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.     

Semi-global exponential stabilization of linear discrete-time systems subject to input saturation via linear feedbacks

In this paper, we show that a linear discrete-time system subject to input saturation is semi-globally exponentially stabilizable via linear state and/or output feedback laws as long as the system in the absence of input saturation is stabilizable and detectable, and has all its poles located inside or on the unit circle. Furthermore, the semi-globally stabilizing feedback laws are explicitly constructed. The results presented here are parallel to our earlier results on the continuous-time counterpart (Lin and Saberi, 1993).     

Semi-global stabilizability of linear null controllable systemswith input nonlinearities

An H_∞-based Lyapunov proof is provided for a result established by Lin and Saberi (1993): if a linear system is asymptotically null controllable with bounded controls then, when subject to input saturation, it is semi-globally stabilizable by linear state feedback. A new result is that if the system is also detectable then it is semi-global stabilizable by completely linear output feedback. Further, an extension which relaxes the requirements on the input characteristic is obtained     

Synthesis of upper-triangular non-linear systems with marginally unstable free dynamics using state-dependent saturation

In this paper we present sufficient conditions under which a fairly large class of single-input non-linear systems including feedforward systems and the well-known ball-and-beam model, are globally asymptotically and locally exponentially stabilizable by smooth state feedback. A nested saturation controller with state-dependent saturation levels is constructed explicitly, using a novel design approach which combines the nested saturation strategy for marginally unstable linear systems subject to input saturation, with the small feedback design technique, developed for global asymptotic stabilization of general non-affine systems with marginally stable free dynamics. The power of the state-dependent saturation design method is demonstrated by solving a number of non-linear control problems, particularly, the global stabilization problem of a class of two-dimensional non-linear systems and the ball-and-beam system.     

Semi-global stabilization of discrete-time linear systems with position and rate-limited actuators

It is shown via explicit construction of feedback laws that, if a discrete-time linear system is asymptotically null controllable with bounded controls, then, when subject to both actuator position and rate saturation, it is semi-globally stabilizable by linear state feedback. If, in addition, the system is also detectable, then it is semi-globally stabilizable via linear output feedback.     

Semiglobal stabilization of linear discrete-time systems subject toinput saturation, via linear feedback-an ARE-based approach

We revisit the problem of semiglobal stabilization of linear discrete-time systems subject to input saturation and give an algebraic Riccati equation (ARE)-based approach to the proof of a fact established earlier (Lin and Saberi, 1995), i.e. a linear discrete-time system subject to input saturation is semiglobally stabilizable via linear feedback as long as the linear system in the absence of the saturation is stabilizable and detectable and all its open-loop poles are located inside or on the unit circle. Moreover, we drastically relax the requirements on the characteristic of the saturation elements as imposed in the earlier work     

Semi-global stabilization of linear systems subject to output saturation

It is established that a SISO linear stabilizable and detectable system subject to output saturation can be semi-globally stabilized by linear output feedback if all its invariant zeros are in the closed left-half plane, no matter where the open loop poles are. This result complements a recent result that such systems can always be globally stabilized by discontinuous nonlinear feedback laws, and can be viewed as dual to a well-known result: a linear stabilizable and detectable system subject to input saturation can be semi-globally stabilized by linear output feedback if all its poles are in the open left-half plane, no matter where the invariant zeros are.     

Constrained exponential stabilizability of marginal and/or unstable linear systems

It has been established that the constrained asymptotic stabilization of marginal and/or unstable linear systems on the constrained asymptotically stabilizable set is possible by non-linear control and that the constrained exponential stabilization on any compact subset of the constrained asymptotically stabilizable set is possible by linear control. However, we show in this paper that the constrained exponential stabilization of such systems on the constrained asymptotically stabilizable set is impossible by any control law.     

H∞-almost disturbance decoupling with internalstability for linear systems subject to input saturation

For a linear system subject to input saturation and input-additive disturbances, we show that: (1) the H_∞-almost disturbance decoupling problem with local asymptotic stability is always solvable via state feedback as long as the system in the absence of saturation is stabilizable, no matter where the open-loop poles are; and (2) the H_∞-almost disturbance decoupling problem with semiglobal asymptotic stability is solvable via state feedback as long as the system in the absence of saturation is stabilizable with all its open-loop poles located in the closed left-half plane. The results generalize those in Lin et al. (1996) by not requiring the disturbance to be bounded by a known bound, or even bounded     

Truncated predictor feedback for linear systems with long time-varying input delays

In this paper we study the problem of stabilizing a linear system with a single long time-varying delay in the input. Under the assumption that the open-loop system is stabilizable and not exponentially unstable, a finite dimensional static time-varying linear state feedback controller is obtained by truncating the predictor based controller and by adopting the parametric Lyapunov equation based controller design approach. As long as the time-varying delay is exactly known and bounded, an explicit condition is provided to guarantee the stability of the closed-loop system. It is also shown that the proposed controller achieves semi-global stabilization of the system if its actuator is subject to either magnitude saturation or energy constraints. Numerical examples show the effectiveness of the proposed approach.     

Anti-windup control design for exponentially unstable LTI systems with actuator saturation

In this paper, a new saturation control technique in the framework of anti-windup compensation is developed for exponentially unstable linear time-invariant systems subject to input nonlinearities. The proposed control algorithm guarantees regional stability in the existence of input saturation, and provides less conservative performance than most existing anti-windup schemes. Moreover, an explicit form of anti-windup controller with its order no more than the order of the plant is derived. An inverted pendulum example is used to demonstrate the advantages of the newly proposed anti-windup control technique.     

Absolute stability and integral control

Absolute stability results of both circle criterion and Popov type are derived for finite-dimensional linear plants with non-linearity in the feedback loop. The linear plant contains an integrator (and so is not asymptotically stable). The (possibly time-varying) non-linearity satisfies a particular sector condition which allows for cases with zero lower gain (such as saturation and deadzone). The conjunction of stable, but not asymptotically stable, linear plants and non-linearities with possibly zero lower gain is a distinguishing feature of the paper. The absolute stability results are invoked in proving convergence and stability properties of low-gain integral feedback control for tracking of constant reference signals in the context of exponentially stable linear systems subject to input and output non-linearities.     

A nonlinear small gain theorem for the analysis of control systemswith saturation

A nonlinear small gain theorem is presented that provides a formalism for analyzing the behavior of certain control systems that contain or utilize saturation. The theorem is used to show that an iterative procedure can be derived for controlling systems in a general nonlinear, feedforward form. This result, in turn, is applied to the control of: 1) linear systems (stable and unstable) with inputs subject to magnitude and rate saturation and time delays; 2) the cascade of globally asymptotically stable nonlinear systems with certain linear systems (those that are stabilizable, right invertible, and such that all of their invariant zeros have nonpositive real part); 3) the inverted pendulum on a cart; and 4) the planar vertical takeoff and landing aircraft     

Limitations on the stabilizability of globally-minimum-phasesystems

The author gives examples showing that, in general, it is possible for a globally-minimum-phase system in normal form to have states that cannot be driven asymptotically to the origin by means of any open-loop control. In particular, this provides counterexamples to a number of recently published stabilization theorems. It is established, by means of examples, that a minimum-phase system in normal form need not be semiglobally stabilizable or small-input semiglobally BIBO stabilizable, even if the zero dynamics is exponentially stable and the completeness condition holds     

Global and semi-global stabilizability in certain cascade nonlinearsystems

This paper addresses the issue of global and semi-global stabilizability of an important class of nonlinear systems, namely, a cascade of a linear, controllable system followed by an asymptotically (even exponentially) stable nonlinear system. Such structure may arise from the normal form of “minimum phase” nonlinear systems that can be rendered input-output linear by feedback. These systems are known to be stabilizable in a local sense. And, in some cases, global stabilizability results have also been obtained. It is also known, however, that when the linear “connection” to the nonlinear system is nonminimum phase, i.e,, it has zeros with positive real part, then global or semi-global stabilizability may be impossible. Indeed, it has been shown that for any given nonminimum phase linear subsystem, there exists an asymptotically stable nonlinear subsystem for which the cascade cannot be globally stabilized. We expand on the understanding of this area by establishing, for a broader class of systems, conditions under which global or semiglobal stabilization is impossible for linear and nonlinear feedback     

输出饱和线性系统的稳定性及L2增益性能

考虑输出饱和线性系统的稳定性以及L     

Output regulation for linear systems via adaptive internal model

Given a stabilizable and detectable linear system with additive disturbances and output references generated by a linear stable exosystem with unknown parameters and known order, the problem of designing a global output feedback regulator which asymptotically achieves output regulation and disturbance rejection is considered. The system is assumed to be known while the frequencies of the exosystem are unknown; all exosystem oscillatory modes are assumed to be excited by the initial condition. A global solution is proposed consisting of a dynamic output feedback controller which includes exponentially convergent estimates of the unknown frequencies.     

Constrained MPC for uncertain linear systems with ellipsoidal target sets

Robustly feasible invariant sets provide a way of identifying stabilizable regions for uncertain/time-varying linear systems with input constraints under fixed state feedback control laws. With the introduction of extra degrees of freedom in the form of perturbed control laws, these stabilizable regions can be enlarged. This was done in Lee and Kouvaritakis (Automatica 36 (2000) 1497–1504) in conjunction with polyhedral invariant sets and the aim here is to extend this work using ellipsoidal target sets. We also extend the analysis to take into account both polytopic and unstructured bounded disturbances, as well as unstructured uncertainties.     

Simultaneous linear and anti-windup controller synthesis using multiobjective convex optimization

We present a method for the synthesis of a control law for input constrained linear systems that incorporates both a traditional linear output-feedback controller as well as a static anti-windup compensator. Unlike traditional two-step anti-windup controller designs in which the linear controller and anti-windup compensator are designed sequentially, our method synthesizes all controller parameters simultaneously. This one-step design retains the anti-windup structure, thus providing structurally ‘a priori’ compensation for saturation. We derive sufficient conditions for guaranteeing global quadratic stability and for satisfying multiple, possibly conflicting, performance objectives on the constrained and unconstrained closed-loop dynamics. The resulting synthesis problem is recast as an optimization over linear matrix inequalities (LMIs). We demonstrate the proposed method on a benchmark problem.     

On the Lp (ℓp) stabilization of open‐loop neutrally stable linear plants with input subject to amplitude saturation

It has been shown that for asymptotically null controllable linear systems with input saturation and non‐input‐additive disturbances, there exist nonlinear control laws that achieve global stabilization and L_p (?_p) stabilization without finite‐gain for any p∈[1,∞). Recently, it also has been shown that for a simple double integrator there is no saturated linear controller that can achieve L_p stabilization for p>2. In this paper, we show that if a linear system is open‐loop neutrally stable and stabilizable then there exist saturated linear control laws that achieve L_p (?_p) stability for any p∈[1,∞) and for arbitrary initial conditions. As a byproduct, we also show that the closed‐loop system with a saturated linear control law has a nice property similar to linear systems, i.e., any vanishing disturbance produces a vanishing state with arbitrary initial condition. Copyright © 2003 John Wiley & Sons, Ltd.