Zongli Lin

Semi-global exponential stabilization of linear systems subject to “input saturation” via linear feedbacks

Abstract It is known that a linear time-invariant system subject to “input saturation” can be globally asymptotically stabilized if it has no eigenvalues with positive real parts. It is also shown by Fuller (1997) and Sussmann and Yang (1991) that in general one must use nonlinear control laws and only some special cases can be handled by linear control laws. In this paper we show the existence of linear state feedback and/or output feedback control laws for semi-global exponential stabilization rather than global asymptotic stabilization of such systems. We explicitly construct linear static state feedback laws and/or linear dynamic output feedback laws that semi-globally exponentially stabilize the given system. Our results complement the “negative result” of Fuller (1977) and Sussmann and Yang (1991).

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

Abstract 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).

A low-and-high gain approach to semi-global stabilization and/or semi-global practical stabilization of a class of linear systems subject to input saturation via linear state and output feedback

In this paper we propose a linear low-and-high gain design technique to solve the semi-global stabilization problem for a class of linear systems subject to input saturation. The central idea behind this new technique is to increase the utilization of the available control capability of the system. The power of this new semi-global design technique is shown by solving the problem of semi-global stabilization with input-additive disturbance rejection which is formulated as semi-global practical stabilization problem.<<ETX>>

Adaptive high-gain stabilization of 'minimum-phase' nonlinear systems

The authors present an adaptive high-gain controller which can stabilize minimum-phase nonlinear systems with strong relative degree. The basic structure of the control method is similar to adaptive control methods in the sense that there are two feedback loops. The inner loop is a high-gain feedback loop with a plant and a controller. The controller is completely specified except for a scalar parameter, the high-gain parameter. The controller is designed such that the stabilization of the closed loop will be achieved if the high-gain parameter is chosen large enough. This feature facilitates a simple procedure of tuning the high-gain parameter by increasing it monotonically until it is large enough. The tuning of the parameter is set up by the outer loop. The high-gain parameter is increased in a stepwise fashion so as to follow a continuous-time parameter which is determined by an adaptation rule. The stepwise incrementation plays a crucial role in simplifying the analysis, since between any two switching points the controller is a simple linear time-invariant one.<<ETX>>

Linear systems toolbox: system analysis and control design in the matlab environment

The software package linear system toolbox in Matlab is described. The Linear Systems Toolbox is aimed at providing some commonly used system analysis functions which are either not available or not reliable in other software packages. This tool box contains about 50 functions. It includes, among other functions, the computation of geometric subspaces and the invariant zeros, the inner-outer factorization of a linear system, the determination of the infinite zero structures, Morses and Saberis indices and strong stability, and the solutions to the decoupling and squaring down problems. Some examples are included to demonstrate the use of the tool box.<<ETX>>

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

This paper extends the results of the authors (1992) on semi-global stabilization of a class of minimum-phase nonlinear systems via linear state feedback. The class of minimum-phase nonlinear systems considered in this paper subsume those of the previous paper. More specifically, the authors show, by explicit construction of the control laws, that a cascade of linear stabilizable and nonlinear asymptotically stable subsystems is semi-globally stabilizable by linear dynamic feedback of the state of the linear subsystem if the linear subsystem is right invertible and has all its invariant zeros in the closed left half s-plane. The authors proposed linear dynamic state feedback control law has a single tunable gain parameter that allows for local asymptotical stability and regulation to the origin for any initial condition in some a priori given (arbitrarily large) bounded set.<<ETX>>

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

The problem of semi-global stabilization of a class of partially linear composite systems is considered. It is shown 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 the linear subsystem is right invertible and has all its invariant zeros in the closed left half s-plane, and the only linear variables entering the nonlinear subsystem are the output of the linear subsystem.<<ETX>>

On the validity of solutions and equilibrium points in a nonlinear network

For constrained differential equations arising in a specific nonlinear network, the inconsistency between two seemingly equivalent unconstrained representations has been long debated in the literature. One of the representations has nonunique solutions which violate the network constraints. Using new analytical tools, this brief proves that the two representations are not equivalent. Also the explanations for the discrepancy which appeared recently are incorrect as such, but can be restated precisely.

A notion of solutions and equilibrium points for non smooth systems

Steady state operation of engineering systems is generally around a stable equilibrium point. The concept of an equilibrium point for a dynamic system is well-established as a constant solution in the time domain. In other words, the system is at steady state at an equilibrium point when all time derivatives are equal to zero. In numerous text books, the standard mathematical definition of an equilibrium point x* for the dynamical system x/spl dot/=f(x) is that it satisfies the equation f(x*)=0. But it has been shown that this definition may be inadequate from an engineering point-of-view. The aim of the paper is to formally propose a rigorous definition for solutions and equilibrium points, which is also intuitively appealing. Here the concept of an equilibrium point is developed mathematically as a constant solution for the dynamic system, by first precisely defining the notion of an order of a solution. The basic motivation for the proposed definition of an order of a solution comes from the continuity concept of solutions in the state space. Order 1 equilibria correspond to the traditional definition of equilibrium points. But such points need not be equilibria in the real system. It is shown that the origin for the classical example x/spl dot/=x2/3 is physically not an equilibrium point, but behaves like a regular point. Other phenomena such as the presence of impasse points and unbounded higher order time derivatives can also be associated with conventional equilibrium points which hence are weak in scope. Several examples are presented in the paper to show the versatility of the new definition. The implications of the definition for special cases such as when the function f is smooth are established and are shown to be consistent with the existing notions. It is hoped that the proposed definitions will allow a rigorous extension of the results for equilibria of smooth systems to non smooth systems, but no such attempt is made in this paper.<<ETX>>

Explicit expressions for cascade factorizations of general non-strictly proper systems

The authors present explicit expressions for two different cascade factorizations of any detectable system which is not necessarily left invertible and which is not necessarily strictly proper. The first is a well-known minimum phase/all-pass factorization by which G(s) is written as G/sub m/(s)V(s), where G/sub m/(s) is left invertible and of minimum phase, while V(s) is a stable right invertible all-pass transfer function matrix which has all unstable invariant zeros of G(s) as its invariant zeros. The second is a generalized cascade factorization by which G(s) is written as G/sub M/(s)U(s), where G/sub M/(s) is left invertible and of minimum-phase with its invariant zeros at desired locations in the open left-half s-plane, while U(s) is a stable right invertible system which has all awkward invariant zeros, including the unstable invariant zeros of G(s), as its invariant zeros, and is asymptotically all-pass.<<ETX>>