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.     

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     

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     

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.     

Semiglobal stabilization of multi-input linear systems with saturated linear state feedback

For multi-input linear systems with eigenvalues in the closed left-half comlex plane, we address the problem of stabilization by a linear state feedback subject to saturation. Using an eigenvalue-generalized eigenvector assignment technique, we prove that such systems can be semiglobally stabilized with a saturated linear state feedback. Based on this result, we propose an algorithm to calculate an -parametrized family of state feedback gain matrices that semiglobally stabilize the system. Two examples are used to illustrate the results.     

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L2增益性能,在输出为状态量的饱和函数量有界的形式下,得出了可稳线性系统为可静态反馈半全局镇定的结论,并且对两类性能函数分别得出静态下系统满足某L2增益性能指标的充分条件。     

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     

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).     

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.     

Application of facial reduction to H ∞ state feedback control problem

One often encounters numerical difficulties in solving linear matrix inequality (LMI) problems obtained from H _∞ control problems. For semidefinite programming (SDP) relaxations for combinatorial problems, it is known that when either an SDP relaxation problem or its dual is not strongly feasible, one may encounter such numerical difficulties. We discuss necessary and sufficient conditions to be not strongly feasible for an LMI problem obtained from H _∞ state feedback control problems and its dual. Moreover, we interpret the conditions in terms of control theory. In this analysis, facial reduction, which was proposed by Borwein and Wolkowicz, plays an important role. We show that the dual of the LMI problem is not strongly feasible if and only if there exist invariant zeros in the closed left-half plane in the system, and present a remedy to remove the numerical difficulty with the null vectors associated with invariant zeros in the closed left-half plane. Numerical results show that the numerical stability is improved by applying it.     

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     

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.     

Stabilizability of multivariable systems and the Ljusternik— nirel'mann category of real Grassmannians

In this paper a numeric criterion, k(m, p) n, is given for stabilizability by constant gain output feedback of the generic linear multivariable system with m inputs, n states, and p outputs. This criterion is defined in terms of a topological invariant of the space of gains, both finite and infinite, arising from an interpretation of the problem of stabilizability as a problem concerning open covers of this space. This invariant, originally considered by Ljusternik and nirel'mann in the calculus of variations, can be estimated from below by using a theorem of Eilenberg together with the methods of the Schubert calculus, thus leading to some very explicit corollaries concerning generic stabilizability. As far as I am aware, these corollaries cannot be derived from existing results concerning pole-assignability. After passing to a problem on coverings of the (compactified) gain space, the main technical problem which remains in a high gain stability analysis — stated here as a High Gain Lemma — which appears to be of independent interest. That is, the topological argument implies that the closed-loop system is ‘stable’ for some gain, possibly infinite. If the root-locus map were defined and continuous at all infinite gains, then the conclusion that the system is stabilizable by finite gain could be deduced from a simple continuity argument. However, it is known that if mp > n then the root-locus map has points of discontinuity at infinity, and, since mp n is now known to be necessary for generic stabilizability, in most cases of interest one requires a far more subtle argument than one would give, for example, in the scalar case.     

Decentralized control of dynamically interconnected systems

Interconnected systems, where the subsystems are interconnected by some dynamic interaction system, are considered. It is shown that this type of system can be stabilized by decentralized dynamic output feedback, if the subsystems are stabilizable by (centralized) dynamic output feedback and the interaction system is stable. The relation to previous results is discussed.     

Minimum norm output feedback design under specified eigenvalue areas

Starting from a new parametric expression for the state feedback controller of a linear, time-invariant system a procedure is derived which reduces the design of output feedback controllers to an unconstrained optimization problem where all free parameters, both the closed-loop eigenvalues and the invariant parameter vectors, are used for its numerical solution by a gradient-based search technique. Thus minimal norm output feedback controllers can be designed that place the closed-loop eigenvalues within specified regions of the eigenvalue plane.     

Proper,denominator assigning feedback compensators equivalent to state variable feedback

For linear, time-invariant, stabilizable multivariable systems, we examine the problem of the existence and computation of proper denominator assigning and internally stabilizing feedback compensators, which give rise to a closed-loop system, whose transfer function matrix is equal to one that can be obtained by the action of state variable feedback. We establish a sufficient condition for the solution of this problem for the class of systems with the same number of inputs and outputs and non-singular transfer function matrix with all its zeros located in the open left half of the complex plane.     

Linear systems with input nonlinearities: Global stabilization by scheduling a family of H∞-type controllers

A new global stabilization algorithm is presented for linear systems that have (A, B) stabilizable, the eigenvalues of A in the closed left-half plane, but where the controls pass through nonlinearities of a general type. This algorithm is based on a semi-global stabilization solution for the same class of systems. The global solution is constructed by dynamically scheduling the adjustable parameter of the semi-global solution according to the size of the state. The semi-global solution is a family of linear control laws generated by a family of H_∞-type algebraic Riccati equations. We show that the closed-loop state feedback system has the property that additive disturbances which converge exponentially to zero cannot produce unbounded states. This fact is used to show that, under a linear detectability assumption, the state feedback results are recovered when the solution is implemented via a standard linear-based observer.