Nonsmooth Robust Stabilization of a Class of Two-Dimensional Nonlinear Systems

In this paper we consider a class of parameterized families of nonlinear systems which cannot be robustly asymptotically stabilized by means of C ¹ feedback. We construct C ⁰ state feedback laws which are smooth away from the origin and which robustly asymptotically stabilize these families of systems. We then show that, in some cases, the regularity of the obtained robust asymptotic stabilizers is “maximum” in the sense that the considered families of systems do not admit any Lipschitz continuous robust asymptotic stabilizer. Date received: June 27, 1997. Date revised: July 28, 1998.     

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.     

Stabilizability by state feedback implies stabilizability by encoded state feedback

Encoded state feedback is a term which refers to the situation in which the state feedback signal is sampled every T units of time and converted (encoded) into a binary representation. In this note stabilization of nonlinear systems by encoded state feedback is studied. It is shown that any nonlinear control system which can be globally asymptotically stabilized by “standard” (i.e. with no encoding) state feedback can also be globally asymptotically stabilized by encoded state feedback, provided that the number of bits used to encode the samples is not less than an explicitly determined lower bound. By means of this bound, we are able to establish a direct relationship between the size of the expected region of attraction and the data rate, under the stabilizability assumption only, a result which—to the best of our knowledge—does not have any precedent in the literature.     

Hybrid state feedback stabilization with l 2 performance for discrete-time switched linear systems

In this paper, the co-design of continuous-variable controllers and discrete-event switching logics, both in state feedback form, is investigated for a class of discrete-time switched linear control systems. It is assumed that none of the subsystems is stabilized through a continuous state feedback alone. However, it is possible to stabilize the whole switched system via carefully designing both the continuous controllers and the switching logics. Sufficient synthesis conditions for this co-design problem are proposed here in the form of bilinear matrix inequalities, which is based on the argument of multiple Lyapunov functions. The closed-loop switched system forms a special class of linear hybrid system, and is shown to be asymptotically stable with a finite l ₂ induced gain.     

Stabilizing controllers design for switched nonlinear systems in strict-feedback form

This paper considers the stabilization problem for a class of switched nonlinear systems under arbitrary switching. Based on the backstepping method and the control Lyapunov function approach, it is shown that, under a simultaneous domination assumption, a switched nonlinear system in strict-feedback form can be globally uniformly asymptotically stabilized by a continuous state feedback controller. A universal formula for constructing stabilizing feedback laws is presented. One example is included for verifying the obtained results.     

Stabilization of feedforward systems approximated by a non-linear chain of integrators

We prove that, for any odd integer p and any strictly positive integer n, feedforward systems which are approximated at the origin by a chain of integrators of degree p and length n can be globally asymptotically stabilized by bounded smooth time-invariant state feedbacks. Our proof is based on the construction of a Lyapunov function and the feedback laws we obtain are given by explicit formulas.     

On passivity-based output feedback global stabilization of euler-lagrange systems

It is well known that in systems described by Euler-Lagrange equations the stability of the equilibria is determined by the potential energy function. Further, these equilibria are asymptotically stable if suitable damping is present in the system. These properties motivated the development of a passivity-based controller design methodology which aims at modifying the potential energy of the closed loop and the addition of the required dissipation. To achieve the latter objective measurement of the generalized velocities is typically required. Our main contribution in this paper is the proof that damping injection without velocity measurement is possible via the inclusion of a dynamic extension provided the system satisfies a dissipation propggation condition. This allows us to determine a class of Euler-Lagrange systems that can be globally asymptotically stabilized with dynamic output feedback. We illustrate this result with the problem of set-point control of elastic joints robots. Our research contributes, if modestly, to the development of a theory for stabilization of nonlinear systems with physical structures which effectively exploits its energy dissipation properties.     

Adaptive λ-tracking for a class of infinite-dimensional systems

For a class of high-gain stabilizable multivariable linear infinite-dimensional systems we present an adaptive control law which achieves approximate asymptotic tracking in the sense that the tracking error tends asymptotically to a ball centred at 0 and of arbitrary prescribed radius λ>0. This control strategy, called λ-tracking, combines proportional error feedback with a simple nonlinear adaptation of the feedback gain. It does not involve any parameter estimation algorithms, nor is it based on the internal model principle. The class of reference signals is W^1,∞, the Sobolev space of absolutely continuous functions which are bounded and have essentially bounded derivative. The control strategy is robust with respect to output measurement noise in W^1,∞ and bounded input disturbances. We apply our results to retarded systems and integrodifferential systems.     

A note on reduced order stabilizing output feedback controllers

In this paper we show that if a certain class of nonlinear systems is globally asymptotically stabilizable through an n-dimensional output feedback controller then it can be always stabilized through an (n − p)-dimensional output feedback controller, where p is the number of outputs and n is the dimension of the state space. This result gives an alternative construction of reduced order controllers for linear systems, and recovers in a more general framework the concept of dirty derivative, used in the framework of rigid and elastic joint robots, and gives an alternative procedure for designing reduced-order controllers for nonlinear systems considered in the existing literature.     

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.     

Robust H∞ Control for Non‐Minimum Phase Switched Cascade Systems with Time Delay

The H_∞ control of a class of the uncertain switched nonlinear cascaded systems with time delay is explored in this paper via the multiple Lyapunov functions. The considered systems are assumed to comprise an inherently nonlinear and a linearizable nonlinear dynamic system that may be non‐minimum phase. A group of partial differential inequalities containing adjustable functions are used in the control design task. The state feedback controllers and a suitable switching law are designed simultaneously so as to achieve the desired disturbance attenuation while preserving asymptotic stability for all admissible uncertainties. The partial differential inequalities are of lower dimension than general Hamilton–Jacobi inequalities, and therefore the solving process is feasible. This particular technique is applicable even if no subsystem is asymptotically stable. The non‐minimum phase property is compensated for by means of an appropriate switching mechanism. A robust H_∞ control for non‐switched cascade system with time delay is obtained in addition. An illustrative example is given to demonstrate the efficiency of the proposed design method.     

Feedback stabilization of strongly non-linear systems using the CBH formula

This paper presents an approach to the construction of stabilizing feedback for strongly non-linear systems. The class of systems of interest includes systems with drift which are affine in the controls and which cannot be stabilized by continuous state feedback. The approach is independent of the selection of a Lyapunov function, but requires the on-line solution of a non-linear programming satisficing problem stated in terms of expressions resulting from the composition of flows via the Campbell–Baker–Hausdorff formula.     

A generalization of Vidyasagar's theorem on stabilizability using state detection

In this paper we generalize the Vidyasagar's well known theorem on the local stabilizability problem of nonlinear systems using state detection [11]. Our purpose is to prove that if a system is weakly detectable and stabilizable by means of a continuous state feedback u = γ(x), for which no differentiability assumption is imposed, then the system is also stabilized by the law u = γ(z), where z is the output of a weak detector for the state x. The result above is applicable to several cases not covered by other works.     

Simultaneous stabilization of a collection of single‐input nonlinear systems based on CLFs

This paper considers the simultaneous stabilization problem of a collection of single‐input nonlinear systems. Based on the technique of control Lyapunov functions (CLFs), a sufficient condition for the existence of a simultaneously stabilizing state feedback controller is proposed. It is shown that a collection of feedback linearizable systems in canonical form can be simultaneously globally asymptotically stabilized by a single state feedback controller. Moveover, for a set of three‐order chaotic dynamical systems, the simultaneous stabilization problem is considered and a similar result is derived. All the proposed simultaneously stabilizing state feedback controllers are explicitly constructed. Numerical examples are provided to illustrate the effectiveness of the proposed schemes. Copyright © 2011 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Positively invariant sets for continuous-time systems with the cone-preserving property

The main objective of this paper is to determine positively invariant and asymptotically stable polyhedral sets for a linear continuous-time system [xdot](t) = Ax(t) for which matrix e ^1A is a cone-preserving matrix, that is, e ^1A K ? K, for some proper cone K. Necessary and sufficient conditions guaranteeing that some bounded sets are positively invariant and contractive are given. These sets are obtained by means of the intersection of shifted cones. First, some results presented under a geometrical form and also in algebraic form allow characterization of systems having the cone-preserving property. Finally, as an application, the proposed results are used to determine a stability domain for a state feedback regulator with constraints on either or both states and controls.     

On the stabilization of controllable and observable systems by an output feedback law

For control systems which can be locally stabilized in small time by means of a dynamic periodic time-varyingstate feedback law, we give a sufficient condition on Lie derivatives of the output for local stabilization in small time by means of a dynamic periodic time-varyingoutput feedback law. If the system is analytic our sufficient condition is also necessary.     

A scaled feedback stabilization of power integrator triangular systems

Using unbounded time-varying scaling of the states we design C¹ feedback laws for power integrator triangular systems which globally asymptotically stabilize (GAS) the origin despite the uncontrollability of the linearization. With bounded scaling the feedback laws achieve global practical stability (GPS). For a trade-off between GAS/GPS of the origin and unboundedness/boundedness of the scaling we construct a dynamic version of these feedback laws.     

Global robust stabilization of nonlinear systems subject to input constraints

Our main purpose in this paper is to further address the global stabilization problem for affine systems by means of bounded feedback control functions, taking into account a large class of control value sets: p, r ‐weighted balls ??^m _r (p), with 1<p?∞, defined via p, r ‐weighted gauge functions. Observe that p=∞ is allowed, so that m‐dimensional r ‐hyperboxes ??^m _r (∞)?[?r₁^?,r₁⁺]×???×[?r_m^?,r_m⁺], r_j^±>0 are also considered. Working along the line of Artstein–Sontag's approach, we construct an explicit formula for a one‐parameterized family of continuous feedback controls taking values in ?? _r ^m(p) that globally asymptotically stabilize an affine system, provided an appropriate control Lyapunov function is known. The designed family of controls is suboptimal with respect to the robust stability margin for uncertain systems. The problem of achieving disturbance attenuation for persistent disturbances is also considered. Copyright © 2002 John Wiley & Sons, Ltd.     

Constructive nonsmooth stabilization of triangular systems

The problem of local asymptotic continuous feedback stabilization of single-input nonlinear systems is considered here. The aim is to explicitly construct a continuous asymptotically stabilizing feedback for a class of nonlinear systems that is known to be continuous feedback asymptotically stabilizable, but for which the known results do not provide an effective method to compute the stabilizer. Computer simulations are included to show practical applicability of our approach.     

H∞ control for a class of cascade switched nonlinear systems

In this paper, we investigate the H_∞ control problem for a class of cascade switched nonlinear systems consisting of two nonlinear parts which are also switched systems using the multiple Lyapunov function method. Firstly, we design the state feedback controller and the switching law, which guarantees that the corresponding closed‐loop system is globally asymptotically stable and has a prescribed H_∞ performance level. This method is suitable for a case where none of the switched subsystems is asymptotically stable. Then, as an application, we study the hybrid H_∞ control problem for a class of nonlinear cascade systems. Finally, an example is given to illustrate the feasibility of our results. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society