The Nyquist robust sensitivity margin for uncertain closed‐loop systems

The Nyquist robust sensitivity margin is proposed as a new scalar indicator of robust stability that also provides a meaningful quantitative assessment of the worst sensitivity realized by the uncertain closed loop. After formulating and discussing in detail the underlying optimization problem required for the calculation of the margin, the approach is applied to the characterization of the robust stability of a closed‐loop featuring a linear system with an affine uncertainty structure and a parametric uncertainty set described by a real rectangular polytope. The capabilities of the methodology are illustrated through examples, which include an approach for quantifying alternative robustness margins, such as a parametric stability margin. The computational algorithm is systematic and can be carried out with high numerical precision. Copyright © 2005 John Wiley & Sons, Ltd.     

Robust stability and instability of biochemical networks with parametric uncertainty

Parameter perturbations in dynamical models of biochemical networks affect the qualitative dynamical behaviour observed in the model. Since this qualitative behaviour is in many cases the key model output used to explain biological function, the robustness analysis of the model’s behaviour with respect to parametric uncertainty is a crucial step in systems biology research. In this paper, we develop a new method for robustness analysis of the dynamical behaviour. As a first step, we provide a characterization of non-robust perturbations as a system of polynomial equalities and inequalities. In the second step, we apply the Positivstellensatz and Handelman representation of polynomials to check for the non-existence of solutions to this system, which can be relaxed to solving a linear program. Thereby, a solution to the linear program yields a robustness certificate for the considered dynamical behaviour. With these robustness certificates, we propose an algorithm to compute a lower robustness bound corresponding to a level of parametric uncertainty up to which no local bifurcations can occur. The applicability of the proposed method to biochemical network models is illustrated by analysing the robustness of oscillations in a model of the NF-κB signalling pathway. The results may be used to define a level of confidence in the observed model behaviour under parametric uncertainty, making them valuable for evaluating dynamical models of biological networks.     

An iterative approach to H−/H∞ Fault Detection Observer Design for Discrete‐Time Uncertain Systems

In this paper, H _?/H _∞ multi‐objective fault detection observer design problem for linear discrete‐time systems is studied. Owing to the model inaccuracy in practical situations, system uncertainty is considered as well as the structure of the uncertainty. Sufficient conditions for the existence of the observers to guarantee the fault sensitivity and disturbance robustness in infinite frequency domain are presented. A new iterative linear matrix inequality (LMI) algorithm for the observers are proposed to obtain the solutions. In order to reduce the conservativeness, the observer design problem in finite frequency domain is also investigated. Simulation results illustrate the effectiveness of the proposed methods.     

RETRACTED ARTICLE: Nonlinear backstepping control design of LSM drive system using adaptive modified recurrent Laguerre orthogonal polynomial neural network

The good control performance of the permanent magnet linear synchronous motor (LSM) drive system is very difficult to achieve using linear controller because of uncertainty effects, such as ending-fictitious force. A backstepping approach is proposed to control the motion of the LSM drive system. With the proposed backstepping control system, the mover position of the LSM drive achieves good transient control performance and robustness. Although favorable tracking responses can be obtained by the backstepping control system, the chattering in the control effort is critical because of the large control gain. Because there are many nonlinear and time-varying uncertainties in the LSM drive systems, the nonlinear backsteping control system, which an adaptive modified recurrent Laguerre orthogonal polynomial neural network (NN) is used to estimate uncertainty, is thus proposed to reduce the chattering in the control effort and thereby enhance the robustness of the LSM drive system. In addition, the on-line parameter training methodology of the modified recurrent Laguerre orthogonal polynomial NN is based on the Lyapunov stability theorem. Furthermore, two optimal learning rates of the modified recurrent Laguerre orthogonal polynomial NN are derived to accelerate parameter convergence. Finally, comparison of the experimental results of the present study demonstrates the high control performance of the proposed control scheme.

     

Nonlinear uncertain systems and robustness optimization

An approach to (adaptive) stabilization is developed for a class of uncertain nonlinearly perturbed linear dynamical systems modelled by a controlled differential inclusion on Rⁿ. The essential features are nonlinear feedback and optimization of robustness with respect to unmatched uncertainty. The approach is geometric: (i) a family ?? of proper subspaces V ? Rⁿ is introduced with the property that {0} ? V is an asymptotically stable equilibrium for the projection, onto V, of the unperturbed linear flow; (ii) for the perturbed system, it is then shown that each V ? ?? can be rendered attractive and (almost) invariant by some choice of feedback determined by the nature of the perturbation and other a priori system information; specifically, under differing hypotheses, we can render V ? ?? almost invariant and attractive, or invariant and finite-time attractive, or adaptively attractive via, respectively, linear high-gain feedback or discontinuous feedback, or adaptive feedback. Associated with each V ? ?? is a measure of robustness with respect to unmatched uncertainty (a stability radius), quantified by a real parameter η. The problem of optimizing this measure is formulated as an optimization problem over ??, solvable (in a supremal sense of η-maximizing sequences (V_j) ??) via solutions of particular algebraic Riccati equations of order k ≤ (n - m), where m ≤ n is the control dimension; typically k = n - m..     

Affinely adjustable robust optimization for a multi-period inventory problem with capital constraints and demand uncertainties

This study focuses on a multi-period inventory problem with capital constraints and demand uncertainties. The multi-period inventory problem is formulated as an optimization model with a joint chance constraint (JCC) requiring the purchase cost for each period not to exceed the available capital with a probability guarantee. To hedge against demand uncertainties, an affinely adjustable robust optimization approach is used to convert the developed model into a robust counterpart. By approximating the JCC under a budgeted uncertainty set to which the demands belong, the robust multi-period inventory model with the JCC is transformed into a linear programming model, which can be solved efficiently. Numerical studies are reported to illustrate the robustness, practicality, and effectiveness of the proposed model and the solution approach. The numerical results show that the proposed model and solution approach outperform the sample average approximation approach. Numerical studies are used further to analyze the impact of the budget coefficient and the upper bound parameter on the inventory costs and the realized capital constraint satisfaction rate. The proposed model and solution approach are further extended to the multi-product case.     

A new model reference control architecture: Stability,performance, and robustness

In this paper, we develop a new model reference control architecture to effectively suppress system uncertainties and achieve a guaranteed transient and steady‐state system performance. Unlike traditional robust control frameworks, only a parameterization of the system uncertainty given by unknown weights with known conservative bounds is needed to stabilize uncertain dynamical systems with predictable system performance. In addition, the proposed architecture's performance is not dependent on the level of conservatism of the bounds of system uncertainty. Following the same train of thought as adaptive controllers that modify a given reference system to improve system performance, the proposed method is inspired by a recently developed command governor theory that minimizes the effect of system uncertainty by augmenting the input signal of the uncertain dynamical and reference systems. Specifically, a dynamical system, called a command governor, is designed such that its output is used to modify the input of both the controlled uncertain dynamical and reference systems. It is theoretically shown that if the command governor design parameter is judiciously selected, then the controlled system approximates the given original, unmodified reference system. The proposed approach is advantageous over model reference adaptive control approaches because linearity of the uncertain dynamical system is preserved through linear control laws, and hence, the closed‐loop performance is predictable for different command spectrums. Additionally, it is shown that the architecture can be modified for robustness improvements with respect to high frequency content due to, for example, measurement noise. Modifications can also be made in order to accommodate actuator dynamics and retain closed‐loop stability and predictable performance. The main contribution of this paper is the rigorous analysis of the stability and performance of a system utilizing the command governor framework. A numerical example is provided to illustrate the effectiveness of the proposed architecture. Copyright © 2015 John Wiley & Sons, Ltd.     

Quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems

This paper presents a novel quadratic optimal neural fuzzy control for synchronization of uncertain chaotic systems via H^∞ approach. In the proposed algorithm, a self-constructing neural fuzzy network (SCNFN) is developed with both structure and parameter learning phases, so that the number of fuzzy rules and network parameters can be adaptively determined. Based on the SCNFN, an uncertainty observer is first introduced to watch compound system uncertainties. Subsequently, an optimal NFN-based controller is designed to overcome the effects of unstructured uncertainty and approximation error by integrating the NFN identifier, linear optimal control and H^∞ approach as a whole. The adaptive tuning laws of network parameters are derived in the sense of quadratic stability technique and Lyapunov synthesis approach to ensure the network convergence and H^∞ synchronization performance. The merits of the proposed control scheme are not only that the conservative estimation of NFN approximation error bound is avoided but also that a suitable-sized neural structure is found to sufficiently approximate the system uncertainties. Simulation results are provided to verify the effectiveness and robustness of the proposed control method.     

Global quadratic stabilization of a class of nonlinear systems

The problem of quadratic stabilization for a class of nonlinear systems is examined in this paper. By employing a well-known Riccati approach, we develop a technique for designing a state feedback control law which quadratically stabilizes the system for all admissible uncertainties. This state feedback control law consists of linear and nonlinear feedback control terms. The linear feedback control term is generalized from a well-known H_∞ result, while the nonlinear term can be viewed as a correcting term for the presence of nonlinear bounded uncertainty. This stabilization result is extended to static output feedback and to systems for which the nonlinear uncertainty satisfies generalized matching conditions. Furthermore, we point out that in the presence of nonlinear uncertainty the global quadratic stability may be destroyed by some arbitrary small mismatched uncertainty in the matrix, and proceed to establish the region of semi-global quadratic stability of the controlled system. © 1998 John Wiley & Sons, Ltd.     

Optimal H2 and H∞ fixed-order dynamic compensation using canonical forms

This paper presents a technique for designing fixed order dynamic compensators in controller canonical form which are robust to both structured and unstructured uncertainty. The formulation uses an approach which combines an H₂ and H_∞ optimization process. Specifically, a quadratic performance index is minimized subject to a constraint on the H_∞ norm of the closed loop transfer function from disturbances to controlled outputs. This provides robustness to unstructured uncertainty. To provide robustness to structured uncertainty, an upper bound is computed for the worst case parameter variations, and it is then included in the derivation of the optimality conditions. A robust controller for a jetfoil boat is used to demonstrate this design technique. For the specific case of no structured uncertainty, the results of this approach using the canonical form compensator are analytically related to the previously published results for a fixed-order dynamic compensator. It is demonstrated that the use of the canonical form greatly simplifies the system of necessary conditions.     

New robust stability criterion and robust controller synthesis

In this paper, a new robust stability criterion for linear systems is proposed by combining the passivity and small gain theorems in different frequency bands. A controller synthesis method based on the new criterion is also developed. The controller can achieve both good performance and robustness in the same frequency band, if the uncertainty in that frequency band is passive or near passive. For processes with near passive lumped uncertainties with large gain in the frequency region in which good performance is required, the proposed controller can have better performance than that of H_∞ control for the same robustness specification. © 1998 John Wiley & Sons, Ltd.     

On Lyapunov stability robustness bounds for linear discrete-time systems with structured time-varying uncertainty

In this paper we analyse the stability robustness of linear discrete-time systems which are described by a state-space model but are perturbed with structured time-varying uncertainty. We present new Lyapunov stability robustness bounds in which the freedom of the matrix Q is utilized more effectively than that used by Kolla et al. (1989) to obtain a larger bound of tolerable time-varying uncertainty, and the similarity transformation is employed more directly and usefully than that proposed by Kolla and Farison (1990) to reduce conservation. Further, the relationship between the matrix Q and the similarity transformation matrix M is given. Improvements are illustrated by an application of our proposed method to a macroeconomic system     

Experimental validation of a model-based robust controller for multi-body mechanisms with flexible links

This paper presents the experimental validation of a model-based robust approach to the synthesis of position and vibration controllers for flexible link mechanisms. The design approach is based on the solution of a mixed-norm (H ₂/H _∞) problem, which allows ensuring, at the same time, satisfactory performances and robustness. A chief advantage of the approach proposed is that it allows designing robust control schemes by incorporating appropriate linear or piece-wise linear observers and regulators, which in turn can be synthesized making use of the accurate dynamic models currently available in literature.     

Analysis of robust performance with multiple objectives

Performance robustness with multiple objectives of linear control systems having structured norm-bounded uncertainty is considered. In order to deal with any two objective functions Ψ₁ and Ψ₂ associated with a single feedback control system, the combined criterion Ψ[Ψ₁ ψ₂]Ψ S 1 is often used. The paper, however, considers Ψ ψ₁ Ψ 1 and Ψ W, 11 1 directly. It is shown that it is possible to assess robust performance of two objectives by a single μ test if repeated nonscalar blocks are adequately introduced in the structure of μ. In other words, a performance robustness problem with multiple objectives is proved to be equivalent to a stability robustness problem with extra repeated uncertainty blocks. This equivalence theorem is applicable to various system and norm setups including sampled-data systems.     

Modeling and Control of Linear Two-time Scale Systems: Applied to Single-Link Flexible Manipulator

This paper deals with the problem of H _∞ control of linear two-time scale systems. The authors’ attention is focused on the robust regulation of the system based on a new modeling approach under the assumption of norm-boundedness of the fast dynamics. In the proposed approach, the fast dynamics are treated as a norm-bounded disturbance (dynamic uncertainty). In this view, the synthesis is performed only for the certain dynamics of the two-time scale system, whose order is less than that of the original system. It should be noted, however, that this scheme is significantly different from the conventional approaches of order reduction for linear two-time scale systems. Specifically, in the present work, explicitly or implicitly, all the dynamics of the system are taken into consideration. In other words, the portion that is treated as a perturbation is incorporated in the design by its maximum possible gain – in the L ₂ sense – over different values of the inputs. One of the advantagesof this approach is that – unlike in the conventional approaches of the order reduction – the reduced-order system still keeps some information of the ‘deleted’ subsystem. Also, we consider the robust stability analysis and stability bound improvement of perturbed parameter (ɛ) in the two-time scale systems by using linear fractional transformations and structured singular values (μ) approach. In this direction, by introducing the parametric uncertainty and dynamic uncertainty in the two-time scale systems, we represent the system as a standard μ-interconnection framework by using linear fractional transformations, and derive a set of new stability conditions for the system in the frequency domain. The exact solution of ɛ-bound is characterized. It is shown that, in spite of the coupling between the dynamic uncertainties and certain dynamics, the designed H _∞ controller stabilizes the overall closed-loop system, in the presence of norm-bounded disturbances. To show the effectiveness of the approach, the modeling of the single-link flexible manipulator and control of the Tip-position of the manipulator by utilizing the mentioned method are presented in the case study.     

Observer design methodology for stochastic and deterministic robustness

A unified observer with stochastic and deterministic robustness is developed in this paper so that an observer is less sensitive to both stochastic and deterministic uncertainties. For stochastic robustness, the norm of the observer gain and the lower bound of the observer decay rate are shown to be design factors which can minimize the upper bound of the estimation error variance. For deterministic robustness, the L ₂ norm-based condition number of the observer eigenvector matrix is utilized to address robust estimation performance against deterministic uncertainties. In order to justify the proposed method, a graphical approach is first introduced, and then a multi-objective optimization problem including linear matrix inequality constraints is formulated to provide the unified robustness.     

Fixed‐order robust H∞ control and control‐oriented uncertainty set shaping for systems with ellipsoidal parametric uncertainty

In this paper, sufficient conditions for robust output feedback controller design for systems with ellipsoidal parametric uncertainty are given in terms of solutions to a set of linear matrix inequalities. A polynomial method is employed to design a fixed‐order controller that assigns closed‐loop poles within a given region of the complex plane and that satisfies an H_∞ performance specification. The main feature of the proposed method is that it can be extended easily for control‐oriented uncertainty set shaping using a standard input design approach. Consequently, the results can be extended to joint robust control/input design procedure whose controller structure and performance specifications are translated into the requirements on the input signal spectrum used in system identification. This way, model uncertainty set can be tuned for the robust control design procedure. The simulation results show the effectiveness of the proposed method. Copyright © 2010 John Wiley & Sons, Ltd.     

Integrated fault estimation and fault‐tolerant control for uncertain Lipschitz nonlinear systems

This paper proposes an integrated fault estimation and fault‐tolerant control (FTC) design for Lipschitz non‐linear systems subject to uncertainty, disturbance, and actuator/sensor faults. A non‐linear unknown input observer without rank requirement is developed to estimate the system state and fault simultaneously, and based on these estimates an adaptive sliding mode FTC system is constructed. The observer and controller gains are obtained together via H_∞ optimization with a single‐step linear matrix inequality (LMI) formulation so as to achieve overall optimal FTC system design. A single‐link manipulator example is given to illustrate the effectiveness of the proposed approach. Copyright © 2016 John Wiley & Sons, Ltd.