitive geometrical interpretation of the result is then given and its practical implications are discussed.

Identification Using Binary Measurements for IIR Systems

This paper studies the identification of infinite impulse response (IIR) systems with binary-valued measurements. The key to our approach is to recast the initial identification problem into an identification problem in the set-membership framework. This enables us to identify the system using an identification algorithm devoted to IIR systems in such a framework. Although the proposed algorithm is designed for noise free data, an analysis is provided in the presence of noise, and illustrative examples are provided to demonstrate the effectiveness of the algorithm.

Distributed Interpolatory Algorithms for Set Membership Estimation

This work addresses the distributed estimation problem in a set membership framework. The agents of a network collect measurements which are affected by bounded errors, thus implying that the unknown parameters to be estimated belong to a suitable feasible set. Two distributed algorithms are considered, based on projections of the estimate of each agent onto its local feasible set. The main contribution of the paper is to show that such algorithms are asymptotic interpolatory estimators, i.e. they converge to an element of the global feasible set, under the assumption that the feasible set associated to each measurement is convex. The proposed techniques are demonstrated on a distributed linear regression estimation problem.

Prognosis of uncertain linear time-invariant discrete systems using unknown input interval observer

In this paper, a model-based prognosis method where the degradation cannot be directly measured but only detectable through the drift of a model parameter is considered. This parameter drift is viewed as an unknown input, whose reconstruction allows the estimation of the degradation state. Model-based prognosis is divided into a filtering step where the current degradation state is estimated, and an uncertainty propagation step where the future degradation state is predicted. During these two steps, model uncertainty and measurement uncertainty are taken into account within the set-membership framework using interval techniques. The filtering step is performed with an interval unknown input observer for linear time-invariant discrete systems. Then, based on a set-membership constraint satisfaction methodology, interval propagation is performed to estimate the bounds that include the future degradation state until some failure threshold is reached, allowing the deduction of the interval including the remaining useful life. In order to demonstrate the efficiency of the proposed model-based prognosis methodology, the degradation of a suspension system is studied.

Low-Complexity Matrix-Based Conjugate Gradient Channel Estimation for Cooperative Wireless Sensor Networks

In this paper, we consider a general cooperative wireless sensor network (WSN) with multiple hops and the problem of channel estimation. A matrix-based conjugate gradient (CG) algorithm is first developed, which is then incorporated with the set-membership filtering framework to improve the performance of complex channel estimation for WSNs. Due to selective update, the proposed set-membership matrix-based CG (SM-MCG) algorithm can reduce the computational complexity significantly and extend the lifetime of WSN. In addition, by introducing an exponentially decaying bound and combining with SM-MCG algorithm, we design an updating criterion to accurately track the optimal maximum iteration number of each data window while reducing complexity further. Simulation results show good performance of our proposed algorithms in terms of convergence speed, steady state mean-square error, and indicate reduced complexity.

Two variants of the IIR spline adaptive filter for combating impulsive noise

It has been pointed out that the nonlinear spline adaptive filter (SAF) is appealing for modeling nonlinear systems with good performance and low computational burden. This paper proposes a normalized least M-estimate adaptive filtering algorithm based on infinite impulse respomse (IIR) spline adaptive filter (IIR-SAF-NLMM). By using a robust M-estimator as the cost function, the IIR-SAF-NLMM algorithm obtains robustness against non-Gaussian impulsive noise. In order to further improve the convergence rate, the set-membership framework is incorporated into the IIR-SAF-NLMM, leading to a new set-membership IIR-SAF-NLMM algorithm (IIR-SAF-SMNLMM). The proposed IIR-SAF-SMNLMM inherits the benefits of the set-membership framework and least-M estimate scheme and acquires the faster convergence rate and effective suppression of impulsive noise on the filter weight and control point adaptation. In addition, the computational burdens and convergence properties of the proposed algorithms are analyzed. Simulation results in the identification of the IIR-SAF nonlinear model show that the proposed algorithms offer the effectiveness in the absence of non-Gaussian impulsive noise and robustness in non-Gaussian impulsive noise environments.

Interval observer-based fault detectability analysis using mixed set-invariance theory and sensitivity analysis approach

ABSTRACTThis paper addresses the characterisation of the minimum detectable fault (MDF) by means of residual sensitivity integrated with the set-invariance theory when using an interval observer-based approach as a fault detection (FD) scheme. Uncertainties (disturbances and noise) are considered as of unknown but bounded nature (i.e. in the set-membership framework). A zonotopic-set representation towards reducing set operations to simple matrix calculations is utilised to bound the state/output estimations provided by the interval observer-based approach. In order to show the connection between sensitivity and set-invariance analyses, mathematical expressions of the MDF are derived when considering different types of faults. Finally, a simulation case study based on a quadruple-tank system is employed to both illustrate and discuss the effectiveness of the proposed approach. The interval observer-based FD scheme is used to test the MDF obtained from the integration of both residual sensitivity analysis and set-invariance theory in the considered case study.

Data-Driven Filtering for Nonlinear Systems With Bounded Noises and Quantized Measurements

This paper focuses on the data-driven filtering problem on nonlinear systems with bounded noises and quantized measurements which are subject to logarithmic quantization. Because the accurate system model is unknown, an almost-optimal data-driven filter is designed from an available data set within the nonlinear set membership framework. The set is composed of system inputs and quantized measurements, which can be more easily obtained than the mathematical model. The logarithmic quantization is dependent on the fact that its quantization error serves as a multiplicative noise which widely exists in many applications. For a further improvement of estimation performance, a preliminary approximation is obtained by neural network technique. Two numerical examples are given to illustrate the effectiveness of the proposed algorithm.