A biased approach to nonlinear robust stability and performance with applications to adaptive control

The nonlinear robust stability theory of Georgiou and Smith [IEEE Trans. Automat. Control, 42 (1997), pp. 1200–1229] is generalized to the case of notions of stability with bias terms. An example from adaptive control illustrates nontrivial robust stability certificates for systems which the previous unbiased theory could not establish a nonzero robust stability margin. This treatment also shows that the bounded-input bounded-output robust stability results for adaptive controllers in French [IEEE Trans. Automat. Control, 53 (2008), pp. 461–478] can be refined to show preservation of biased forms of stability under gap perturbations. In the nonlinear setting, it also is shown that in contrast to linear time invariant systems, the problem of optimizing nominal performance is not equivalent to maximizing the robust stability margin

Operator inclusions and operator-differential inclusions

In Chapter 2, we first introduce a generalized inverse differentiability for set-valued mappings and consider some of its properties. Then, we use this differentiability, Ekeland's Variational Principle and some fixed point theorems to consider constrained implicit function and open mapping theorems and surjectivity problems of set-valued mappings. The mapping considered is of the form F(x, u) + G (x, u). The inverse derivative condition is only imposed on the mapping x F(x, u), and the mapping x G(x, u) is supposed to be Lipschitz. The constraint made to the variable x is a closed convex cone if x F(x, u) is only a closed mapping, and in case x F(x, u) is also Lipschitz, the constraint needs only to be a closed subset. We obtain some constrained implicit function theorems and open mapping theorems. Pseudo-Lipschitz property and surjectivity of the implicit functions are also obtained. As applications of the obtained results, we also consider both local constrained controllability of nonlinear systems and constrained global controllability of semilinear systems. The constraint made to the control is a time-dependent closed convex cone with possibly empty interior. Our results show that the controllability will be realized if some suitable associated linear systems are constrained controllable. In Chapter 3, without defining topological degree for set-valued mappings of monotone type, we consider the solvability of the operator inclusion y0 N1(x) + N2 (x) on bounded subsets in Banach spaces with N1 a demicontinuous set-valued mapping which is either of class (S+) or pseudo-monotone or quasi-monotone, and N2 is a set-valued quasi-monotone mapping. Conclusions similar to the invariance under admissible homotopy of topological degree are obtained. Some concrete existence results and applications to some boundary value problems, integral inclusions and controllability of a nonlinear system are also given. In Chapter 4, we will suppose u A (t,u) is a set-valued pseudo-monotone mapping and consider the evolution inclusions x' (t) + A(t,x((t)) f (t) a.e. and (d)/(dt) (Bx(t)) + A (t,x((t)) f(t) a.e. in an evolution triple (V,H,V*), as well as perturbation problems of those two inclusions

Generalized Gap Metrics and Robust Stability of Nonlinear Systems

A gap metric of Georgiou and Smith (IEEE Trans. Auto. Control, 42(9):1200--1229, 1997), which does not need causal and surjective mapping between graphs to define, is studied and generalized based on the notion of biased norm, the corresponding robust stability theorem is presented in the notion of stability with bias terms. The obtained results are then applied to studied the stability of linear system realizations, semilinear systems with bounded nonlinearities and a nonlinear system with time delay in the inputs

Complexity reduction of Nonlinear Systems

A common problem in nonlinear control is the need to consider systems of high complexity. Here we consider systems, which although may be low order, have high complexity due to a complex right hand side of a differential equation (e.g. a right hand side which has many terms – such systems arise from coordinate transformations in constructive nonlinear control designs). This contribution develops a systematic method for the reduction of this complexity, complete with error bounds. In the case when the underling nonlinear system input/output operator is stable and differentiable, the operator Taylor expansion, truncated after a finite number of terms, is taken as the approximation. If the nonlinear system i/o operator is not stable, but admits a coprime factorizations, the Taylor approximation is made to both coprime factors. By bounding the gap between the polynomial system and the original nominal plant, and applying gap robust stability approaches, it is proved that local stability of approximation implies the local stability of the underlining nonlinear systems, and explicit robust stability margins and performance bounds obtained. For systems specified by a finite dimensional first order differential equation, the first order approximant is the system linearisation and the higher order approximants have greater state dimension but with polynomial right hand sides

Graph Topologies, Gap Metrics and Robust Stability for Nonlinear Systems

Graph topologies for nonlinear operators which admit coprime factorisations are defined w.r.t. a gain function notion of stability in a general normed signal space setting. Several metrics are also defined and their relationship to the graph topologies are examined. In particular relationships between nonlinear generalisations of the gap and graph metrics, Georgiou-type formulae and the graph topologies are established. Closed loop robustness results are given w.r.t. the graph topology, where the role of a coercivity condition on the nominal plant is emphasised

Robust Stability of Interconnections

We begin by observing that the graph topology with its various metrizations plays a fundamental role in the theory of robust stability for classical LTI systems([1, 2, 6]. The contribution of this note is to develop the basic theory of robust stability involving the gap-distance directly from a behavioural perspective, observing that recent approache

An intrinsic behavioural approach to the gap metric

An intrinsic trajectory level approach without any recourse to an algebraic structure of a representation is utilized to develop a behavioural approach to robust stability. In particular it is shown how the controllable behaviour can be constructed at the trajectory level via Zorn's Lemma, and this is utilized to study the controllable-autonomous decomposition. Stability concepts are defined and the relation between this framework and the well known difficulties of classical input-output approaches to systems over the doubly infinite time-axis are discussed. The gap distance is generalised to the behavioural setting via a trajectory level definition; and a basic robust stability theorem is established for linear shift invariant behaviours. The robust stability theorem is shown to provide an explicit robustness interpretation to the behavioural H∞ synthesis of Willems and Trentelmann

Solutions of implicit evolution inclusions with pseudo-monotone mappings

Existence results are given for the implicit evolution inclusions $(Bx(t))'+A(t, x(t))\ni f(t)$ and $(Bx(t))'+A(t, x(t))-G(t, x(t))\ni f(t)$ with $B$ a bounded linear operator, $A(t,\cdot)$ a bounded, coercive and pseudo-monotone set-valued mapping and $G$ a set-valued mapping of non-monotone type. Continuity of the solution set of first inclusion with respect to $f$ is also obtained which is used to solve the second inclusion

An intrinsic behavioural approach to the double time axis paradox

Behavioural theory is typically developed on the double time axis. On the other hand it is known that there are intrinsic difficulties with double time axis theorems in the input output context. This paper illustrates how the behavioural approach avoids these intrinsic difficulties