On the stabilization of linear discrete time systems subject to input saturation

In this paper, an exponentially unstable linear discrete time system subject to input saturation is shown to be exponentially stabilizable on any compact subset of the constrained asymptotically stabilizable set by a linear periodic variable structure controller. We also point out that any marginally stable system² subject to input saturation can be globally asymptotically stabilized via linear feedback.     

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.     

Stabilization of exponentially unstable discrete-time linear systems by truncated predictor feedback

Predictor state feedback solves the problem of stabilizing a discrete-time linear system with input delay by predicting the future state with the solution of the state equation and thus rendering the closed-loop system free of delay. The solution of the state equation contains a term that is the convolution of the past control input with the state transition matrix. Thus, the implementation of the resulting predictor state feedback law involves iterative calculation of the control signal. A truncated predictor feedback law results when the convolution term in the state prediction is discarded. When the feedback gain is constructed from the solution of a certain parameterized Lyapunov equation, the truncated predictor feedback law has been shown to achieve asymptotic stabilization of a system that is not exponentially unstable in the presence of an arbitrarily large delay by tuning the value of the parameter small enough. In this paper, we extend this result to exponentially unstable systems. Stability analysis leads to a bound on the delay and a range of the values of the parameter for which the closed-loop system is asymptotically stable as long as the delay is within the bound. The corresponding output feedback result is also derived.     

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.     

Consensus of high-order multi-agent systems with large input and communication delays

We study in this paper the consensus problem for multi-agent systems with agents characterized by high-order linear systems with time delays in both the communication network and inputs. Provided that the open-loop dynamics of the agents is not exponentially unstable, but may be polynomially unstable, and the communication topology contains a directed spanning tree, a truncated predictor feedback approach is established to solve the consensus problem. It is shown that, if the delays are constant and exactly known, the consensus problems can be solved by both full state feedback and observer based output feedback protocols for arbitrarily large yet bounded delays. If it is further assumed that the open-loop dynamics of the agents only contains zero eigenvalues, the delays are allowed to be time-varying and unknown. Numerical examples are worked out to illustrate the effectiveness of the proposed approaches.     

Stabilization of linear systems with distributed input delay and input saturation

This paper is concerned with stabilization of a linear system with distributed input delay and input saturation. Both constant and time-varying delays are considered. In the case that the input delay is constant, under the stabilizability assumption on an auxiliary system, it is shown that the system can be stabilized by state feedback for an arbitrarily large delay as long as the open-loop system is not exponentially unstable. In the case that the input delay is time-varying, but bounded, it is shown that the system can be stabilized by state feedback if the non-asymptotically stable poles of the open-loop system are all located at the origin. In both cases, stabilizing controllers are explicitly constructed by utilizing the parametric Lyapunov equation based low gain design approach we recently developed. It is also shown that in the presence of actuator saturation and under the same assumptions on the system, these controllers achieve semi-global stabilization. Some discussions on the assumptions we impose on the system are given. A numerical example illustrates the effectiveness of the proposed stabilization approach.     

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     

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     

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.     

Output Feedback Control of Bilinear Systems via a Bilinear LTR Observer

In the literature, most observer-based output feedback controls for bilinear systems are only applicable to open-loop (neutrally) stable systems. This paper proposes a new observer-based output feedback control that can be applied to open-loop unstable systems. The key component of the new control is an exponentially stable bilinear loop transfer recovery (LTR) observer that derives from the linear LTR observer.     

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.     

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.     

A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback

It is known that if a system can be (robustly) globally asymptotically stabilized by means of a feedback that is driven by functions that are uniformly completely observable (UCO), then this system can be practically semiglobally stabilized by means of (possibly dynamic) output feedback. This papers discusses a significant structural hypothesis under which the existence of a dynamic feedback driven by UCO functions is guaranteed. The class of systems which satisfy this hypothesis includes any stabilizable and detectable linear system and any relative degree one nonlinear system which is stabilizable by dynamic output feedback. In particular, the hypothesis does not require the system to be minimum phase.     

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     

Exponentially stabilizing division controllers for dyadic bilinear systems

It is difficult to asymptotically stabilize a dyadic bilinear system with only multiplicative control inputs when the open-loop dynamics are unstable. The previous approach of cascading a division controller with a constant-size dead zone can only stabilize but not asymptotically stabilize the system. This note proposes a new control design which cascades a division controller with a modified dead zone whose size is proportional to the modulus of the system state. It is shown that this new division controller can globally and exponentially stabilize any open-loop unstable dyadic bilinear system as long as it is controllable.     

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     

An LMI approach to stabilization of linear discrete-time periodic systems

The problem of stabilization of linear discrete-time periodic systems is considered. LMI based conditions for stabilization via static periodic state feedback as well as via static periodic output feedback are presented. In the case of state feedback, the conditions are necessary and sufficient whereas for output feedback the result is only sufficient as it depends on the particular state-space representation used to describe the system. The problem of quadratic stabilization in the presence of either norm-bounded or polytopic parameter uncertainty is also treated. As an application of the output feedback stabilization technique, we consider the problem of designing a stabilizing (respectively, quadratically stabilizing) static periodic output feedback controller for linear time-invariant discrete-time systems which are not stabilizable (respectively, quadratically stabilizable) by static constant output feedback.     

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.     

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.