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.     

Critical dwell time of switched linear systems

In this paper, we consider the relation between the switching dwell time and the stabilization of switched linear control systems. First of all, a concept of critical dwell time is given for switched linear systems without control inputs, and the critical dwell time is taken as an arbitrary given positive constant for a switched linear control systems with controllable switching models. Secondly, when a switched linear system has many stabilizable switching models, the problem of stabilization of the overall system is considered. An on-line feedback control is designed such that the overall system is asymptotically stabilizable under switching laws which depend only on those of uncontrollable subsystems of the switching models. Finally, when a switched system is partially controllable (While some switching models are probably unstabilizable), an on-line feedback control and a cyclic switching strategy are designed such that the overall system is asymptotically stabilizable if all switching models of this uncontrollable subsystems are asymptotically stable. In addition, algorithms for designing switching laws and controls are presented.     

Critical dwell time of switched linear systems

In this paper, we consider the relation between the switching dwell time and the stabilization of switched linear control systems. First of all, a concept of critical dwell time is given for switched linear systems without control inputs, and the critical dwell time is taken as an arbitrary given positive constant for a switched linear control systems with controllable switching models. Secondly, when a switched linear system has many stabilizable switching models, the problem of stabilization of the overall system is considered. An on-line feedback control is designed such that the overall system is asymptotically stabilizable under switching laws which depend only on those of uncontrollable subsystems of the switching models. Finally, when a switched system is partially controllable （While some switching models are probably unstabilizable）, an on-line feedback control and a cyclic switching strategy are designed such that the overall system is asymptotically stabilizable if all switching models of this uncontrollable subsystems are asymptotically stable. In addition, algorithms for designing switching laws and controls are presented.     

Constrained stabilization of discrete-time systems

Based on the growth rate of the set of states reachable with unit-energy inputs, we show that a discretetime controllable linear system is globally controllable to the origin with constrained inputs if and only if all of its eigenvalues lie in the closed unit disk. These results imply that the constrained Infinite-Horizon Model Predictive Control algorithm is stabilizing for a sufficiently large number of control moves if and only if the controlled system is stabilizable and all its eigenvalues lie in the closed unit disk. In the second part of the paper, we propose an implementable Model Predictive Control algorithm and show that with this scheme a discrete-time linear system with n poles on the unit disk (with any multiplicity) can be globally stabilized if the number of control moves is larger than n. For pure integrator systems, this condition is also necessary. Moreover, we show that global asymptotic stability is preserved for any asymptotically constant disturbance entering at the plant input.     

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     

Adaptive stabilization using a nonlinear time-varying controller

Byrnes et al. (1986) showed that there is no smooth, finite-dimensional, nonlinear time-invariant (NLTI) controller which asymptotically stabilizes every finite-dimensional, stabilizable and detectable, linear time-invariant (LTI) plant (with a fixed number of inputs and outputs). Here we construct a finite-dimensional nonlinear time-varying (NLTV) controller which does exactly that; we treat both the discrete-time and continuous-time cases. With p equal to one in the discrete-time case and the number of plant outputs in the continuous-time case, we first show that for every stabilizable and detectable plant, there exists a p-dimensional linear time-varying (LTV) compensator which provides exponential stabilization; we then construct a (p+1)-dimensional NLTV controller which asymptotically stabilizes every admissible plant by switching between a countable number of such LTV compensators     

Semi-global stabilization of partially linear composite systems via feedback of the state of the linear part

We consider the problem of semi-global stabilization of a class of partially linear composite systems. We show, by explicit construction of the control laws, that a cascade of linear stabilizable and nonlinear asymptotically stable subsystems is semi-globally stabilizable by a dynamic feedback of the state of the linear subsystem if (a) the linear subsystem is right invertible and has all its invariant zeros in the closed left half s-plane, and (b) the only linear variables entering the nonlinear subsystem are the output of the linear subsystem. Our work generalizes previous results by C.I. Byrnes and A. Isidori (1991), H.J. Sussmann and P.V. Kokotovic (1991), and A.R. Teel (1992).     

Exponential stabilization of discrete-time switched linear systems

This article studies the exponential stabilization problem for discrete-time switched linear systems based on a control-Lyapunov function approach. It is proved that a switched linear system is exponentially stabilizable if and only if there exists a piecewise quadratic control-Lyapunov function. Such a converse control-Lyapunov function theorem justifies many of the earlier synthesis methods that have adopted piecewise quadratic Lyapunov functions for convenience or heuristic reasons. In addition, it is also proved that if a switched linear system is exponentially stabilizable, then it must be stabilizable by a stationary suboptimal policy of a related switched linear-quadratic regulator (LQR) problem. Motivated by some recent results of the switched LQR problem, an efficient algorithm is proposed, which is guaranteed to yield a control-Lyapunov function and a stabilizing policy whenever the system is exponentially stabilizable.     

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     

Hybrid static output feedback stabilization of two-dimensional LTI systems: a geometric method

For two-dimensional linear time-invariant (LTI) systems which are not stabilizable via a single static output feedback, we propose a hybrid stabilization strategy based on a geometric method. More precisely, we design two static output feedback gains and a switching law between the feedback gains so that the entire closed-loop system is asymptotically stable. The proposed switching law is composed of output-dependent switching and time-controlled switching. We demonstrate the hybrid control method with various examples for different cases.     

Bounded smooth state feedback and a global separation principle for non-affine nonlinear systems

This paper develops sufficient conditions for a general nonlinear control system to be locally (resp. globally) asymptotically stabilizable via smooth state feedback. In particular, it is shown that as in the case of affine systems, this is possible if the unforced dynamic system of ∑₁ is Lyapunov stable and appropriate controllability-like rank conditions are satisfied. Our results incorporate a series of well-known stabilization theorems proposed in the literature for affine control systems and extend them to nonaffine nonlinear control systems.     

Stabilizing switching design for switched linear systems: A state-feedback path-wise switching approach

This paper addresses the problem of switching stabilization for discrete-time switched linear systems. Based on the abstraction-aggregation methodology, we propose a state-feedback path-wise switching law, which is a state-feedback concatenation from a finite set of switching paths each defined over a finite time interval. We prove that the set of state-feedback path-wise switching laws is universal in the sense that any stabilizable switched linear system admits a stabilizing switching law in this set. We further develop a computational procedure to calculate a stabilizing switching law in the set.     

Output feedback stabilization of the angular velocity of a rigid body

The problem of stabilization of the angular velocity of a rigid body using only two control signals and partial state information is addressed. It is shown that if any two (out of three) states are measured the system is not asymptotically stabilizable with (continuous) dynamic output feedback. Nevertheless, we prove that practical stability is achievable if the measurable states fulfill a certain structural property, and that, under the same structural condition, a hybrid control law yielding exponential convergence can be constructed. Finally, we also study some geometric features of the Euler’s equations and the connection between local strong accessibility and local observability.     

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.     

Robust Stabilizability Verification of Polynomially Uncertain LTI Systems

This paper investigates the stabilizability of uncertain linear time-invariant (LTI) systems via structurally constrained controllers. First, an LTI uncertain system is considered whose state-space matrices depend polynomially on the uncertainty vector, defined over some region. It is shown that if the system is stabilizable by a structurally constrained controller in one point belonging to the region, then it is stabilizable by a controller with the same structure in all points belonging to the region, except for those located on an algebraic variety. Thus, if a system is stabilizable via a constrained controller at the nominal point, then it is almost always stabilizable at any operating point around the nominal model. It is also shown how this algebraic variety (or the dominant subvariety of it) can be computed efficiently. The results obtained in this paper encompass a broad range of the existing results in the literature on robust stability of the LTI systems, in addition to new ones.     

Time-varying feedback control of nonaffine nonlinear systems without drift

Sufficient conditions are presented under which a general nonlinear system without drift is globally asymptotically stabilizable by time-varying state feedback. A novel approach is developed for the design of a time-varying smooth state feedback controller. The controller is explicitly constructed by using the bounded state feedback strategy (Lin, 1995, 1996) combined with Lyapunov technique as well as lossless systems theory. This work incorporates earlier global stabilization results (Coron, 1992; Pomet, 1992) for controllable affine systems without drift, which are known not to be smoothly stabilizable via any time-invariant state feedback.     

Stabilization of second-order LTI switched systems

This paper studies and solves the problem of asymptotic stabilization of switched systems consisting of unstable secondorder linear time-invariant (LTI) subsystems. Necessary and sufficient conditions for asymptotic stabilizability are first obtained. If a switched system is asymptotically stabilizable, then the conic switching laws proposed in the paper are used to construct a switching law that asymptotically stabilizes the system. Switched systems consisting of two subsystems with unstable foci are studied first and then the results are extended to switched systems with unstable nodes and saddle points. The results are applicable to switched systems that consist of more than two subsystems.     

Numerical schemes for nonlinear predictor feedback

This paper focuses on a specific aspect of the implementation problem for predictor-based feedback laws: the problem of the approximation of the predictor mapping. It is shown that the numerical approximation of the predictor mapping by means of a numerical scheme in conjunction with a hybrid feedback law that uses sampled measurements can be used for the global stabilization of all forward complete nonlinear systems that are globally asymptotically stabilizable and locally exponentially stabilizable in the delay-free case. Explicit formulae are provided for the estimation of the parameters of the resulting hybrid control scheme.     

Some results on the stabilization of switched systems

This paper deals with the stabilization of switched systems with respect to (w.r.t.) compact sets. We show that the switched system is stabilizable w.r.t. a compact set by means of a family of switched signals if and only if a certain control affine system whose admissible controls take values in a polytope is asymptotically controllable to that set. In addition we present a control algorithm that based on a family of open-loop controls which stabilizes the aforementioned control system, a model of the system and the states of the switched system, generates switching signals which stabilize the switched system in a practical sense. We also give results about the convergence and the robustness of the algorithm.     

Stabilization of discrete systems with multiple-time scales

A two-stage method is developed for the stabilization of linear time-invariant discrete systems with multiple-time scales. It is shown that the feedback gains are completely independent. The closed-loop system is asymptotically stable for all sufficiently small singular perturbation parameters which are constrained to be in a bounded set.