Correct-by-Design Control of Parametric Stochastic Systems

This work proposes a methodology to estimate parameters for linear and nonlinear dynamical systems, with partial state measurement, that satisfy the property of parameter linearity. This methodology is experimental, offline, and recursive. It uses discontinuous state observers to estimate all state variables and the disturbance terms needed in the estimation processes. Because the equivalent output injection corresponds to the disturbances produced by the parameter uncertainties, the methodology allows us to obtain the best parameter estimation by minimising an index related to the power of the equivalent output injection; a smaller value represents a better estimation. With this parameter estimation, we can establish a model that facilitates the design and implementation of many control algorithms, including robust controllers. We validate the methodology through numerical simulations and experiments with linear, nonlinear, and discontinuous systems. Based on the experimental results, we conclude that the proposed algorithm's performance is better than other methodologies.

Adaptive control has been developed for decades, and now it has become a rigorous and mature discipline which mainly focuses on dealing parametric uncertainties in control systems, especially linear parametric systems. Nonparametric uncertainties were seldom studied or addressed in the literature of adaptive control until new areas on exploring limitations and capability of feedback control emerged in recent years. Comparing with the approach of robust control to deal with parametric or nonparametric uncertainties, the approach of adaptive control can deal with relatively larger uncertainties and gain more flexibility to fit the unknown plant because adaptive control usually involves adaptive estimation algorithms which play role of “learning” in some sense. This chapter will introduce a new challenging topic on dealing with both parametric and nonparametric internal uncertainties in the same system. The existence of both two kinds of uncertainties makes it very difficult or even impossible to apply the traditional recursive identification algorithms which are designed for parametric systems. We will discuss by examples why conventional adaptive estimation and hence conventional adaptive control cannot be applied directly to deal with combination of parametric and nonparametric uncertainties. And we will also introduce basic ideas to handle the difficulties involved in the adaptive estimation problem for the system with combination of parametric and nonparametric uncertainties. Especially, we will propose and discuss a novel class of adaptive estimators, i.e. information-concentration (IC) estimators. This area is still in its infant stage, and more efforts are expected in the future for gainning comprehensive understanding to resolve challenging difficulties. Furthermore, we will give two concrete examples of semi-parametric adaptive control to demonstrate the ideas and the principles to deal with both parametric and nonparametric uncertainties in the plant. (1) In the first example, a simple first-order discrete-time nonlinear system with both kinds of internal uncertainties is investigated, where the uncertainty of non-parametric part is characterized by a Lipschitz constant L, and the nonlinearity of parametric part is characterized by an exponent index b. In this example, based on the idea of the IC estimator, we construct a unified adaptive controller in both cases of b = 1 and

Designing minimum variance controllers (MVC) for nonlinear systems is confronted with many difficulties. The methods which are able to identify MIMO nonlinear systems are scarce, and linear models are not accurate in modeling nonlinear systems. In this paper, Vector ARX (VARX) models are proposed for designing MVC and generalized minimum variance controller (GMVC) for linear and nonlinear systems, and the accuracy of these models in approximating the nonlinear MIMO system is studied. However, the VARX is a linear model. It is shown that this model can identify some kinds of nonlinear systems with any desired accuracy. Therefore, the controller designed by the VARX is accurate, even for these nonlinear systems. The proposed controller is tested on a both linear system and a nonlinear four-tank benchmark process. In spite of the simplicity of designing GMVCs for the VARX models, the results show that the proposed method is accurate and implementable.

An increasingly large number of safety-critical embedded systems rely on software to prevent and mitigate hazards occurring due to design errors and unexpected interactions of the system with its users and the environment. Implementing a safety instrumented function in the way advocated by the traditional software methods requires an intimate understanding and thorough validation of a complex ecosystem of programming languages, compilers, operating systems and hardware. We propose to consider an alternative where a system designer, for each individual problem, creates in a correct-by-construction manner both the design of a system and its compilation and execution infrastructure. This permits an uninterrupted chain of a formal correctness argument spanning from formalised requirements all the way to the gate-level characterisation of an execution environment. The past decade of advances in verification technology turned the mechanical verification of large-scale models into a reality while the pressure of certification makes the cost of a formally verified development routine increasingly acceptable. The proposal fits the Grand Challenge for Computer Research posed by Hoare in 2003, namely, development of a Verifying Compiler which not only mechanically translates a given program from one language to another but also verifies its correctness according to a formal specification. This allows meeting the most stringent software certification requirements such as SIL 4. We illustrate the vision with a small case-study developed using the Event-B modelling notation and tools.

This paper proposes a novel variant of Statecharts, called Safecharts, especially for use in the specification and the design of safety critical systems. The objective is to provide a sharper focus on safety issues and a systematic approach to deal with them. This is achieved by making a clear separation between functional and safety requirements. A novel feature of Safecharts is the safety annotation, which proposes an explicit ordering of states according to risk level. Transitions are classified according to their risk nature and given a new priority scheme for their execution in the event of any non-determinism. A railway signalling system, a well-known case study, is used as an example to demonstrate some features and semantics of Safecharts.

Designing safety critical systems is a complex task due to the need of guaranteeing that the resulting model can cope with all the functional and non-functional requirements of the system. Obtaining such guarantees is only possible with the use of model verification techniques. This paper presents an approach aimed to fulfill the needs of critical system design. The proposed approach is based on the Architecture Analysis and Design Language (AADL), which is suitable to describe the system's architecture. A sequence of model transformations facilitates the verification of the designed AADL model and so assures its correctness. It must be highlighted that this is not performed in a single step, as it is possible to verify AADL models with different abstraction levels, which allows successive refinements in a top-down approach.

This work presents a new strategy in design optimization of nonlinear vibration absorbers with continuous and discontinuous motions. A cluster-based harmonic balance aided optimization technique using force balance or energy balance as the basis is generalized and adapted for nonlinear systems. It is found that optimal design parameters form a cluster in the parameter space and points from the parameter space inside the cluster satisfies design considerations. One of the main disadvantages of using existing optimization methods in nonlinear systems is that the parameter regimes, which provide periodic solutions, are not known beforehand, so one has to first do bifurcation studies to arrive at periodic regimes and optimization has to be conducted in the range. Proposed method combines these two steps as it converges to periodic clusters alone. Since the method admits only periodic solutions, occurrence of conditions such as chaos and quasi periodicity can be eliminated from the dynamics of the system. The proposed method can also be used to find the optimal parameters of both linear and nonlinear dynamical systems.

Presented here is an amplitude and frequency modulation method (AFMM) for extracting damage-induced nonlinear characteristics and intermittent transient responses by processing steady-state/transient responses using the empirical mode decomposition, Hilbert-Huang transform (HHT), and nonlinear dynamic characteristics derived from perturbation analysis. A sliding-window fitting (SWF) method is derived to show the physical implication of the proposed method and other methods for time-frequency signal decomposition. Similar to the short-time Fourier transform and wavelet transform the SWF uses windowed regular harmonics and function orthogonality to extract time-localized regular and/or distorted harmonics. On the other hand the HHT uses the apparent time scales revealed by the signal's local maxima and minima to sequentially sift components of different time scales, starting from high-frequency to low-frequency ones. Because HHT does not use predetermined basis functions and function orthogonality for component extraction, it provides more accurate instant amplitudes and frequencies of extracted components for accurate estimation of system characteristics and nonlinearities. Moreover, because the first component extracted from HHT contains all original discontinuities, its time-varying amplitude and frequency are excellent indicators for pinpointing times and locations of impulsive external loads and damages that cause intermittent responses. However, the discontinuity-induced Gibbs' effect makes HHT analysis inaccurate around the two data ends. On the other hand, the SWF analysis is not affected by Gibbs' effect, but it cannot extract accurate time-varying frequencies and amplitudes. Numerical results show that the proposed AFMM can provide accurate estimations of softening and hardening effects, different orders of nonlinearity, linear and nonlinear system parameters, and time instants of intermittent transient responses for damage detection and estimation.

This paper presents a method for recursively estimating the static parameters of linear or nonlinear stochastic dynamic systems given the systems' inputs and outputs. The paper accomplishes this objective by combining polynomial chaos theory with maximum likelihood estimation. The parameter estimates are calculated in a recursive or iterative manner. To the best of the author's knowledge, this is the first paper to address recursive maximum likelihood parameter estimation using polynomial chaos theory. The proposed approach is demonstrated on two systems: a linear 2nd order system with unknown damping and natural frequency, and a nonlinear Van der Pol oscillator with an unknown nonlinear damping coefficient. Because this recursive estimator is applicable to nonlinear systems, the authors portend that this novel formulation will be useful for a broad range of estimation problems.

This note deals with the tracking control of multiple-input-multiple-output (MIMO) nonlinear parametric strict-feedback systems in the presence of additive disturbances and parametric uncertainties. For such systems, C/sup 0/ robust adaptive controllers usually cannot ensure asymptotic tracking or even regulation. In this work, under the assumption the disturbances are C/sup 2/ with bounded time derivatives, we present a C/sup 0/ robust adaptive control construction that guarantees the tracking error is asymptotically driven to zero.

Nonlinear systems play a significant role in numerous scientific and engineering disciplines, and comprehending their behavior is crucial for the development of effective control and prediction strategies. This paper introduces a novel data-driven approach for accurately modeling and estimating parameters of nonlinear systems utilizing trust region optimization. The proposed method is applied to three well-known systems: the Van der Pol oscillator, the Damped oscillator, and the Lorenz system, which find broad applications in engineering, physics, and biology. The results demonstrate the efficacy of the approach in accurately identifying the parameters of these nonlinear systems, enabling a reliable characterization of their behavior. Particularly in chaotic systems like the Lorenz system, capturing the dynamics on the attractor proves to be crucial. Overall, this article presents a robust data-driven approach for parameter estimation in nonlinear dynamical systems, holding promising potential for real-world applications.Graphic

The concept of stability is one of the most fundamental in the study of natural sciences. The complexity of the stability problem depends on the degree of mathematical idealization of the system. In general, time invariant systems are much easier to handle than time varying systems. Here, first a bird’s eye view is presented on the important methods in constant parameter linear and nonlinear systems. Following this is a quick review of stability analysis of time varying systems touching upon periodic systems, second order systems, arbitrarily time varying systems, the absolute stability problem and the functional analytic approach. The paper also includes a summary of some important results of the authors’ work on the stability of second order multidimensional linear arbitrarily time varying systems. The theorems derived generalize some existing theorems. Application of one of them to the damped Mathieu equation yields better results than those existing in literature for some values in the parameter space. The asymptotic behaviour and boundedness of almost constant coefficient systems have also been studied and some new theorems derived.

This paper deals with the tracking control of multi-inut/multi-output nonlinear parametric strict-feedback systems in the presence of additive disturbances and parametric uncertainties. For such systems, robust adaptive controllers usually cannot ensure asymptotic tracking or even regulation. In this work, under the assumption the disturbances are C2with bounded time derivatives, we present a C0robust adaptive control construction that guarantees the tracking error is asymptotically driven to zero.

The on-line or adaptive identification of parameters in abstract linear and nonlinear infinite-dimensional dynamical systems is considered. An estimator in the form of an infinite-dimensional linear evolution system having the state and parameter estimates as its states is defined. Convergence of the state estimator is established via a Lyapunov estimate. The finite-dimensional notion of a plant being sufficiently rich or persistently excited is extended to infinite dimensions. Convergence of the parameter estimates is established under the additional assumption that the plant is persistently excited. A finite-dimensional approximation theory is developed, and convergence results are established. Numerical results for examples involving the estimation of both constant and functional parameters in one-dimensional linear and nonlinear heat or diffusion equations and the estimation of stiffness and damping parameters in a one-dimensional wave equation with Kelvin--Voigt viscoelastic damping are presented.

Knowledge management in the design of safety-critical systems addresses the question of how designers can share, capitalize, and reuse knowledge in an effective and reliable way. Knowledge management is situated in groups, organizations, and communities, playing different roles in the design process. Design of safety-critical systems has specific properties, such as dealing with complexity, traceability, maturity of knowledge, interaction between experts, awareness of the status of information, and trust in knowledge. Documentation is of crucial importance in design processes, ensuring that these properties are taken care of in a proper and reliable way. However, writing is not an easy task for engineers, and support is needed. Several knowledge management solutions, both tools and organizational setups, are available to support design work, such as active notification of changes, personal and team workspaces, active design documents and knowledge portal solutions.