Stabilization of exponentially unstable linear systems withsaturating actuators

We study the problem of stabilizing exponentially unstable linear systems with saturating actuators. The study begins with planar systems with both poles exponentially unstable. For such a system, we show that the boundary of the domain of attraction under a saturated stabilizing linear state feedback is the unique stable limit cycle of its time-reversed system. A saturated linear state feedback is designed that results in a closed-loop system having a domain of attraction that is arbitrarily close to the null controllable region. This design is then utilized to construct state feedback laws for higher order systems with two exponentially unstable poles     

Stabilization of linear strict-feedback systems with delayed integrators

The problem of compensation of input delays for unstable linear systems was solved in the late 1970s. Systems with simultaneous input and state delay have remained a challenge, although exponential stabilization has been solved for systems that are not exponentially unstable, such as chains of delayed integrators and systems in the ‘feedforward’ form. We consider a general system in strict-feedback form with delayed integrators, which is an example of a particularly challenging class of exponentially unstable systems with simultaneous input and state delays, and design a predictor feedback controller for this class of systems. Exponential stability is proven with the aid of a Lyapunov-Krasovskii functional that we construct using the PDE backstepping approach.     

The anti-windup problem for discrete-time linear systems: Definition and solutions

The anti-windup problem for discrete-time linear systems is formalized in terms of the l₂ norm of the deviation of the actual response of the system with saturation and anti-windup compensation from the (ideal) unconstrained response. We show that, paralleling continuous-time results [A.R. Teel, N. Kapoor, The anti-windup problem: its definition and solution, in: Proceedings of the 4th ECC, Brussels, Belgium, July 1997], the problem is globally solvable if and only if the plant is non-exponentially unstable and it is robustly globally solvable if and only if the plant is exponentially stable. We provide a constructive solution whenever the problem is solvable. Also offered is a high-performance global solution for exponentially stable plants based on receding horizon control. Illustrative simulations are included.     

Truncated predictor feedback control for exponentially unstable linear systems with time-varying input delay

The stabilization of exponentially unstable linear systems with time-varying input delay is considered in this paper. We extend the truncated predictor feedback (TPF) design method, which was recently developed for systems with all poles on the closed left-half plane, to be applicable to exponentially unstable linear systems. Assuming that the time-varying delay is known and bounded, the design approach of a time-varying state feedback controller is developed based on the solution of a parametric Lyapunov equation. An explicit condition is derived for which the stability of the closed-loop system with the proposed controller is guaranteed. It is shown that, for the stability of the closed-loop system, the maximum allowable time-delay in the input is inversely proportional to the sum of the unstable poles in the plant. The effectiveness of the proposed method is demonstrated through numerical examples.     

Convergence of the Kalman filter gain for a class of nondetectable signal extraction problems

Convergence of the filter gain is proved for a class of problems with undetectable modes on the unit circle. This situation arises when the signal is an unstable ARMA process corrupted by ARMA noise with the same unstable modes, since modes common to signal and noise are unobservable.     

Global exponential regulation of discrete time linear systems with unknown neutrally stable exosystems

Discrete time, linear, stabilizable and detectable systems with known parameters are considered: the regulation problem is addressed when the reference output and/or the disturbances contain sinusoidal terms generated by a linear exosystem with unknown parameters. Only an upper bound on the number of unknown sinusoids is supposed to be known. A constructive algorithm is proposed to drive the regulation error exponentially to zero on the basis of its measurement only, under the same necessary and sufficient conditions which are required when the exosystem is known. The control strategy includes an online detector for the number of excited frequencies and exponentially converging global estimates of the exosystem unknown parameters. An illustrative example containing a variable number of frequencies is worked out and simulated.     

A note on the differential regulator equation for non-minimum phase linear systems with time-varying exosystems

Some control problems in practice are often formulated as a linear output regulation problem with time-varying exosystems. Although this problem has been studied recently, an explicit and constructive solution has been given only for minimum phase systems. This paper presents a solution for non-minimum phase systems whose zero dynamics is hyperbolic (or has an exponentially dichotomic split in the case of time-varying zero dynamics). The idea is inspired by the non-causal stable inversion.     

Output feedback design for saturated linear plants using deadzone loops

In this paper, we present an LMI-based synthesis approach on output feedback design for input saturated linear systems by using deadzone loops. Algorithms are developed for minimizing the upper bound on the regional L₂ gain for exogenous inputs with L₂ norm bounded by a given value, and for minimizing this upper bound with a guaranteed reachable set or domain of attraction. The proposed synthesis approach will always lead to regionally stabilizing controllers if the plant is exponentially unstable, to semi-global results if the plant is non-exponentially unstable, and to global results if the plant is already exponentially stable, where the only requirement on the linear plant is detectability and stabilizability. The effectiveness of the proposed techniques is illustrated with one example.     

On Asymptotic Stabilizability of Linear Systems With Delayed Input

This paper examines the asymptotic stabilizability of linear systems with delayed input. By explicit construction of stabilizing feedback laws, it is shown that a stabilizable and detectable linear system with an arbitrarily large delay in the input can be asymptotically stabilized by either linear state or output feedback as long as the open-loop system is not exponentially unstable (i.e., all the open-loop poles are on the closed left-half plane). A simple example shows that such results would not be true if the open-loop system is exponentially unstable. It is further shown that such systems, when subject to actuator saturation, are semiglobally asymptotically stabilizable by linear state or output feedback.     

A Remark on an Example By Teel–Hespanha With Applications to Cascaded Systems

The properties of a system, proposed by Teel and Hespanha, which is globally exponentially stable but with state that can be driven to infinity by an arbitrarily small exponentially decaying disturbance, are discussed in detail. These are used to propose a family of systems with a similar property and to argue that unstable behavior may be nongeneric and not detected by means of simulations. Finally, sufficient conditions for the existence of unbounded trajectories in cascaded systems are given