Maximal controllability of input constrained unstable systems by the addition of implicit constraints

The null controllable set of a system is the largest set of states that can be controlled to the origin. Control systems that have a region of attraction equal to the null controllable set are said to be maximally controllable closed-loop systems. In the case of open-loop unstable plants with amplitude constrained control it is well known that the null controllable set does not cover the entire state-space. Further the combination of input constraints and unstable system dynamics results in a set of state constraints which we call implicit constraints. It is shown that the simple inclusion of implicit constraints in a controller formulation results in a controller that achieves maximal controllability for a class of open-loop unstable systems.     

On the performance of high-gain observers with gain adaptation under measurement noise

We address the problem of state observation for a system whose dynamics may involve poorly known, perhaps even nonlocally Lipschitz functions and whose output measurement may be corrupted by noise. It is known that one way to cope with all these uncertainties and noise is to use a high-gain observer with a gain adapted on-line. The proposed method, while presented for a particular case, relies on a “generic” analysis tool based on the study of differential inequalities involving quadratic functions of the error system in two coordinate frames plus the gain adaptation law. We establish that, for bounded system solutions, the estimated state and the gain are bounded. Moreover, we provide an upper bound for the mean value of the error signals as a function of the observer parameters. Since due to perturbations the gain adaptation law may drive the observer/plant interconnection to nearby boundary of its stability region, oscillatory behavior may emerge. To overcome this issue, we suggest an adaptive procedure based on a space averaging technique involving several copies of the observer.     

Lossless convexification of a class of optimal control problems with non-convex control constraints

We consider a class of finite time horizon optimal control problems for continuous time linear systems with a convex cost, convex state constraints and non-convex control constraints. We propose a convex relaxation of the non-convex control constraints, and prove that the optimal solution of the relaxed problem is also an optimal solution for the original problem, which is referred to as the lossless convexification of the optimal control problem. The lossless convexification enables the use of interior point methods of convex optimization to obtain globally optimal solutions of the original non-convex optimal control problem. The solution approach is demonstrated on a number of planetary soft landing optimal control problems.     

Time-local formulation and identification of implicit Volterra models by use of diffusive representation

We present a time-continuous identification method for nonlinear dynamic Volterra models of the form HX=f(u,X)+v with H, a causal convolution operator. It is mainly based on a suitable parameterization of H deduced from the so-called diffusive representation, which is devoted to state representations of integral operators. Following this approach, the complex dynamic nature of H can be summarized by a few numerical parameters on which the identification of the dynamic part of the model will focus. The method is validated on a physical numerical example.     

A parametric periodic Lyapunov equation with application in semi-global stabilization of discrete-time periodic systems subject to actuator saturation

This paper is concerned with semi-global stabilization of discrete-time linear periodic systems subject to actuator saturation. Provided that the open loop characteristic multipliers are within the closed unit circle, a low gain feedback design approach is proposed to solve the problem by state feedback. Our approach is based on the solution to a parametric discrete-time periodic Lyapunov equation. The proposed approaches not only generalize the corresponding results for time-invariant systems to periodic systems, but also reveal some important intrinsic properties of this class of periodic matrix equations. A numerical example is worked out to illustrate the effectiveness of the proposed approaches.     

Robustness analysis of genetic regulatory networks affected by model uncertainty

A fundamental problem in systems biology consists of investigating robustness properties of genetic regulatory networks (GRNs) with respect to model uncertainty. This paper addresses this problem for GRNs where the coefficients are rationally affected by polytopic uncertainty, and where the saturation functions are not exactly known. First, it is shown that a condition for ensuring that the GRN has a globally asymptotically stable equilibrium point for all admissible uncertainties can be obtained in terms of a convex optimization problem with linear matrix inequalities (LMIs), hence generalizing existing results that mainly consider only the case of GRNs where the coefficients are linearly affected by the uncertainty and the regulatory functions are in SUM form. Second, the problem of estimating the worst-case convergence rate of the trajectories to the equilibrium point over all admissible uncertainties is considered, and it is shown that a lower bound of this rate can be computed by solving a quasi-convex optimization problem with LMIs. Third, the paper considers the problem of estimating the set of uncertainties for which the GRN has a globally asymptotically stable equilibrium point. This problem is addressed, first, by showing how one can compute estimates with fixed shape by solving a quasi-convex optimization problem with LMIs, and second, by deriving a procedure for computing estimates with variable shape. Numerical examples illustrate the use of the proposed techniques.     

Interval observers for linear time-invariant systems with disturbances

It is shown that, for any time-invariant exponentially stable linear system with additive disturbances, time-varying exponentially stable interval observers can be constructed. The technique of construction relies on the Jordan canonical form that any real matrix admits and on time-varying changes of coordinates for elementary Jordan blocks which lead to cooperative linear systems. The approach is applied to detectable linear systems.     

Stabilizing composite control for a class of linear systems modeled by singularly perturbed Ito differential equations

In this paper, the problem of robust H_∞ control is investigated for sampled-data systems with probabilistic sampling. The parameter uncertainties are time-varying norm-bounded and appear in both the state and input matrices. For the simplicity of technical development, only two different sampling periods are considered whose occurrence probabilities are given constants and satisfy Bernoulli distribution, which can be further extended to the case with multiple stochastic sampling periods. By applying an input delay approach, the probabilistic sampling system is transformed into a continuous time-delay system with stochastic parameters in the system matrices. By linear matrix inequality (LMI) approach, sufficient conditions are obtained, which guarantee the robust mean-square exponential stability of the system with an H_∞ performance. Moreover, an H_∞ controller design procedure is then proposed. An illustrative example is included to demonstrate the effectiveness of the proposed techniques.     

Attainability of the minimum data rate for stabilization of linear systems via logarithmic quantization

This paper investigates the attainability of the minimum average data rate for stabilization of linear systems via logarithmic quantization. It is shown that a finite-level logarithmic quantizer suffices to approach the well-known minimum average data rate for stabilizing an unstable linear discrete-time system under two basic network configurations. In particular, we derive explicit finite-level logarithmic quantizers and the corresponding controllers to approach the minimum average data rate.     

Approaches to extended non-quadratic stability and stabilization conditions for discrete-time Takagi–Sugeno fuzzy systems

This paper provides simple and effective linear matrix inequality (LMI) characterizations for the stability and stabilization conditions of discrete-time Takagi–Sugeno (T–S) fuzzy systems. To do this, more general classes of non-parallel distributed compensation (non-PDC) control laws and non-quadratic Lyapunov functions are presented. Unlike the conventional non-quadratic approaches using only current-time normalized fuzzy weighting functions, we consider not only the current-time fuzzy weighting functions but also the l-step-past (l?0) and one-step-ahead ones when constructing the control laws and Lyapunov functions. Consequently, by introducing additional decision variables, it can be shown that the proposed conditions include the existing ones found in the literature as particular cases. Examples are given to demonstrate the effectiveness of the approaches.