Semi-global stabilization of minimum phase nonlinear systems in special normal forms

We semi-globally stabilize certain minimum phase nonlinear systems which are in a normal form where the nonlinear subsystem is driven by an output of a linear system that possesses (possibly) nonzero peaking exponents. We eliminate the peaking phenomenon by stabilizing part of the linear system with a high-gain linear control and part of the linear system with a small, bounded control. The interpretation of this approach will be that we are redefining the outputs to add asymptotically stable nonlinear zeros to the system in a manner that allows the new composite zero dynamics to be asymptotically stable on arbitrarily large compact sets.     

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).     

Optimal simultaneous stabilization of linear single-input systems via linear state feedback control

Simultaneous stabilization of a collection of linear time-invariant systems is considered. The aim is to develop a practical method for the design of a single feedback controller such that all the systems involved are stabilized with good transient responses. Hurwitz's necessary and sufficient conditions are used as the required set of constraints for simultaneous stability. For transient behaviour, we introduce the usual quadratic objective function as performance index for each system. Their sum is the objective function for the collection of systems and the problem is to minimize it by choosing a feedback gain vector subject to boundedness and the Hurwitz stability constraints. A computational technique is proposed for solving this problem. The results obtained give good transient behaviour. For illustration, two numerical examples are solved.     

Semi-global stabilization and output regulation of constrained linear plants via measurement feedback

Semi-global stabilization and output regulation of linear systems subject to state and/or input constraints have been studied in our earlier work by using state feedback. For the same problems, observer based measurement feedback control designs are the topics of this paper. High-gain observers are used in the feedback design in order to obtain accurate estimates of the state so that the constraint violation can be avoided. Due to the peaking phenomenon associated with a high-gain observer, a special saturation protection is built in the control laws to avoid possible constraint violation. The results in this paper show that the semi-global stabilization and semi-global output regulation problems for constrained linear systems are solvable via measurement feedback under solvability conditions similar to those in the state feedback.     

Asymptotic stabilization of minimum phase nonlinear systems

How a class of multivariable nonlinear systems can be stabilized about an equilibrium via smooth state feedback is shown. More precisely, conditions under which, for every compact set of initial states, it is possible to design a feedback law which drives to the equilibrium all initial states in this compact set are described. The theory includes the development of globally defined transformations of the system equations to their global normal form     

Simultaneous stabilization via linear state feedback control

The problem of simultaneously stabilizing a finite collection of plants via a single compensator utilizing full state information is considered. The authors assume that the complete state can be measured and seek a (static) linear state feedback control law. Sufficient conditions for such full state gains to exist are given and they show via several examples how stabilizing compensators can be designed     

Non-regular feedback linearization of nonlinear systems via a normal form algorithm

In this paper, the problem of non-regular static state feedback linearization of affine nonlinear systems is considered. First of all, a new canonical form for non-regular feedback linear systems is proposed. Using this form, a recursive algorithm is presented, which yields a condition for single input linearization. Then the left semi-tensor product of matrices is introduced and several new properties are developed. Using the recursive framework and new matrix product, a formula is presented for normal form algorithm. Based on it, a set of conditions for single-input (approximate) linearizability is presented.     

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).     

Asymptotic regulation of minimum phase nonlinear systems usingoutput feedback

We consider a single-input/single-output (SISO) nonlinear system which has a well-defined normal form with asymptotically stable zero dynamics. We allow the system's equation to depend on constant uncertain parameters and disturbance inputs which do not change the relative degree. Our goal is to design an output feedback controller which regulates the output to a constant reference. The integral of the regulation error is augmented to the system equation, and a robust output feedback controller is designed to bring the state of the closed-loop system to a positively invariant set. Once inside this set, the trajectories approach a unique equilibrium point at which the regulation error is zero. We give regional as well as semiglobal results     

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.     

Simultaneous stabilization of linear single-input systems by linear state feedback control

A necessary and sufficient condition for the existence of a linear state feedback controller which will simultaneously stabilize a finite collection of single-input linear systems is presented. This condition can be checked numerically through the solution of a non-linear programming problem presented here and can be used to determine whether or not a particular simultaneous stabilization problem has a solution. If the stabilization problem has a solution, the method provides a means of constructing a suitable controller. The proposed method is very versatile. It is a simple matter to limit the size of elements of the feedback gain vector and thus ensure that the control action is not excessive. In addition, for simultaneous stabilization problems which have a range of possible solutions, the proposed method makes it possible to select one which yields good performance.     

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.     

Exponential stabilization of linear systems with time-varying delayed state feedback via partial spectrum assignment

We consider the problem of controlling a linear system when the state is available with a known time-varying delay (delayed-state feedback control) or the actuator is affected by a delay. The solution proposed in this paper consists in partially assigning the spectrum of the closed-loop system to guarantee the exponential zero-state stability with a prescribed decay rate by means of a finite-dimensional control law. A non conservative bound on the maximum allowed delay for the prescribed decay rate is presented, which holds for both cases of constant and time-varying delays. An advantage over recent and similar approaches is that differentiability or continuity of the delay function is not required. We compare the performance of our approach, in terms of delay bound and input signal, with another recent approach.     

Semi-global output regulation for linear systems with input saturation by composite nonlinear feedback control

To improve transient performance of output response, this paper applies composite nonlinear feedback (CNF) control technique to investigate semi-global output regulation problems for linear systems with input saturation. Based on a linear state feedback control law for a semi-global output regulation problem, a state feedback CNF control law is constructed by adding a nonlinear feedback part. The extra nonlinear feedback part can be applied to improve the transient performance of the closed-loop system. Moreover, an observer is designed to construct an output feedback CNF control law that also solves the semi-global output regulation problem. The sufficient solvability condition of the semi-global output regulation problem by CNF control is the same as that by linear control, but the CNF control technique can improve the transient performance. The effectiveness of the proposed method is illustrated by a disturbance rejection problem of a translational oscillator with rotational actuator system.     

Global stabilization of high-order nonlinear time-delay systems by state feedback

We consider the problem of global stabilization by state feedback for a class of high-order nonlinear systems with time-delay. By developing a novel dynamic gain-based backstepping approach, a state feedback controller independent of the time-delay is explicitly constructed with the help of appropriate Lyapunov–Krasovskii functionals. The precise knowledge (even the upper bound) of the time-delay is not required. It is proved that the states of the nonlinear time-delay systems can be regulated to the origin while all the closed loop signals are globally bounded. Finally, both physical and academic examples are given to illustrate the applications of the proposed scheme.     

Global stabilization of a class of delay discrete-time nonlinear systems via state and output feedback

This paper deals with stabilization of a class of delay discrete-time nonlinear systems through state and output feedback. We provide an explicit bounded state feedback law as an extension of the Jurdjevic-Quinn method, from nonlinear theory, to this class of systems. Next, we present a useful and systematic approach to design an observer for the same class of systems. Then, we show how the global stabilization problem via dynamic output feedback can be solved by using the two previous results. Finally, numerical examples are given to illustrate the effectiveness of the proposed design method.     

Markov跳变系统的有限时间状态反馈镇定

讨论一类含有限能量未知扰动的线性Markov跳变系统的有限时间镇定问题.针对连续系统和离散系统两种情况,利用构造的Lyapunov-Krasovskii函数,并结合线性矩阵不等式方法,分别证明并给出了跳变系统有限时间镇定控制器有解的充分条件.采用该方法设计的镇定控制器可使连续系统和离散系统对所有满足条件的未知扰动是有限时问有界和有限时间镇定的.最后通过数值示例表明了该设计方法的有效性.     

A theorem on global stabilization of nonlinear systems by linear feedback

In this paper we investigate the global stabilizability problem for a wide class of single-input nonlinear systems whose the linearization at the equilirrium is controllable. We show that under general assumptions there exists a linear feedback law which globally exponentially stabilizes the system at its equilibrium. The proof of our main theorem is based on some ideas from a previous paper. We use the theorem to recover a recent result of Gauthier et al. concerning the observer design problem.     

Further results on state feedback stabilization of stochastic high-order nonlinear systems

In this paper,a combined homogeneous domination and sign function design approach is presented to state feedback control for a class of stochastic high-order nonlinear systems with time-varying delay.The use of the combined approach relaxes the restriction on nonlinear functions and makes the closed-loop system globally asymptotically stable in probability.