Rather than requiring the objectives of system performance to be met accurately, it is quite common for practical systems design to have the individual performance objective being described in terms of the desired probability of attaining that objective. In this context, constrained probability analysis, filtering and control (CPAFC) problems are of significant engineering importance, mainly for two reasons: (1) it is almost impossible to ensure that certain performances are achieved with probability 1 because of uncontrolled external forces and unavoidable modelling errors; and (2) in some cases, it is satisfactory if certain performances are achieved with an acceptable probability.
CPAFC problems can find applications in various engineering applications. For example, in the control of large space structures, the vibration level at multiple points on the structure must be kept within specified bounds with certain guaranteed probability. Other examples include the paper making control problem, telescope pointing problem, robot arm pointing problem, spacecraft intercept problem and mean-time-between-failures control problem. These kinds of engineering problems place increasing demands on systems analysis and design because the problems under consideration generally involve multiple objectives, that is, the probability restrictions on performance requirements of the system outputs. Traditional system design techniques, such as LQG, H ∞ and L 1 designs, do not give a direct solution to the above CPAFC problems. For instance, LQG controllers minimise a linear quadratic performance index, which is actually the expectation (or average) of the performance evaluated by means of the H 2 norm, leading to a calculable output variance. As such, LQG design lacks guaranteed probability constraints with respect to individual system outputs or performances.
It should be pointed out that, recently, there has been a growing research interest in studying conventional filtering and control problems within a probabilistic framework by using a variety of optimisation approaches. The CPAFC has already become an ideal research area for control engineers, mathematicians, and computer scientists to manage, analyse, interpret and synthesise probabilistic information from real-world systems under stochastic disturbances. Sophisticated system theories and computing algorithms have been exploited or emerged in the general area of CPAFC, such as analysis and control of probability density functions, analysis and control with randomised algorithms, probability-constrained predictive control, as well as networked system analysis with probability constraints.
This special issue aims to bring together the latest approaches to understanding filtering and control for complex systems with probabilistic performance constraints. Topics include, but are not limited to the following aspects: (1) systems analysis with probability constraints; (2) probabilistic parameter identification of stochastic systems; (3) robustness with probability constraints; (4) methods and algorithms for randomised dynamics analysis; and (5) probability-constrained estimation, filtering and control for networked control systems. We have solicited submissions to this special issue from electrical engineers, control engineers, mathematicians and computer scientists. After a rigorous peer review process, 16 papers have been selected that provide overviews, solutions or early promises, to manage, analyse and interpret dynamical behaviours of complex systems with probability constraints. These papers cover both the practical and theoretical aspects of control and filtering with probabilistic performance constraints in the broad areas of dynamical systems, mathematics, statistics, operational research and engineering.
The standard robust analysis and control assures (only) a performance bound over the uncertainty region, which provides no information regarding the performance variation (or dispersion) over the uncertainty region. In a probabilistic robust performance context, it might be interesting and useful to address, for example, the ‘mean’ (to be defined) performance over the uncertainty region. If a better mean and a better bound are contradicting goals, one may sometimes be prepared to accept a slightly worse performance bound in order to obtain significantly better ‘mean’ performance. In the article, ‘Polytopic best-mean H ∞ performance analysis’ by Boyarski, the probabilistic versions of standard polytopic convex robust H ∞ analysis and state-feedback synthesis problems are studied for asymptotically stable infinite-horizon linear systems. This article focuses on the performance distribution over the uncertainty region (rather than just on the performance bound, as is customary in current robust control), and on optimising the ‘mean’ performance over the uncertainty region. The proposed approach exploits the convexity of the uncertainty polytope by imposing different disturbance attenuation levels (DALs) at the polytope's vertices, rather than assigning only a different Lyapunov function (as customary) to each vertex. These additional ‘degrees of freedom’ enable analysis and control designs that directly address the two main statistical properties of the resulting performance population, mean and variance. The proposed approach can be used in conjunction with practically any robust convex analysis and control method. A Monte-Carlo analysis verifies correct statistics of the resulting closed-loop ‘pointwise’ H ∞-norms over the uncertainty region.
The problem of optimal state estimation and parameter identification for stochastic systems with unknown parameters has been receiving systematic treatment. In the article, ‘Joint state and parameter estimation for uncertain stochastic nonlinear polynomial systems’ by Basin, Loukianov and Hernandez-Gonzalez, the joint filtering and parameter identification problem is investigated for uncertain stochastic nonlinear polynomial systems with unknown parameters in the state equation over nonlinear polynomial observations. The unknown parameters are incorporated into the extended polynomial state vector, which should be mean-square estimated over polynomial observations. The obtained filtering problem is further reduced to the filtering problem for polynomial system states over direct linear observations, assuming the nonlinear drift components in the observation equation as more additional states and including them in the extended state vector. The designed mean-square filter for the extended state vector also serves as the identifier for the unknown parameters. The simulation results show that the proposed method is effective. In the article, ‘Advances and applications of chance-constrained approaches to systems optimisation under uncertainty’ by Geletu, Klöppel, Zhang and Li, the state-of-the-art in chance-constrained nonlinear optimisation problems is given. It is indicated that, there is a growing interest in the application of chance-constrained optimisation (CCOPT) methods in various fields of engineering. The use of the chance-constrained nonlinear model predictive (CCNMPC) scheme is an advancement of traditional nonlinear model predictive strategies to cope with possible constraint. In the CCNMPC scheme, it is required to solve the chance constrained nonlinear optimisation problem on each prediction horizon. The type of algorithm to be used to solve the CCOPT depends on the structural properties of the chance constraints. Two case-studies, one with Gaussian normal and the other with product-beta distributed uncertainties, show applicability of chance-constrained models in the solution of modern engineering problems.
In many practical situations, data packets may suffer transmission delays for numerous reasons, such as network congestion, random failures in the transmission device, accidental loss of some measurements or data inaccessibility at certain times. In the article, ‘Linear estimation based on covariances for networked systems featuring sensor correlated random delays’ by Caballero-Águila, Hermoso-Carazo and Linares-Pérez, the covariance information is used to derive recursive algorithms for the least-squares (LS) linear estimation problem from delayed observations coming from multiple sensors with different delay characteristics. The filtering and smoothing (fixed-point and fixed-interval) problems are also considered. The recursive algorithms are derived using the innovation technique, which simplifies substantially the derivation of the algorithms. The precision of the LS estimators is measured by the estimation error covariance matrices, which provide a global measurement of the performance of the estimators. The feasibility of the proposed estimators is illustrated by a simulation example. In recent years, multi-sensor estimation fusion techniques have attracted increasing attention in many practical applications. In the article, ‘Robust distributed fusion for system with randomly uncertain sensor estimation error cross-covariance’ by Wu, Zhou, Hu and Li, the aim is to fuse different sensor estimates with partial knowledge of the sensor estimation error cross-covariance matrix, and employ a random matrix to describe the uncertainty of cross-covariance matrix. To derive a satisfied estimation fusion, the MSE only on the most favourable realisations of the aforementioned random matrix is minimised, and is modelled as an optimisation problem with chance constraint. Using an appropriate relaxation strategy, a robust fusion method is presented. Furthermore, the relaxation gap is discussed. The simulation shows that the presented fusion estimation has a smaller estimation error compared with the covariance intersection (CI) filter.
With the rapid development of communication technology, there have been more and more control systems exchanging data via communication channels, giving rise to networked systems (NSs). In the article, ‘State estimation for networked systems with randomly occurring quantisations’ by He, Wang, Ji and Zhou, the H ∞ state estimation problem is studied for a class of discrete-time NSs with randomly occurring quantisations (ROQs). The plant considered is described by using a simple discrete-time linear system model and all quantisers are assumed to be in the logarithmic type with different quantisation laws. A Bernoulli distributed stochastic sequence is employed to determine which quantiser is used at a certain time instant. A sufficient condition ensuring the existence of the desirable estimator is obtained by using the Lyapunov function approach. A networked three-tank system is constructed to illustrate the effectiveness of that proposed. In the article, ‘Probability-dependent H ∞ filtering for nonlinear stochastic systems with missing measurements and randomly occurring communication delays’ by Che, Shu, Yang and Ding, the H ∞ filtering problem is investigated for a class of nonlinear stochastic systems subject to missing measurements and randomly occurring communication delays. By constructing a new Lyapunov–Krasovskii functional, a sufficient condition is presented to ensure that the filtering error system is exponentially mean-square stable and the filtering error satisfies the H ∞-norm requirement. An illustrative example is presented to demonstrate the validity of the proposed scheme. In the article, ‘State estimation for networked systems: an extended IMM algorithm’ by Wu and Ye, the state estimation problem is studied for NSs with three kinds of observation uncertainties (i.e. missing measurements, packet delays and packet dropouts). Both the measurement state and network transmission state are assumed to follow a Markov process, which can capture the temporal correlation nature of the measurement process and network channels. By modifying the widely adopted interacting multiple models (IMM) algorithm, an extended IMM algorithm for the state estimation of the Markov jump linear system is proposed. The effectiveness and advantage of the proposed method are verified by simulation.
Markov jump systems (MJSs) have attracted much attention due to their comprehensive application in many systems, such as in manufacturing systems, economic systems, electrical systems and communication systems. In the article, ‘Passivity-based output feedback control of Markovian jump systems with discrete and distributed time-varying delays’ by Karimi, the problem of output feedback control is investigated for Markovian jump systems subject to both discrete and distributed time-varying delays. A Lyapunov–Krasovskii function is constructed to establish new sufficient conditions for ensuring exponentially mean-square stability and the passivity criteria. The delay-dependent output feedback control gains can be obtained from the linear matrix inequality (LMI) formulations. Numerical examples are given to illustrate the effectiveness of the proposed results. In the article, ‘H ∞ scheduling control on stochastic neutral systems subject to actuator nonlinearity’ by Yin, Shi, Liu and Song, the gain-scheduled robust H ∞ controller is designed for a class of stochastic time-varying neutral systems subject to time-varying delay and actuator saturation. At different selected time points, one can obtain a series of time-constant neutral system models with Markov jumping parameters. The continuous gain-scheduled controller design method is employed, and an entire time-varying controller is designed for the complete working region. A simulation example is given to illustrate the effectiveness of the developed techniques. In the article, ‘Guaranteed cost control for discrete-time Markovian jump linear system with time delay’ by Li, Sun and Gao, the problems of stability analysis and guaranteed cost control are studied for a discrete-time Markovian jump system with time-varying delay. The guaranteed cost controller existence criterion is proposed such that the whole interconnection system is robustly stochastically stable. Two illustrative examples are exploited to demonstrate the usefulness of the developed result.
Control and filtering for stochastic systems has been of interest of many researchers during the past three decades. In the article, ‘Probabilistic tracking control for non-Gaussian stochastic process using novel iterative learning algorithms’ by Yi, Sun and Guo, the real-time optimisation probabilistic tracking control problem is discussed. The iterative learning process is used to tune the parameters that decide the shape and location of the basis functions. The computation problem for iterative learning laws of neural network (NN) parameters can be transformed into a set of LMIs. The stability, tracking control and robustness can also be considered simultaneously by the designed convex ILC optimisation algorithms. It is noted that the proposed approach has the independent significance in complex stochastic distribution control (SDC) fields, and also has potential applications in other ILC optimisation problems with nonlinear performance functions. Jump Markov systems modelled by continuous-time stochastic processes are a very important class of models appearing in dynamical phenomena, such as signal processing, target tracking, and econometrics. Since the density evolution method for predictor equations satisfies the Fokker–Planck–Kolmogorov Equation (FPKE) in Bayes estimation, the FPKE in conjunction with Bayes’ conditional density update formula can provide optimal estimation for a general continuous-discrete nonlinear filtering problem. In the article, ‘Particle filter for state estimation of jump Markov nonlinear system with application to multi-targets tracking’ by Han, Ding, Hao and Hu, the state estimation problem is studied for jump JMNSs. A particle filter is designed to achieve Bayes estimation for jump Markov nonlinear systems (JMNSs). To test the viability of the proposed algorithm, the multiple targets tracking in video surveillance is enforced. The experiment results show that the proposed filter performs well in the multiple targets tracking.
In recent years, the model predictive control (MPC) problem has attracted increasing research attention. The optimisation of predicted control policies in MPC enables the use of information on future disturbance inputs which can be known at a future point on the prediction horizon. In the article, ‘On prediction strategies in stochastic MPC’ by Munoz-Carpintero, Cannon and Kouvaritakis, a stochastic MPC (SMPC) algorithm is proposed which makes use of the disturbance information available to a predicted control policy in order to reduce constraint tightening. Unlike the existing approaches, the compensation is applied over the entire horizon, thereby leading to a significant constraint relaxation which makes more control authority available for the optimisation of performance. In addition, the compensation has a striped lower triangular dependence on the uncertainty. The benefits of the proposed algorithm are achieved through an improvement of constraint tightening that makes use of the information on the bounds as well as the distribution of the uncertainty. In the article, ‘Probability-based constrained MPC for structured uncertain systems with state and random input delays’ by Lu, Li and Xi, a synthesis approach to design constrained MPC is presented for structured uncertain systems with fixed state delays and a random input delay, where the input delay process is governed by a discrete-time finite-state Markov chain. By invoking an appropriate augmented state, the considered system is transformed into a standard structured uncertain time-delay Markov jump linear system (MJLS) without input delay. A multi-step feedback control law is adopted by the proposed design to minimise an upper bound on the expected performance, which serves as the objective function of the MPC optimisation problem at each time instant. The closed-loop system with the proposed design is proven to be stable in the mean square sense. A numerical example is given to illustrate the feasibility of the proposed results.
Batch processes are characterised by a prescribed processing of raw materials into final products for a finite duration. Batch processes have received more attention in the chemical industry during the last decades on the demands of producing desired qualities of high-added-value products, such as polymers, pharmaceuticals and bio-chemicals. Process dynamics and stochastic disturbances are inherent characteristics of batch processes, which cause monitoring of batch processes a challenging problem in practice. Hence, a simple method without complicated algorithms is desired from the application viewpoint to reduce the design and engineering efforts. In the article, ‘Data-driven monitoring for stochastic systems and its application on batch process’ by Yin, Ding, Sari and Hao, a subspace-aided data-driven approach is given for batch process monitoring. Without the availability of the process model, the parameters of the process monitoring system, which can be realised by the parity space and diagnostic observer, can be directly identified from the test data. Issues such as process dynamics and stochastic disturbances are taken into consideration within the framework of the proposed scheme. In addition, to cope with non-Gaussian residual signals, the thresholds for fault detection purposes are determined according to the estimated probability density function by the kernel density estimation technique. Finally, a benchmark of a two-phase fed-batch fermentation of penicillin production process is given to show the effectiveness of the proposed method.
This special issue is a timely reflection of research progress in the area of control and filtering for complex systems with probabilistic performance constraints. Finally, we would like to acknowledge all of the authors for their efforts in submitting high-quality papers. We are also very grateful to the reviewers for their thorough and on-time reviews of the papers. Last, but not least, our deepest gratitude goes to Professor Peter Fleming, Editor-in-Chief of International Journal of Systems Science for his consideration, encouragement, and advice to publish this special issue.