Introduces the notion of string stability of a countably infinite interconnection of a class of nonlinear systems. Intuitively, string stability implies uniform boundedness of all the states of the interconnected system for all time if the initial states of the interconnected system are uniformly bounded. It is well known that the input output gain of all the subsystems less than unity guarantees that the interconnected system is input-output stable. The authors derive sufficient ("weak coupling") conditions which guarantee the asymptotic string stability of a class of interconnected systems. Under the same "weak coupling" conditions, string-stable interconnected systems remain string stable in the presence of small structural/singular perturbations. In the presence of parameter mismatch, these "weak coupling" conditions ensure that the states of all the subsystems are all uniformly bounded when a gradient-based parameter adaptation law is used and that the states of all the systems go to zero asymptotically.

String stability of interconnected systems

In this paper we introduce the notion of string stability of a countably infinite interconnection of a class of nonlinear systems. Intuitively, string stability implies uniform boundedness of all the states of the interconnected system for all time if the initial states of the interconnected system are uniformly bounded. It is well known that the I/O gain of all the subsystems less than unity guarantees that the interconnected system is I/O stable. We derive sufficient ("weak coupling") conditions which guarantee asymptotic string stability of a class of interconnected systems. Under the same "weak coupling" conditions, string stable interconnected systems remain string stable in the presence of small structural/singular perturbations. In the presence of parameter mismatch, these "weak coupling" conditions ensure that the states of all the subsystems are all uniformly bounded when gradient based parameter adaptation law is used and that they go to zero asymptotically.

Nowadays, throughput has become a limiting factor in road transport. An effective means to increase the road throughput is to employ a small intervehicle time gap using automatic vehicle-following control systems. String stability, i.e., the disturbance attenuation along the vehicle string, is considered an essential requirement for the design of those systems. However, the formal notion of string stability is not unambiguous in literature, since both stability and performance interpretations exist. Therefore, a novel definition for string stability of nonlinear cascaded systems is proposed, using input-output properties. This definition is shown to result in well-known string stability conditions for linear cascaded systems. The theoretical results are experimentally validated using a platoon of six passenger vehicles equipped with cooperative adaptive cruise control.

/pdf/lp-string-stability-of-cascaded-systems-application-to-2qer765sh8.pdf

Lp String Stability of Cascaded Systems: Application to Vehicle Platooning

SUMMARY This article takes up methods for Bayesian inference in a linear model in which the disturbances are independent and have identical Student-t distributions. It exploits the equivalence of the Student-t distribution and an appropriate scale mixture of normals, and uses a Gibbs sampler to perform the computations. The new method is applied to some well-known macroeconomic time series. It is found that posterior odds ratios favour the independednt Student- linear model over the normal linear model inodl, and that the posterior odds ratio in favour of difference stationarity over trend stationarity is often substantially less in the favoured Student-t models. The possibility of leptokurtic disturbances is a common concern of econometricians and other users of the linear model. This article takes up methods for Bayesian inference in a linear model in which the disturbances are independent and have identical Student-t distributions. It exploits the equivalence of the Student-t distribution and an appropriate scale mixture of normals, and uses a Gibbs sampler to perform the computations. The main contribution is to provide a simple and stable computational method for full Bayesian inference in the independent Student-t linear model. The new method is applied to some well-known macroeconomic time series. It is found that posterior odds ratios favour the independent Student-t linear model over the normal linear model, and that the posterior odds ratio in favour of difference stationarity over trend stationarity is often substantially less in the favoured Student-t models. The work reported here builds on Bayesian treatments of heteroscedasticity, which began with hierarchical models for the analysis of variance (Lindley, 1965, 1971). With the adaptation of a prior distribution making cell means linear in cofactors (Lindley and Smith, 1972), this treatment was effectively extended to the linear regression model. Lindley (1971) took up the conjugate prior in which the inverses of the variances are x2(v), up to a factor of proportionality with an improper prior. It is shown in Section 2.1 that this is equivalent to the specification of an independent Student-t linear model with known degrees of freedom. In an important variant on this model, Leonard (1975) used a prior in which the log variances are linear functions of cofactors, and constructed an approximation to the posterior density. This article extends these developments in two ways. First, it shows how to construct the exact posterior distribution, whereas the earlier work was confined to obtaining the posterior mode of an approximate posterior. (Interestingly, however, the algorithm in the appendix of

Bayesian treatment of the independent student-t linear model

Starting with the vector observation model y = Hx + v , robust Bayesian estimates \hat{x} of the vector x are constructed for the following two distinct situations: 1) the state x is Gaussian and the observation error v is (heavy-tailed) non-Gaussian and 2) the state is heavy-tailed non-Gaussian and the observation error is Gaussian. Bounds with respect to broad symmetric non-Gaussian families are derived for the error covariance matrix of these estimates. These "one-step" robust estimates are then used to obtain robust estimates for the Kalman filter setup y_{k}= H_{k}x_{k}+ v_{k}, x_{k+1}=\Phi_{k}x_{k}+w_{k} . Monte Carlo results demonstrate the robustness of the proposed estimation procedure, which might be termed a robustified Kalman filter.

Robust bayesian estimation for the linear model and robustifying the Kalman filter

A random variable is said to have elliptical distribution if its probability density is a function of a quadratic form. This class includes the Gaussian and many other useful densities in statistics. It is shown in this paper that this class of densities can be expressed as integrals of a set of Gaussian densities. This property is not changed under linear transformation of the random variables. It is also proved in this paper that the conditional expectation is linear with exactly the same form as the Gaussian case. Many estimation results of the Gaussian case can be readily extended. Problems of computing optimal estimation, filtering, stochastic control, and team decisions in various linear systems become tractable for this class of random processes.

Estimation and decision for linear systems with elliptical random processes

Decentralized regulation for a string of coupled linear systems is studied. Available to each controller is localized information, represented by linear functions of the state. Necessary conditions for the optimal time-invarient linear regulators are established. The theory can be applied for designing simple on-board computers for regulating a string of moving vehicles.

Optimal dencentralized regulation for a string of coupled systems

An on-line optimal environmental/economic dispatch methodology for electric power generation is developed in this paper. Aside from the conventional economic dispatch, constraints on air quality (such as those specified by the U.S. Environmental Protection Agency) are added to the minimum fuel cost problem. Using the integrated Gaussian puff model based on the statistical turbulent theory, rapid dynamic features of pollutant dispersion and its forecast surrounding the plants are emphasized. By applying a convex programming algorithm repeatedly, a set of marginal environmental imposts for the power plants at different times are obtained. Such imposts are incorporated with the fuel cost in the ordinary short-term economic dispatching program to indirectly account for the environmental impact of power generation on the quality of the ambient air. The approach is specifically taken to have little modification for existing economic dispatch programs and be implemented for real power networks. The proposed approach has been simulated in a power system with three plants and three monitoring points.

An approach to on-line power dispatch with ambient air pollution constraints

This paper investigates the problem of unconstrained optimization when there is only partial information on the random parameters in the objective function. The relation between the optimization performance and the available information is established. The best information structure design with fixed rank is described. The designing procedure is set up in such a way that successive information augmentation or deletion can be considered. The procedure can also be extended to multiperson decision problems.

Kai-ching Chu

Papers

Estimation and decision for linear systems with elliptical random processes

Optimal dencentralized regulation for a string of coupled systems

An approach to on-line power dispatch with ambient air pollution constraints

Designing information structures for quadratic decision problems

Designing information structures for quadratic decision problems