Identifiability of discrete-time nonlinear systems: The local state isomorphism approach

A new theorem is provided to test the identifiability of discrete-time systems with polynomial nonlinearities. That extends to discrete-time systems the local state isomorphism approach for continuous-time systems. Two examples are provided to illustrate the approach.     

Controller design for flexible structure vibration suppression with robustness to contacts

Model-based feedback control of vibration in flexible structures can be complicated by the possibility that interaction with an external body occurs. If not accounted for, instability or poor performance may result. In this paper, a method is proposed for achieving robust vibration control of flexible structures under contact. The method uses robust linear state feedback, coupled with a state estimation scheme utilizing contact force measurement. Uncertain contact characteristics are modelled by a sector-bounded non-linear function, such that state feedback gains can be synthesized using a matrix inequality formulation of the Popov stability criterion. A separation theorem is used to establish a robust H₂ cost bound for the closed loop system. Experimental results from a multi-mode flexible structure testbed confirm that vibration attenuation and stability can be maintained over a broad range of contact characteristics, in terms of compliance and clearance.     

Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolic PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T_n of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid”, 0≤ξ_n≤ξ_n−1?≤ξ₁≤x≤1. We illustrate our design method with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing blow up in open loop, and stabilization for large initial conditions in closed loop. In a companion paper we give a theoretical study of the properties of the transformation, showing global convergence of the transformation and of the control law nonlinear Volterra operators, and explicitly constructing the inverse of the feedback linearizing Volterra transformation; this, in turn, allows us to prove L² and H¹ local exponential stability (with an estimate of the region of attraction where possible) and explicitly construct the exponentially decaying closed loop solutions.     

A new delay system approach to network-based control

This paper presents a new delay system approach to network-based control. This approach is based on a new time-delay model proposed recently, which contains multiple successive delay components in the state. Firstly, new results on stability and H_∞ performance are proposed for systems with two successive delay components, by exploiting a new Lyapunov-Krasovskii functional and by making use of novel techniques for time-delay systems. An illustrative example is provided to show the advantage of these results. The second part of this paper utilizes the new model to investigate the problem of network-based control, which has emerged as a topic of significant interest in the control community. A sampled-data networked control system with simultaneous consideration of network induced delays, data packet dropouts and measurement quantization is modeled as a nonlinear time-delay system with two successive delay components in the state and, the problem of network-based H_∞ control is solved accordingly. Illustrative examples are provided to show the advantage and applicability of the developed results for network-based controller design.     

Optimal experimental design and some related control problems

This paper traces the strong relations between experimental design and control, such as the use of optimal inputs to obtain precise parameter estimation in dynamical systems and the introduction of suitably designed perturbations in adaptive control. The mathematical background of optimal experimental design is briefly presented, and the role of experimental design in the asymptotic properties of estimators is emphasized. Although most of the paper concerns parametric models, some results are also presented for statistical learning and prediction with nonparametric models.     

New method for identifying finite degree Volterra series

In this paper, the identification of a class of nonlinear systems which admits input-output maps described by a finite degree Volterra series is considered. In actual fact, it appears that this class can model many important nonlinear multivariable processes not only in engineering, but also in biology, socio-economics, and ecology.To solve this identification problem, we propose a method based on local gradient search in a local parameterization of the state-space realization of finite degree Volterra series with infinite horizon. Using the local parameterization not only reduces the amount of the gradient calculations to the minimal value, but also overcomes the nonuniqueness problem of the optimal solution.Moreover, we propose a sequential projection method to provide an initial estimation of the parameters of finite degree Volterra series realization. This initial estimation is used to initialize the gradient search method.     

Nonlinear vehicle side-slip estimation with friction adaptation

A nonlinear observer for estimation of the longitudinal velocity, lateral velocity, and yaw rate of a vehicle, designed for the purpose of vehicle side-slip estimation, is modified and extended in order to work for different road surface conditions. The observer relies on a road-tire friction model and is therefore sensitive to changes in the adhesion characteristics of the road surface. The friction model is parametrized with a single friction parameter, and an update law is designed. The adaptive observer is proven to be uniformly globally asymptotically stable and uniformly locally exponentially stable under a persistency-of-excitation condition and a set of technical assumptions, using results related to Matrosov's theorem. The observer is tested on recorded data from two test vehicles and shows good results on a range of road surfaces.     

Geometric characterization on the solvability of regulator equations

The solvability of the regulator equation for a general nonlinear system is discussed in this paper by using geometric method. The ‘feedback’ part of the regulator equation, that is, the feasible controllers for the regulator equation, is studied thoroughly. The concepts of minimal output zeroing control invariant submanifold and left invertibility are introduced to find all the possible controllers for the regulator equation under the condition of left invertibility. Useful results, such as a necessary condition for the output regulation problem and some properties of friend sets of controlled invariant manifolds, are also obtained.     

Controller certification

A modern problem from aerospace control involves the certification of a large set of potential controllers with either a single plant or a fleet of potential plant systems, with both plants and controllers being MIMO and, for the moment, linear. Experiments on a limited number of controller/plant pairs should establish the stability and a certain level of margin of the complete set. We consider this certification problem for a set of controllers and provide algorithms for selecting an efficient subset for testing. This is done for a finite set of candidate controllers and, at least for SISO plants, for compact infinite set. In doing this, the ν-gap metric will be the main tool. Computational examples are given, including one of certification of an aircraft engine controller. The overarching aim is to introduce truly MIMO margin calculations and to understand their efficacy in certifying stability over a set of controllers and in replacing legacy single-loop gain and phase margin calculations.     

Krylov projection framework for Fourier model reduction

This paper analyzes the Fourier model reduction (FMR) method from a rational Krylov projection framework and shows how the FMR reduced model, which has guaranteed stability and a global error bound, can be computed in a numerically efficient and robust manner. By monitoring the rank of the Krylov subspace that underlies the FMR model, the projection framework also provides an improved criterion for determining the number of Fourier coefficients that are needed, and hence the size of the resulting reduced-order model. The advantages of applying FMR in the rational Krylov projection framework are demonstrated on a simple example.