Universal stabilization of a class of nonlinear systems with homogeneous vector fields

Under relative-degree-one and minimum-phase assumptions, it is well known that the class of finite-dimensional, linear, single-input (u), single-output (y) systems (A,b,c) is universally stabilized by the feedback strategy u = Λ(λ)y, λ = y², where Λ is a function of Nussbaum type (the terminology “universal stabilization” being used in the sense of rendering /s(0/s) a global attractor for each member of the underlying class whilst assuring boundedness of the function λ(·)). A natural generalization of this result to a class ^k of nonlinear control systems (a,b,c), with positively homogeneous (of degree k 1) drift vector field a, is described. Specifically, under the relative-degree-one (cb ≠ 0) and minimum-phase hypotheses (the latter being interpreted as that of asymptotic stability of the equilibrium of the “zero dynamics”), it is shown that the strategy u = Λ(λ)/vby/vb^k−1y, assures ^k-universal stabilization. More generally, the strategy u = Λ(λ)exp(/vby/vb)y, assures -universal stabilization, where = _{k 1} ^k.     

Remarks on nonlinear planar control systems which are linearizable by feedback

The possibility that certain nonlinear systems can be linearized by the nonlinear feedback group has recently attracted a great deal of attention in the literature. At least for mechanical systems, the apparent ubiquity of linearizable systems is often accounted for by the claim that scalar control systems defined on R² are generically linearizable. Since there are several possible interpretations of the term ‘generic’ and since, until now, no rigorous proof of genericity for any use of the term has been offered, I thought it worthwhile to investigate in what sense, if any, this ‘folklore theorem’ is valid. Contrary to the prevailing intuition, linearizable systems fail to be dense in any topology for which evaluation of vector fields at points is a continous operation. This includes, for example, both the weak and strong C^∞-topologies.What, then, are the global properties of the class of linearizable systems? In this note, it is also shown that this class has no interior points in any reasonable, complete topology making feedback operations continuous. In particular, linearizable systems in R² are far from being structurally stable in the weak C^∞-topology. In the other hand, one of the main results presented here is that globally linearizable systems are open in the strong, or Whitney, topology. Thus, globally linearizable planar systems are structurally stable. If the well-known local conditions actually implied global linearizability, this result would be trivial. Recent work of Boothby shows, however, that global linearization is far more subtle. For this reason, in the course of the proof it is necessary to modify the existing global linearization criteria. In particular, a ‘nonvanishing’ criterion for irjectivity of smooth maps on R″, sharpening the so-called ‘ratio condition’, is presented. This result appears to be of independent interest and leads to improvements of the global linearization condition in R″, due to Hunt, Su and Meyer, and of the global non-linear observability conditions in R″, due to Kou, Elliott and Tarn.     

Feedback stabilization of a class of distributed parameter systems with control constraints

The feedback stabilization problem for a class of infinite-dimensional linear systems with control constraints is investigated. The approach is developed using a state space system framework which is based on a semigroup formulation. In contrast to the previous works in this direction, it is not assumed that the C₀ semigroup we deal with is a semigroup of contractions. The main result links the problem with the positive invariance of appropriate polyhedral sets. Verifiable necessary and sufficient conditions for this latter property are stated.     

-bounded robust control of nonlinear cascade systems

In this paper, we consider the L^∞-bounded robust control problem for a class of nonlinear cascade systems with disturbances. Sufficient conditions are provided under which a hard bound is imposed on the system performance measure. The backstepping approach is used for controller design. A practical example is provided to illustrate the method.     

Stabilization and H control of symmetric systems: an explicit solution

In this paper we address the H^∞ control analysis, the output feedback stabilization, and the output feedback H^∞ control synthesis problems for state-space symmetric systems. Using a particular solution of the Bounded Real Lemma for an open-loop symmetric system we obtain an explicit expression to compute the H^∞ norm of the system. For the output feedback stabilization problem we obtain an explicit parametrization of all asymptotically stabilizing control gains of state-space symmetric systems. For the H^∞ control synthesis problem we derive an explicit expression for the optimally achievable closed-loop H^∞ norm and the optimal control gains. Extension to robust and positive real control of such systems are also examined. These results are obtained from the linear matrix inequality formulations of the stabilization and the H^∞ control synthesis problems using simple matrix algebraic tools.     

Flow regularity and optimality conditions with controls inL p

In this paper we study theC ^p regularity of the flow of a nonlinear nonautonomous control system with respect to control maps belonging toL ^p withp≥r. The results obtained are applied to get first- and second-order optimality conditions when the control space isL ^p . The problem which we consider is in the Mayer form and includes endpoint constraints. We present first-order necessary conditions for a wide class of control systems. Moreover, we show that the usual second-order sufficient conditions are effective only if the mapf that defines the control system is a polynomial of degree two in the control variable and the controls belong toL ².     

Normal forms for multi‐input flat systems of minimal differential weight

We present normal forms for nonlinear control systems that are the closest to static feedback linearizable ones, that is, for systems that become feedback linearizable via the simplest dynamic feedback, which is the one‐fold prolongation of a suitably chosen control. They form a particular class of flat systems, namely those of differential weight n + m + 1, where n is the number of states and m is the number of controls. We also show that the dynamic feedback may create singularities in the control space depending on the state and we discuss them. We also address the issue of the normalization of the system only versus that of the system together with a flat output. Finally, we illustrate our results by several examples.     

Coprime factorization over a class of nonlinear systems

In this paper, we consider the problem of generalizing elements of linear coprime factorization theory to a nonlinear context. The idea is to work with a suitably wide class of nonlinear systems to cover many practical situations, yet not cope with so broad a class as to disallow useful generalizations to the linear results. In particular, we work with nonlinear systems characterized in terms of (possibly time-varying) state-dependent matrices A(x), B(x), C(x), D(x) and an initial state x₀. (This class clearly does contain the class of finite-dimensional linear (time-varying) systems.) We achieve first right coprime factorizations for idealized situations. To achieve stable left factorizations we specialize to the case where the matrices are output-dependent. Alternatively, we work with systems, perhaps augmented by a direct feedthrough term, where the input is reconstructible from the output. For nonlinear feedback control systems, with plant and controller having stable left factorizations, then under appropriate regularity-conditions earlier results have allowed the generation of the class of stabilizing controllers for a system in terms of an arbitrary stable system (parameter). Plant uncertainties, including unknown initial conditions are modelled by means of a Yula–Kucera-type parametrization approach developed for nonlinear systems. Certain robust stabilization results are also shown, and simulations demonstrate the regulation of nonlinear plants using the techniques developed. All the results are presented in such a way that specialization for the case of linear systems is immediate.     

On computing the worst-case peak gain of linear systems

Based on the bounds due to Doyle and Boyd, we present simple upper and lower bounds for the l¹-norm of the ‘tail’ of the impulse response of finite-dimensional discrete-time linear time-invariant systems. Using these bounds, we may in turn compute the l^∞-gain of these systems to any desired accuracy. By combining these bounds with results due to Khammash and Pearson, we derive upper and lower bounds for the worst-case l^∞-gain of discrete-time systems with diagonal perturbations.     

On the weighted sensitivity minimization problem for delay systems

We consider the H^∞-optimal sensitivity problem for delay systems. In particular, we consider computation of μ:= inf {|W-φq|_∞ : q ε H^∞(j )} where W(s) is any function in RH^∞(j ), and φ in H^∞(j ) is any inner function. We derive a new explicit solution in the pure delay case where φ = e^−sh, h > 0.     

Global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws

This paper deals with the global exact boundary controllability of a class of quasilinear hyperbolic systems of conservation laws with linearly degenerate characteristics. It is shown that the system with nonlinear boundary conditions is globally exactly boundary controllable in the class of piecewise C¹ functions. Some physical and mechanical applications are given.     

General quadratic performance analysis and synthesis of differential algebraic equation (DAE) systems

In this paper, control of linear differential-algebraic-equation systems, subject to general quadratic constraints, is considered. This setup, especially, includes the H_∞ control problem and the design for strict passivity. Based on linear matrix inequality (LMI) analysis conditions, LMI synthesis conditions for the existence of linear output feedback controllers are derived by means of a linearizing change of variables. This approach is constructive: a procedure for the determination of controller parameterizations is given on the basis of the solution of the LMI synthesis conditions. A discussion of the possible applications of the presented results concludes the paper.     

Non-smooth simultaneous stabilization of nonlinear systems: An interpolation method

In this paper, we introduce a method which enables us to construct a continuous simultaneous stabilizer for pairs of systems in which cannot be simultaneously stabilized by means of C¹ feedback. We extend this method to higher-dimensional systems and show that any pair of asymptotically stabilizable nonlinear systems can be simultaneously stabilized (not asymptotically) by means of continuous feedback.     

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.     

Index set: A practical indexing scheme for object database systems

Efficient indexing in a class hierarchy is essential for the achievement of high performance in query evaluation for object database management systems. In this paper, we present a practical indexing scheme, index set, which provides good index configuration for any real database environment. The proposed scheme considers the distribution of key values, as well as query patterns such as query weight on each class. The index set can easily be applied to any database system, since it uses the well-known B⁺-tree structure. We develop a cost model and, through experiments, demonstrate the performance of the proposed scheme over various class hierarchies.     

C3连续的单位四元数插值样条曲线

目的 构造一类C³连续的单位四元数插值样条曲线,证明它的插值性和连续性,并把它应用于刚体关键帧动画设计中。方法 利用R³空间中插值样条曲线的5次多项式调配函数的累和形式构造了S³空间中单位四元数插值样条曲线,它不仅能精确通过一系列给定的方向,而且能生成C³连续的朝向曲线。结果 与Nielson的单位四元数均匀B样条插值曲线的迭代构造方法相比,所提方法避免了为获取四元数B样条曲线控制顶点对非线性方程组迭代求解的过程,提高了运算效率;与单位四元数代数三角混合插值样条曲线的构造方法（Su方法）相比,所提方法只用到多项式基,运算速度更快。本例中创建关键帧动画所需的时间与Nielson方法和Su方法相比平均下降了73%和33%。而且,相比前两种方法,所提方法产生的四元数曲线连续性更高,由C²连续提高到C³连续,这意味着动画中刚体的朝向变化更加自然。结论 仿真结果表明,本文方法对刚体关键帧动画设计是有效的,对实时性和流畅性要求高的动画设计场合尤为适用。     

On the state space and frequency domain characterization of H-norm of sampled-data systems

This paper studies the problem of characterization and computation of the H^∞-norm of sampled-data systems using the time-invariant function space model via lifting. With the advantage of time-invariance, the treatment gives an eigenvalue-type characterization, first in the operator form in the frequency domain and then in the Hamiltonian-type finite-dimensional form. The form obtained can be adopted for use with the bisection algorithm for actual computation.     

The l-control problem for time-varying discrete systems

This paper deals with the l²-problem for time-varying discrete systems. The problem considered is a deterministic counterpart of the well-known LQG problem. An adequate l²-seminorm for linear bounded input-output operators is introduced in order to express the optimality criterion. The solution is obtained by successively reducing the original problem to simpler ones putting in evidence the Separation Principle of the optimal controller.     

Stabilization of a class of generalized bilinear systems

This paper studies local stabilization of a class of analytic nonlinear systems in terms of, which includes ordinary bilinear systems as its subset, ż=f(z)+g(z)u, f(0)=0, g(0)=0, z∈R² which can be achieved via a feedback control law u=u(z) with u(0)=0. Following the theoretical results a potential application, stabilization of non-minimum phase systems, is investigated. Copyright © 1999 John Wiley & Sons, Ltd.