Continuous Affine Research Articles

On the Ryll-Nardzewski fixed point property in convex uniform spaces

In this paper, we introduce the concept of a convex uniform space as a natural generalization of locally convex spaces, and a fixed point property as a semigroup formulation of the famous Ryll-Nardzewski fixed point theorem, and study its non-linear counterpart in this framework. The Nardzewski's fixed point theorem is a beautiful result in fixed point theory that ensures the existence of a common fixed point for any noncontracting semigroup of continuous affine self-mappings on a non-void weakly compact convex set in a separated locally convex space. We establish that in a separated convex uniform space satisfying a certain property (P), any noncontracting finite semigroup of contraction mappings of a non-empty bounded closed convex subset possesses a common fixed point. In case where the underlying space is admissible, we prove that this result is extendable to arbitrary noncontracting semigroups without condition (P) if we assume that the phase space is weakly compact and convex.

Stability Analysis of Hybrid Integrator-Gain Systems using Linear Programming

Stability and performance of hybrid integrator-gain systems (HIGS) are mostly analyzed using linear matrix inequalities (LMIs) to construct continuous piecewise quadratic (CPQ) Lyapunov functions. To create additional methods for the analysis of HIGS and other conewise linear systems, a method based on linear programming (LP) for constructing continuous piecewise affine (CPA) Lyapunov functions is investigated. In this paper, it is shown how linear programming can be used to prove input-to-state stability and calculate upper bounds on the L1-gain and H1-norm. The numerical efficiency of this CPA (LP) method will be compared to that of the CPQ (LMI) method on a numerical example involving HIGS.

Fast eye centre localization using combined unsupervised technics

Eye movements offer precious information about persons? state. Video surveillance, marketing, driver fatigue as well as medical diagnosis assistance applications manage eye behavior. We propose a new method for efficiently detecting eye movement. In this paper, we combine circle eye model with eye feature method to improve the accuracy. A set of detectors estimate the eyes centers to increase the localization rate. As a pre-processing stage, the mean of the edges yields the center of the two eye regions. Image treatment operations reduce the ROI. A Circle Hough Transform (CHT) algorithm is adopted in a modified version as a detector to find the circle eye in the image; the circle center found represents the eye's pupil estimation. We introduced the Maximally Stable Extremal Region (MSER) as a second detector, which has never been used for eye localization. Invariant to continuous geometric transformations and affine intensity changes and detected at several scales, MSERs efficiently detect regions of interest, in our case eye regions, and precisely, their centers. Ellipses fit MSERs, and their centroid estimation match eyes center. We demonstrate that the true eye centers can be found by combining these methods. The validation of the proposed method is performed on a very challenging BioID base. The proposed approach compares well with existing state-of-the-art techniques and achieves an accuracy of 82.53% on the BioID database when the normalized error is less than 0.05, without prior knowledge or any learning model.

Unitary groups, -theory, and traces

Abstract We show that continuous group homomorphisms between unitary groups of unital C*-algebras induce maps between spaces of continuous real-valued affine functions on the trace simplices. Under certain $K$ -theoretic regularity conditions, these maps can be seen to commute with the pairing between $K_0$ and traces. If the homomorphism is contractive and sends the unit circle to the unit circle, the map between spaces of continuous real-valued affine functions can further be shown to be unital and positive (up to a minus sign).

Isometric actions on Lp-spaces: dependence on the value of p

Answering a question by Chatterji–Druţu–Haglund, we prove that, for every locally compact group $G$, there exists a critical constant $p_G \in [0,\infty ]$ such that $G$ admits a continuous affine isometric action on an $L_p$ space ($0< p<\infty$) with unbounded orbits if and only if $p \geq p_G$. A similar result holds for the existence of proper continuous affine isometric actions on $L_p$ spaces. Using a representation of cohomology by harmonic cocycles, we also show that such unbounded orbits cannot occur when the linear part comes from a measure-preserving action, or more generally a state-preserving action on a von Neumann algebra and $p>2$. We also prove the stability of this critical constant $p_G$ under $L_p$ measure equivalence, answering a question of Fisher.

Triangulations and Error Estimates for Interpolating Lyapunov Functions

The CPA method to compute Lyapunov functions depends on a triangulation of the relevant part of the state space. In more detail, a CPA (Continuous and Piecewise Affine) function is affine on each simplex of a given triangulation and is determined by the values at the vertices of the triangulation. Two important aspects in the proof that the CPA method is always able to generate a CPA Lyapunov function if the triangulation is sufficiently fine, are (a) the geometry of the simplices of the triangulation and (b) error estimates of CPA interpolations of functions. In this paper the aspect (a) is tackled by extending the notion of (h, d)-boundedness, which so far has depended on the order of the vertices in each simplex, and it is shown that it is essentially independent of the order and can be expressed in terms of the condition number of the shape matrix. Concerning (b), existing error estimates are generalised to other norms to increase the flexibility of the CPA method. In particular, when the CPA method is used to verify Lyapunov function candidates generated by other methods. Parts of the results were presented in Giesl and Hafstein (Uniformly regular triangulations for parameterizing Lyapunov functions. In: Proceedings of the 18th International Conference on Informatics in Control, Automation and Robotics (ICINCO), 549–557, 2021).

Common Lyapunov Functions for Switched Linear Systems: Linear Programming-Based Approach

We study the stability of an equilibrium of arbitrarily switched, autonomous, continuous-time systems through the computation of a common Lyapunov function (CLF). The switching occurs between a finite number of individual subsystems, each of which is assumed to be linear. We present a linear programming (LP) based approach to compute a continuous and piecewise affine (CPA) CLF and compare this approach with different methods in the literature. In particular we compare it with the prevalent use of linear matrix inequalities (LMIs) and semidefinite optimization to parameterize a quadratic common Lyapunov function (QCLF) for the linear subsystems.

Robust Data-Driven Control Barrier Functions for Unknown Continuous Control Affine Systems

In this letter, we introduce robust data-driven control barrier functions (CBF-DDs) to guarantee robust safety of unknown continuous control affine systems despite worst-case realizations of generalization errors from prior data under various continuity assumptions. To achieve this, we leverage non-parametric data-driven approaches for learning guaranteed upper and lower bounds of an unknown function from the data set to formulate/obtain a safe input set for a given state. By incorporating the safe input set into an optimization-based controller, the safety of the system can be ensured. Moreover, we present several complexity reduction approaches including providing subproblems that can be solved in parallel and downsampling strategies to improve computational performance.

A piecewise ellipsoidal reachable set estimation method for continuous bimodal piecewise affine systems

In this work, the issue of estimation of reachable sets in continuous bimodal piecewise affine systems is studied. A new method is proposed, in the framework of ellipsoidal bounding, using piecewise quadratic Lyapunov functions. Although bimodal piecewise affine systems can be seen as a special class of affine hybrid systems, reachability methods developed for affine hybrid systems might be inappropriately complex for bimodal dynamics. This work goes in the direction of exploiting the dynamical structure of the system to propose a simpler approach. More specifically, because of the piecewise nature of the Lyapunov function, we first derive conditions to ensure that a given quadratic function is positive on half spaces. Then, we exploit the property of bimodal piecewise quadratic functions being continuous on a given hyperplane. Finally, linear matrix characterizations of the estimate of the reachable set are derived.

Output-only identification of self-excited systems using discrete-time Lur'e models with application to a gas-turbine combustor

A self-excited system is a nonlinear system with the property that a constant input yields a bounded, nonconvergent response. Nonlinear identification of self-excited systems is considered using a Lur'e model structure, where a linear model is connected in feedback with a nonlinear feedback function. To facilitate identification, the nonlinear feedback function is assumed to be continuous and piecewise affine (CPA). The present paper uses least-squares optimisation to estimate the coefficients of the linear dynamics and the slope vector of the CPA nonlinearity, as well as mixed-integer optimisation to estimate the order of the linear dynamics and the breakpoints of the CPA function. The proposed identification technique requires only output data, and thus no measurement of the constant input is required. This technique is illustrated on a diverse collection of low-dimensional numerical examples as well as data from a gas-turbine combustor.

Centre of a compact convex set

We introduce the notion centre of a convex set and study the space of continuous affine functions on a compact convex set with a centre. We show that these spaces are precisely the dual of a base normed space in which the underlying base has a (unique) centre. We also characterize the corresponding base norm space. We obtain a condition on a compact, balanced, convex subset of a locally convex space, so that the corresponding space of continuous affine functions on the convex set is an absolute order unit space. Similarly, we characterize a condition on the base with a centre of a base normed space, so that the latter becomes an absolutely base normed space.

Special affine multiresolution analysis and the construction of orthonormal wavelets in L 2(ℝ)

The aim of this study is to construct both the continuous and discrete special affine wavelets in . Firstly, we introduce the continuous special affine wavelet transform via the convolution operations in the special affine Fourier domain. Subsequently, we investigate all the fundamental properties of the proposed transform using special affine Fourier transforms. Secondly, the discrete orthonormal special affine wavelets are constructed by introducing the notion of the special affine multiresolution analysis (MRA) in . Besides, a fast wavelet transform associated with special affine MRA is also presented. Towards the culmination, the formulation of special affine MRA starting from a given scaling function is briefly studied.

On minimal representations of shallow ReLU networks

The realization function of a shallow ReLU network is a continuous and piecewise affine function f:Rd→R, where the domain Rd is partitioned by a set of n hyperplanes into cells on which f is affine. We show that the minimal representation for f uses either n, n+1 or n+2 neurons and we characterize each of the three cases. In the particular case, where the input layer is one-dimensional, minimal representations always use at most n+1 neurons but in all higher dimensional settings there are functions for which n+2 neurons are needed. Then we show that the set of minimal networks representing f forms a C∞-submanifold M and we derive the dimension and the number of connected components of M. Additionally, we give a criterion for the hyperplanes that guarantees that a continuous, piecewise affine function is the realization function of an appropriate shallow ReLU network.

Completely bounded homomorphisms of the Fourier algebra revisited

Abstract Assume that A ⁢ ( G ) A(G) and B ⁢ ( H ) B(H) are the Fourier and Fourier–Stieltjes algebras of locally compact groups 𝐺 and 𝐻, respectively. Ilie and Spronk have shown that continuous piecewise affine maps α : Y ⊆ H → G \alpha\colon Y\subseteq H\to G induce completely bounded homomorphisms Φ : A ⁢ ( G ) → B ⁢ ( H ) \Phi\colon A(G)\to B(H) and that, when 𝐺 is amenable, every completely bounded homomorphism arises in this way. This generalised work of Cohen in the abelian setting. We believe that there is a gap in a key lemma of the existing argument, which we do not see how to repair. We present here a different strategy to show the result, which instead of using topological arguments, is more combinatorial and makes use of measure-theoretic ideas, following more closely the original ideas of Cohen.

Representations of continuous piecewise affine functions

Continuous piecewise affine functions are widely used in computational mathematics. In one-dimensional case such functions are called broken lines. The article analyzes the analytical representations of broken lines in the forms accepted in the theory of polynomial splines as well as in the form of the difference of the maxima of two finite families of affine functions. The connection between these representation is being determined.

Construction of Continuous and Piecewise Affine Feedback Stabilizers for Nonlinear Systems

In this article, two methods for constructing continuous and piecewise affine (CPA) feedback stabilizers for nonlinear systems are presented. First, a construction based on a piecewise affine interpolation of Sontag's “universal” formula is developed. Stability of the corresponding closed-loop system is verified a posteriori by means of a CPA control Lyapunov function and subsequently solving a feasibility problem. Second, we develop a procedure for computing CPA feedback stabilizers via linear programming, which allows for the optimization of a control-oriented criterion in the synthesis procedure. Stability conditions are a priori specified in the linear program, which removes the necessity for a posteriori verification of closed-loop stability. We illustrate the developed methods via two application-inspired examples considering the stabilization of an inverted pendulum and the stabilization of a healthy equilibrium of the hypothalamic-pituitary-adrenal axis.

A Computational Analysis of the Tradeoff in the Estimation of Different State Space Specifications of Continuous Time Affine Term Structure Models

A Computational Analysis of the Tradeoff in the Estimation of Different State Space Specifications of Continuous Time Affine Term Structure Models

Markov-Kakutani Theorem on Hyperspace of a Banach Space

Suppose $X$ is a Banach space and $K$ is a compact convex subset of $X$. Let $\mathcal{F}$ be a commutative family of continuous affine mappings of $K$ into $K$. It follows from Markov-Kakutani Theorem that $\mathcal{F}$ has a common fixed point in $K$. Suppose now $(CC(X), h)$ is the corresponding hyperspace of $X$ containing all compact, convex subsets of $X$ endowed with Hausdorff metric $h$. We shall prove the above version of Markov-Kakutani Theorem is valid on the hyperspace $(CC(X), h)$.

Optimistic optimization for continuous nonconvex piecewise affine functions

This paper considers global optimization of a continuous nonconvex piecewise affine (PWA) function over a polytope. This type of optimization problem often arises in the context of control of continuous PWA systems. In literature, it has been shown that the given problem can be formulated as a mixed integer linear programming (MILP) problem, the worst-case complexity of which grows exponentially with the number of polyhedral subregions in the domain of the PWA function. In this paper, we propose a solution approach that is more efficient for continuous PWA functions with a large number of polyhedral subregions. The proposed approach is based on optimistic optimization, which uses hierarchical partitioning of the feasible set and which can guarantee bounds on the suboptimality of the returned solution with respect to the global optimum given a prespecified finite number of iterations. Since the function domain is a polytope with arbitrary shape, we introduce a partitioning approach by employing Delaunay triangulation and edgewise subdivision. Moreover, we derive the analytic expressions for the core parameters required by optimistic optimization for continuous PWA functions. The numerical example shows that the resulting algorithm is faster than MILP solvers for PWA functions with a large number of polyhedral subregions.

Computation and verification of contraction metrics for exponentially stable equilibria

The determination of exponentially stable equilibria and their basin of attraction for a dynamical system given by a general autonomous ordinary differential equation can be achieved by means of a contraction metric. A contraction metric is a Riemannian metric with respect to which the distance between adjacent solutions decreases as time increases. The Riemannian metric can be expressed by a matrix-valued function on the phase space.The determination of a contraction metric can be achieved by approximately solving a matrix-valued partial differential equation by mesh-free collocation using Radial Basis Functions (RBF). However, so far no rigorous verification that the computed metric is indeed a contraction metric has been provided.In this paper, we combine the RBF method to compute a contraction metric with the CPA method to rigorously verify it. In particular, the computed contraction metric is interpolated by a continuous piecewise affine (CPA) metric at the vertices of a fixed triangulation, and by checking finitely many inequalities, we can verify that the interpolation is a contraction metric. Moreover, we show that, using sufficiently dense collocation points and a sufficiently fine triangulation, we always succeed with the construction and verification. We apply the method to two examples.