On quadratic stability of systems with structured uncertainty

This paper considers the problem of stability robustness with respect to a class of nonlinear time-varying perturbations which are bounded in a component-wise rather than aggregated manner. A family of robustness bounds is parameterized in terms of a non-singular symmetric matrix. It is shown that the problem of computing the largest robustness bound over the set of non-singular symmetric matrices can be approximated by a smooth minimization problem over a compact set. A convergent algorithm for computing an optimal robustness bound is proposed in the form of a gradient flow     

Robustness measure bounds for optimal model matching controldesigns

The theorems introduced by R.V. Patel et al. (1977) for multivariable linear quadratic state feedback design in the presence of perturbations are extended to optimal model matching control system designs. A robustness measure bound is introduced for optimal model matching control systems. Bounds are obtained for allowable nonlinear time-varying perturbations such that the resulting closed-loop system remain stable. Bounds on a special, but important class of perturbations, in which the perturbation is linear, are also derived     

On the robustness of multivariable linear feedback systems in state-space representation

A singular values-based approach to specify the robustness of a multivariable linear feedback system in state-space representation is investigated. The robustness measure which is considered is the largest spectral norm of an additive uncertainty in the closed-loop system matrix, for which stability is guaranteed. It is shown that under the constraint of prescribed pole placement, a lower bound for the robustness measure is maximized, when the Frobenius norm of the closed-loop system matrix is minimized.     

Nominally Robust Model Predictive Control With State Constraints

In this paper, we present robust stability results for constrained discrete-time nonlinear systems using a finite-horizon model predictive control (MPC) algorithm for which we do not require the terminal cost to have any particular properties. We introduce a definition that attempts to characterize the robustness properties of the MPC optimization problem. We assume the systems under consideration satisfy this definition (for which we give sufficient conditions) and make two further assumptions. These are that the value function is bounded by a K_infin function of a state measure (related to the distance from the state to some target set) and that this measure is detectable from the stage cost used in the MPC algorithm. We show that these assumptions lead to stability that is robust to sufficiently small disturbances. While in general the results are semiglobal and practical, when the detectability and upper bound assumptions are satisfied with linear K_infin functions, the stability and robustness are either semiglobal or global (with respect to the feasible set). We discuss algorithms employing terminal inequality constraints and also provide a specific example of an algorithm that employs a terminal equality constraint.     

Performance deterioration in optimum linear systems

Uncertainty in dynamical systems may arise due to inaccuracies in modelling, parameter variations and external disturbances. As a result of this uncertainty, the performance index of an optimal control system deviates from its optimum value, which is referred to as the deterioration. A technique is presented to find a bound on the deteriorated performance index of optimum linear systems subject to bounded uncertainty. Uncertainty is incorporated as a forcing term in the system equations. To find the deteriorated performance bound, the performance index subject to the uncertain system is to be maximized within a specified time interval. The interchange theorem is used to interchange the maximizing and integral operations in the performance index functional to obtain a pointwise problem. Then, a Lyapunov technique, used to find reachable sets of uncertain systems, is applied to find the pointwise maximum values. The method serves as a measure of performance robustness or a measure of sensitivity to uncertainties. The problem is analytically solved for first-order systems. Finally, examples concerning first- and second-order systems are given as applications of the technique.     

Stability of Networked Control Systems Under a Multiple-Packet Transmission Policy

This paper is concerned with stability analysis of discrete-time networked control systems subject to packet loss under a multiple-packet transmission policy with the packet dropping probability of the communication channel bounded from above. Necessary and sufficient conditions for stability are obtained. In addition, the packet dropping margin as a measure of stability robustness of a system against packet loss is defined and its formula is derived. A design method is proposed for enhancing stability robustness subject to the constraint of a set of prescribed nominal closed-loop poles.      

Stability robustness analysis of digital control systems in state-space models

The stability robustness of discrete linear time-invariant systems in state-space models is analysed. Based on a root-locus approach and from the stable-eigenvalue viewpoint, fundamental criteria for testing the stability robustness of autonomous systems are derived and applied to the robustness analysis of multivariable feedback systems. Both the norm bound and the element bounds for the allowed perturbations are obtained. A stability robustness index is denned which is useful both for the analysis and synthesis of control systems.     

Robust stabilization of stochastic systems based on the LQ controller

The robust exponential stability in mean square for a class of linear stochastic uncertain control systems is dealt with. For the uncertain stochastic systems, we have designed an optimal controller which guarantees the exponential stability of the system. Actually, we employed Lyapunov fimction approach and the stochastic algebraic Riccati equation (SARE) to have shown the robusmess of the linear quadratic(LQ) optimal control law.And the algebraic criteria for the exponential stability on the linear stochastic uncertain closed-loop systems are given.     

Robust stabilization of stochastic systems based on the LQ cont roller

The robust exponential stability in mean square for a class of linear stochastic uncertain control systems is dealt with. For the uncertain stochastic systems ,we have designed an optimal controller which guarantees the exponential stability of the system. Actually ,we employed Lyapunov function approach and the stochastic algebraic Riccati equation (SARE) to have shown the robustness of the linear quadratic (LQ) optimal control law. And the algebraic criteria for the exponential stability on the linear stochastic uncertain closed- loop systems are given.     

Stability robustness of networked control systems with respect to packet loss

This paper is concerned with stability analysis of discrete-time networked control systems over a communication channel subject to packet loss whose behavior is modeled by an i.i.d Bernoulli process with a packet dropping probability bounded by a constant. A necessary and sufficient condition for stability is obtained. A packet dropping margin is introduced as a measure of stability robustness of a system against packet dropping, and a formula for it is derived. A design method is proposed for achieving a large margin subject to a constraint that the system has a set of prescribed nominal closed-loop poles.     

Improved measures of stability robustness for linear state space models

In this paper, the aspect of "stability robustness" of linear systems is analyzed in the time domain. A bound on the structured perturbation of an asymptotically stable linear system is obtained to maintain stability using a Lyapunov matrix equation solution. The resulting bound is shown to be an improved bound over the ones recently reported in the literature. Also, special cases of the nominal system matrix are considered, for which the bound is given in terms of the nominal matrix, thereby, avoiding the solution of the Lyapunov matrix equation. Examples given include comparison of the proposed approach with the recently reported results.     

Transient performance and robustness of direct adaptive control

It is shown that a discrete adaptive controller with a fixed gain gradient estimator can be applied to linear systems that are undermodeled provided that the mismatch is small and the estimated parameters are projected into a compact convex set which contains stabilizing control parameters and excludes the possibility of dividing with small numbers. The latter implies that the sign and a lower bound for the high-frequency gain of the modeled part of the system are known. The performance of the adaptive system is related to the size of the external perturbations, and the ideal result as these tend to zero is obtained. The main conclusion is that while convergence, stability, and robustness may be achieved, the transient performance may not be acceptable and the gradient estimator should probably not be used unless modifications are added. Design guidelines are established. The results extend readily to indirect adaptive laws like pole assignment, linear quadratic optimal, and predictive controllers     

On optimal predefined‐time stabilization

This paper addresses the problem of optimal predefined‐time stability. Predefined‐time stable systems are a class of fixed‐time stable dynamical systems for which the minimum bound of the settling‐time function can be defined a priori as an explicit parameter of the system. Sufficient conditions for a controller to solve the optimal predefined‐time stabilization problem for a given nonlinear system are provided. These conditions involve a Lyapunov function that satisfies a certain differential inequality for guaranteeing predefined‐time stability. It also satisfies the steady‐state Hamilton–Jacobi–Bellman equation for ensuring optimality. Furthermore, for nonlinear affine systems and a certain class of performance index, a family of optimal predefined‐time stabilizing controllers is derived. This class of controllers is applied to optimize the sliding manifold reaching phase in predefined time, considering both the unperturbed and perturbed cases. For the perturbed case, the idea of integral sliding mode control is jointly used to ensure robustness. Finally, as a study case, the predefined‐time optimization of the sliding manifold reaching phase in a pendulum system is performed using the developed methods, and numerical simulations are carried out to show their behavior. Copyright © 2017 John Wiley & Sons, Ltd.     

Global stabilization of complex networks with digraph topologies via a local pinning algorithm

This paper concerns the global stability of controlling a complex network with digraph topology to a homogeneous trajectory of the uncoupled system by the local pinning control algorithm. The derived stability condition indicates that the smallest real part of eigenvalues of the Laplacian sub-matrix corresponding to the unpinned vertices can be used to measure the stabilizability of a digraph with a given pinned vertex set. A pinned vertex set can stabilize a directed network to some unstable trajectories, for instance, to a chaotic trajectory of the uncoupled systems, if and only if the pinned vertex set can access all other vertices in the digraph. Furthermore, in the bigraph case, the analytical estimation of the stabilizability’s lower bound suggests that an optimal pinning strategy should take not only the vertex degree, but also the shortest path between pairs of vertices into considerations.     

On continuity properties of the two-block ℓ1 optimal design

This paper investigates continuity properties of the two-block ℓ₁ optimization resulting from the optimal design of BIBO stability robustness for discrete time systems in the presence of coprime factor perturbations. In general, the two-block ℓ₁ optimal design might lack continuity properties, since it might have no finite dimensional solution. However, this paper shows that for a suitable given truncated order, the two-block ℓ₁ suboptimal design is continuous as a map from the plant to the suboptimal closed loop solution, if the plant has no zeros on the unit circle and has a unique suboptimal solution. Furthermore, if the set of plants is compactness, we show that there exists a uniform bound on the degree of suboptimal solution for all plants in the set such that their deviation of suboptimal value from the optimal one have the same bound, and the two-block ℓ₁ suboptimal design is uniformly continuous. In particular, in the case in which the plant has nonunique solution, a procedure is also developed on how to choose a solution that preserves continuity. The results enable the stability of robust ℓ₁ adaptive scheme to hold in the presence of coprime factor perturbations.     

An improved closed-loop stability related measure forfinite-precision digital controller realizations

The pole-sensitivity approach is employed to investigate the stability issue of the discrete-time control system, where a digital controller implemented with finite word length (FWL), is used. A stability related measure is derived, which is more accurate in estimating the closed-loop stability robustness of an FWL implemented controller than some existing measures for the pole-sensitivity analysis. This improved stability measure thus provides a better criterion to find the optimal realizations for a generic controller structure that includes output-feedback and observer-based controllers. A numerical example is used to verify the theoretical analysis and to illustrate the design procedure     

Optimal exponential feedback stabilization of planar systems

This paper solves the problem of finding an optimal feedback control ensuring the maximal rate of convergence of system solutions to the origin for a general class of planar control systems including switched, bilinear systems and ones described by differential inclusions, etc. The prescribed control set is assumed to be compact but not necessarily convex. The developed approach is based on finding the minimal Lyapunov exponent of the system with an open loop control which provides an upper bound for the optimal convergence rate of the closed loop system. Then an optimal feedback controller is constructed for which the obtained bound is attained.     

具有结构不确定性的输出反馈线性系统的鲁棒稳定性分析

基于向量的幂变换方法，对具有结构不确定性的输出反馈线性系统的鲁棒稳定性问题作了分析。单参数摄动时给出了闭环系统鲁棒稳定的充要条件，多参数摄动时得到了保证系统鲁棒稳定的充分条件，导出了闭环系统鲁棒稳定区域的一种代数表达形式。最后给出了实例。     

Conditions under which suboptimal nonlinear MPC is inherently robust

We address the inherent robustness properties of nonlinear systems controlled by suboptimal model predictive control (MPC), i.e., when a suboptimal solution of the (generally nonconvex) optimization problem, rather than an element of the optimal solution set, is used for the control. The suboptimal control law is then a set-valued map, and consequently, the closed-loop system is described by a difference inclusion. Under mild assumptions on the system and cost functions, we establish nominal exponential stability of the equilibrium, and with a continuity assumption on the feasible input set, we prove robust exponential stability with respect to small, but otherwise arbitrary, additive process disturbances and state measurement/estimation errors. These results are obtained by showing that the suboptimal cost is a continuous exponential Lyapunov function for an appropriately augmented closed-loop system, written as a difference inclusion, and that recursive feasibility is implied by such (nominal) exponential cost decay. These novel robustness properties for suboptimal MPC are inherited also by optimal nonlinear MPC. We conclude the paper by showing that, in the absence of state constraints, we can replace the terminal constraint with an appropriate terminal cost, and the robustness properties are established on a set that approaches the nominal feasibility set for small disturbances. The somewhat surprising and satisfying conclusion of this study is that suboptimal MPC has the same inherent robustness properties as optimal MPC.     

Stochastic Uncertain Systems Subject to Relative Entropy Constraints: Induced Norms and Monotonicity Properties of Minimax Games

Entropy and relative entropy are fundamental concepts on which information theory is founded on, and in general, telecommunication systems design. On the other hand, dissipation inequalities, minimax strategies, and induced norms are the basic concepts on which robustness of uncertain control and estimation of systems are founded on. In this paper, the precise relation between these notions is investigated. In particular, it will be shown that the higher the dissipation the higher the entropy of the system, which has implications in computing the induced norm associated with robustness. These connections are obtained by considering stochastic optimal uncertain control systems, in which uncertainty is described by a relative entropy constraint between the nominal and uncertain measures, while the pay-off is a linear functional of the uncertain measure. This is a minimax game, in which the controller measure seeks to minimize the pay-off, while the disturbance measure aims at maximizing the pay-off. Salient properties of the minimax solution are derived, including a characterization of the optimal sensitivity reduction, computation of the induced norm, monotonicity properties of minimax solution, and relations between dissipation and relative entropy of the system. The theory is developed in an abstract setting and then applied to nonlinear partially observable continuous-time uncertain controlled systems, in which the nominal and uncertain systems are described by conditional distributions. In addition, existence of the optimal control policy among the class of policies known as wide-sense control laws is shown, and an explicit formulae for the worst case conditional measure is derived. The results are applied to linear-quadratic-Gaussian problems