An efficient methodology for robust control of dynamic systems with interval matrix uncertainties

Interval models are frequently used for dealing with uncertainties of control systems. However, it is well known that direct analysis and synthesis of a controlled dynamic system with interval matrix uncertainties may be a NP-hard problem. In this work, an efficient methodology for robustness analysis and robust control design of dynamic systems with interval matrix uncertainties is presented systematically, in which the uncertainties appearing in the controlled plant and controller realisation are taken into account simultaneously in an integrated framework. The fundamental problems, such as quadratic stability, guaranteed cost control and H_∞ control of uncertain systems are taken as examples to show the methodology. Necessary and sufficient conditions for linear dynamic systems with interval matrices are derived by transforming all the interval matrices into some more tractable forms. The whole reasoning process is logical and rigorous, and NP-hard problem is successfully avoided. The presented formulations are within the framework of linear matrix inequality and can be implemented conveniently. In contrast to existing vertex-set methods, in which the vertices of interval matrices need to be constructed and checked, the presented methods are more efficient. Three numerical examples are investigated to demonstrate the effectiveness and feasibility of the presented method.     

Estimating confidence interval of software reliability with adaptive testing strategy

Software reliability assessment is a critical problem in safety-critical and mission-critical systems. In the reliability assessment of such a system, both an accurate reliability estimate and a tight confidence interval are required. Adaptive testing (AT) is an on-line testing framework, which dynamically selects test cases from different subdomains to achieve some optimization object. Although AT has been proved effective in minimizing reliability estimator variance, its performance on providing the corresponding confidence interval has not been investigated. In order to address this issue, an AT strategy combined with Bayesian inference (AT-BI) is proposed in this study. The novel AT-BI strategy is expected to be effective in providing both a low-variance estimator and a tight confidence interval. Experiments are set up to validate the effectiveness of the AT-BI strategy.     

Optimal interval estimation fusion based on sensor interval estimates with confidence degrees

The interval estimation fusion method based on sensor interval estimates and their confidence degrees is developed. When sensor estimates are independent of each other, a combination rule to merge sensor estimates and their confidence degrees is proposed. Moreover, two optimization criteria: minimizing interval length with an allowable minimum confidence degree, or maximizing confidence degree with an allowable maximum interval length are suggested. In terms of the two criteria, an optimal interval estimation fusion can be obtained based on the combined intervals and their confidence degrees. Then we can extend the results on the combined interval outputs and their confidence degrees to obtain a conditional combination rule and the corresponding optimal fault-tolerant interval estimation fusion in terms of the two criteria. It is easy to see that Marzullo's fault-tolerant interval estimation fusion [Marzullo, (1990). Tolerating failures of continuous-valued sensors. ACM Transactions on Computer System, 8(4), 284-304] is a special case of our method.     

An ellipsoid algorithm for probabilistic robust controller design

In this paper, a new iterative approach to probabilistic robust controller design is presented, which is applicable to any robust controller/filter design problem that can be represented as an LMI feasibility problem. Recently, a probabilistic Subgradient Iteration algorithm was proposed for solving LMIs. It transforms the initial feasibility problem to an equivalent convex optimization problem, which is subsequently solved by means of an iterative algorithm. While this algorithm always converges to a feasible solution in a finite number of iterations, it requires that the radius of a non-empty ball contained into the solution set is known a priori. This rather restrictive assumption is released in this paper, while retaining the convergence property. Given an initial ellipsoid that contains the solution set, the approach proposed here iteratively generates a sequence of ellipsoids with decreasing volumes, all containing the solution set. At each iteration a random uncertainty sample is generated with a specified probability density, which parameterizes an LMI. For this LMI the next minimum-volume ellipsoid that contains the solution set is computed. An upper bound on the maximum number of possible correction steps, that can be performed by the algorithm before finding a feasible solution, is derived. A method for finding an initial ellipsoid containing the solution set, which is necessary for initialization of the optimization, is also given. The proposed approach is illustrated on a real-life diesel actuator benchmark model with real parametric uncertainty, for which a robust state-feedback controller is designed.     

2D systems based robust iterative learning control using noncausal finite-time interval data

This paper uses a 2D system setting in the form of repetitive process stability theory to design an iterative learning control law that is robust against model uncertainty. In iterative learning control the same finite duration operation, known as a trial over the trial length, is performed over and over again with resetting to the starting location once each is complete, or a stoppage at the end of the current trial before the next one begins. The basic idea of this form of control is to use information from the previous trial, or a finite number thereof, to compute the control input for the next trial. At any instant on the current trial, data from the complete previous trial is available and hence noncausal information in the trial length indeterminate can be used. This paper also shows how the new 2D system based design algorithms provide a setting for the effective deployment of such information.     

On the explicit dead-time compensation for robust model predictive control

Computational simplicity is one of the most important aspects to take into account in robust model predictive control (MPC). In dead-time processes, it is common to use an augmented state-space representation in order to apply robust MPC strategies but, this procedure may affect computational aspects. In this paper, explicit dead-time compensation will be used to avoid augmented representation. This technique will be analyzed in terms of robust stability and constraint satisfaction for discrete-time linear systems. The results of this discussion will be applied to a robust tube-based MPC strategy which is able to guarantee robust stability and constraint satisfaction of a dead-time system by considering a prediction model without dead-time. Moreover, taking advantage of the proposed scheme, the robust MPC will be particularized for first-order plus dead-time models which simplifies significantly controller synthesis. The proposed dead-time compensation method will be applied to different robust MPC strategies in two case studies: (i) a simulated quadruple-tank system, and (ii) an experimental scaled laboratory heater process.     

区间离散广义系统状态反馈鲁棒H∞控制

讨论了一类区间离散广义系统的状态反馈鲁棒H_∞控制问题.在给出区间离散广义系统的等价描述之后,基于系统参数矩阵不等式,得到了问题可解的充分条件,并给出了状态反馈控制器显式表示.所得的控制器保证闭环系统正则,具有因果关系,稳定并且满足给定的H_∞性能指标.数值例子说明了该方法的正确性.     

On the robust adaptive control of a contour-following robotic system

To demonstrate the force sensing and control, a possible model of a contour-following system is represented by a fourth-order linear continuous-time time-invariant system, in which stiffness k_t, robot natural frequency, ω_n, linear accommodation gain k_x, and angular accommodation gain k_φ are all constants obtained by measurement and experiment. This model works well for following a short contour. To satisfactorily follow a longer contour, k_t, ω_n, k_x and k_φ can be treated as unknown constants or time-varying variables. When they are considered as unknown constants, a robust model reference adaptive controller can be used to achieve both stability and tracking without having to know or find the true values of those constants, given bounded input or output disturbances and stable unmodeled dynamics. If ω_n is assumed to be a given constant but K_t, k_x, and k_φ are assumed to be unknown variables, then one has a linear time-varying plant and other types of model reference adaptive controllers have to be used to achieve the same purpose. In this paper, the schemes from various robust model reference adaptive control design will be studied and comparison and suggestions will also be made based on the simulation results for the contour-following robotic system mentioned above.     

On robust stability of multivariate interval plants

In this work we consider the stability property of the feedback connection of a multivariate interval plant with a fixed compensator. Copyright © 2003 John Wiley & Sons, Ltd.     

Using interval arithmetic for robust state feedback design

A problem of a modal P-regulator synthesis for a linear multivariable dynamical system with uncertain (interval) parameters in state space is considered. The designed regulator has to place all coefficients of the system characteristic polynomial within assigned intervals. We have developed the approach proposed earlier in Dugarova and Smagina (Avtomat. i Telemech. 11 (1990) 176) and proved a direct correlation between interval system controllability and existence of robust modal P-regulator.     

Reliability-based robust Pareto design of linear state feedback controllers using a multi-objective uniform-diversity genetic algorithm (MUGA)

In this paper, fuzzy threshold values, instead of crisp threshold values, have been used for optimal reliability-based multi-objective Pareto design of robust state feedback controllers for a single inverted pendulum having parameters with probabilistic uncertainties. The objective functions that have been considered are, namely, the normalized summation of rising time and overshoot of cart (S_RO–C) and the normalized summation of rising time and overshoot of pendulum (S_RO–P) in the deterministic approach. Accordingly, the probabilities of failure of those objective functions are also considered in the reliability-based design optimization (RBDO) approach. A new multi-objective uniform-diversity genetic algorithm (MUGA) is presented and used for Pareto optimum design of linear state feedback controllers for single inverted pendulum problem. In this way, Pareto front of optimum controllers is first obtained for the nominal deterministic single inverted pendulum using the conflicting objective functions in time domain. Such Pareto front is then obtained for single inverted pendulum having probabilistic uncertainties in its parameters using the statistical moments of those objective functions through a Monte Carlo simulation (MCS) approach. It is shown that multi-objective reliability-based Pareto optimization of the robust state feedback controllers using MUGA with fuzzy threshold values includes those that may be obtained by various crisp threshold values of probability of failures and, thus, remove the difficulty of selecting suitable crisp values. Besides, the multi-objective Pareto optimization of such robust feedback controllers using MUGA unveils some very important and informative trade-offs among those objective functions. Consequently, some optimum robust state feedback controllers can be compromisingly chosen from the Pareto frontiers.     

On interval goal programming switching surface robust design for Integral Sliding Mode Control

This paper presents a systematic technique for Integral Sliding Mode Control design applied to linear time-invariant systems, subject to disturbances, whose uncertain parameters belong to closed intervals. This approach considers a computational optimization-based procedure for the switching surface design, formulated as an interval goal programming problem, equivalent to the interval Ackermann׳s equation for the robust eigenvalue placement problem. The design objective is the minimization of the overall deviation from the specified performance for the transient response of the controlled system, which is represented by a polytope of characteristic polynomials. The controllability of interval systems is also numerically treated within an interval analysis framework. An experimental implementation of this switching surface design technique was compared with a classic robust LMI-based design procedure of the switching surface in the position control of a DC motor, coupled to different loads, to illustrate its main features and efficiency.     

A practical procedure to obtain confidence intervals for the Bernoulli parameter

The paper proposes a practical procedure for obtaining a confidence interval (CI) for the parameter π of the Bernoulli distribution. Let x be the observed number of successes of a random sample of size n from this distribution. The procedure is as follows: use Table 1 to determine whether the given pair (n,x) is a small or a large sample pair. If the small sample situation applies then use Table 2 which gives the Sterne–Crow CI. Otherwise, use the Anscombe CI for which practical formulas are given.     

On robust stability of discrete-time adaptive nonlinear control

We consider adaptive control of discrete-time nonlinear systems with a single unknown parameter in this paper. We demonstrate that the necessary and sufficient condition for the existence of a feedback stabilizer that is robust to bounded noise is that the nonlinear growth rate of the system dynamics is less than 4. This result further confirms the conclusion of Guo [On critical stability of discrete-time adaptive nonlinear control, IEEE Trans. Automat. Control 42 (1997) 1488–1499] which addresses unbounded noise in a stochastic setting. Also in our worst-case approach, we find that much simpler adaptive stabilizers can be constructed when the nonlinear growth rate is less than 4.     

A study on robust design optimization of layered plate bonding process considering uncertainties

Design optimization of layered plate bonding process is conducted by considering uncertainties in a manufacturing process, to reduce the crack failure arising due to the difference of thermal expansion coefficients of the adherents. Robust optimization is performed to minimize the mean and variance of the residual stress, which is the major cause of the failure, while constraining the distortion and the instantaneous maximum stress to the allowable limits. In this optimization, the dimension reduction (DR) method is employed to quantify the uncertainty of the responses in the bonding process. It is expected that the DR method benefits the optimization from the perspectives of efficiency, accuracy, and simplicity. Response surface method (RSM) combined with sequential approximate optimization (SAO) technique is employed as an optimization tool. The obtained robust optimal solution is verified by the Monte Carlo simulation.     

Optimization over state feedback policies for robust control with constraints

This paper is concerned with the optimal control of linear discrete-time systems subject to unknown but bounded state disturbances and mixed polytopic constraints on the state and input. It is shown that the class of admissible affine state feedback control policies with knowledge of prior states is equivalent to the class of admissible feedback policies that are affine functions of the past disturbance sequence. This implies that a broad class of constrained finite horizon robust and optimal control problems, where the optimization is over affine state feedback policies, can be solved in a computationally efficient fashion using convex optimization methods. This equivalence result is used to design a robust receding horizon control (RHC) state feedback policy such that the closed-loop system is input-to-state stable (ISS) and the constraints are satisfied for all time and all allowable disturbance sequences. The cost to be minimized in the associated finite horizon optimal control problem is quadratic in the disturbance-free state and input sequences. The value of the receding horizon control law can be calculated at each sample instant using a single, tractable and convex quadratic program (QP) if the disturbance set is polytopic, or a tractable second-order cone program (SOCP) if the disturbance set is given by a 2-norm bound.     

Tube-based robust economic model predictive control

In this paper, we develop a tube-based economic MPC framework for nonlinear systems subject to unknown but bounded disturbances. Instead of simply transferring the design procedure of tube-based stabilizing MPC to an economic MPC framework, we rather propose to consider the influence of the disturbance explicitly within the design of the MPC controller, which can lead to an improved closed-loop average performance. This will be done by using a specifically defined integral stage cost, which is the key feature of our proposed robust economic MPC algorithm. Furthermore, we show that the algorithm enjoys similar properties as a nominal economic MPC algorithm (i.e., without disturbances), in particular with respect to bounds on the asymptotic average performance of the resulting closed-loop system, as well as stability and optimal steady-state operation.     

Nonparametric confidence intervals for population variance of one sample and the difference of variances of two samples

Confidence intervals for the population variance and the difference in variances of two populations based on the ordinary t-statistics combined with the bootstrap method are suggested. Theoretical and practical aspects of the suggested techniques are presented, as well as their comparison with existing methods (methods based on Chi-square statistics and F-statistics). In addition, application of presented methods in domain of insurance property data set is described and analyzed. For data from exponential distribution confidence intervals, which are calculated using described methods (based on transformation of the t-statistics and bootstrap technique), give consistent and best coverage in comparison with other methods.     

Adaptive robust control for servo manipulators

In this paper, an adaptive robust control scheme is developed which is suitable for the control of a class of uncertain nonlinear systems, typical of many servo manipulators. The control scheme is comprised of a model reference adaptive controller (MRAC) augmented with a nonlinear compensator based on an adaptive radial basis function (RBF). The RBF compensator is used to neutralise the effects of uncertain and possibly nonlinear dynamics, so that the equivalent system as seen by the MRAC is reduced to one without significant unstructured modelling errors. A stability analysis is provided to show the uniform stability and the asymptotic tracking capabilities of the proposed control system. Real-time experiment results verify the effectiveness of the control scheme.