Piecewise Affine Approximation Research Articles

Hybrid modelling and control of a synchronous DC-DC converter

In this paper, the problem of hybrid modelling and control of a fixed frequency DC-DC converter, namely the Non-inverting Buck-Boost converter is addressed. The v-resolution method is used to capture the hybrid nature of the system. Two mixed logical dynamical (MLD) models are obtained for the converter using hybrid systems description language (HYSDEL). In the process of using v-resolution concept, the problem of non-linear terms in state update equations is solved using piecewise affine approximation. Three PWA models are obtained using least squares fitting over non-linear dynamics of the system. A suitable model in terms of accuracy and complexity is chosen for controller design. Model predictive control is used as the control strategy and explicit solution to the control problem is obtained. Finally, the simulation results are presented. The accuracy and complexity of the models are compared and the performance of the controller is evaluated over three different scenarios which are common in practice.

Approximation of the curvature of Alexandrov surfaces using dual polyhedra

It is shown that a sequence of pairs of locally polar polyhedra makes it possible to construct piecewise-affine approximation to the spherical Gauss map and to construct convergent pointwise approximations to mean and Gauss curvatures of regular surfaces. Any DC surface (representable as a difference of convex functions, A. D. Alexandrov, 1949) possesses a second differential and a tangent paraboloid almost everywhere, hence approximation by dual polyhedra allows us to approximate the curvature of such surfaces.

Explicit Model Predictive Control for Systems with Linear Parameter-Varying State Transition Matrix

In this paper we propose a closed-loop min-max MPC algorithm based on dynamic programming, to compute explicit control laws for systems with a linear parameter-varying state transition matrix. This enables the controller to exploit parameter information to improve performance compared to a standard robust approach where no uncertainty knowledge is used, while keeping the benefits of fast online computations. The off-line computational burden is similar to what is required for computing explicit control laws for uncertain or nominal LTI systems. The proposed control strategy is applied to the controlled Hénon map to draw a comparison, in terms of complexity and control performance, with a controller based on a piecewise affine approximation.

PIECEWISE AFFINE SYSTEM MODELING AND CONTROL OF PWM CONVERTERS

The averaged switch modeling approach is a powerful method for representing the behavior of a wide variety of converters through equivalent circuits. The model is not linear and it is common to perform a small signal linearization about an operating point and design a linear controller. Models obtained with such method involve considerable approximation and produce results that are limited for high performance controller designs. In this paper a piecewise affine approximation technique is introduced for modeling PWM converters. This model is much more precise in predicting the dynamic response of averaged nonlinear model comparing the linear model. This paper also presents a piecewise linear controller synthesis method for PWM converters described by the proposed PWA model. The proposed controller is very efficient and effective. The design method is well suited for converters having a wide range of variation about their operating point. A simulation example on buck-boost converter is presented to demonstrate the performance of the proposed method for modeling the dynamics of the converter and designing the appropriate controller.

ASSESSMENT OF NONLINEAR PREDICTIVE CONTROL TECHNIQUES FOR DC-DC CONVERTERS

This article focuses on the application of two nonlinear predictive control strategies to fixed frequency switch-mode dc-dc converters. The first approach uses an averaged nonlinear model for predictions and provides an a priori robust stability guarantee for the resulting closed-loop system. The implementation of this scheme requires solving a single linear program on-line. The second approach employs a piecewise affine (PWA) approximation of the converter equations as prediction model and yields an explicit controller off-line, which allows for an a posteriori stability check. An assessment of the synthesis procedures and methodological characteristics is given and simulation results are shown to substantiate the effectiveness of the proposed approaches.

HYBRID IDENTIFICATION OF NONLINEAR BIOCHEMICAL PROCESSES

Disentangling the complexity of biochemical networks requires knowledge about the quantitative relationships between the individual components. However, the nonlinear dynamics of biochemical processes are difficult to identify with traditional identification methods. We therefore propose to model the complex nonlinear biochemical processes with several simpler systems (modes), together with a switching rule that determines which mode is active, i.e. with a hybrid system. We consider the example of a nonlinear biochemical oscillator, and propose a simple piecewise affine (PWA) approximation. Qualitative analysis shows that the PWA model is able to capture the dynamics of the nonlinear oscillator. Hybrid identification procedure is subsequently applied to identify the parameters of the PWA model.

EXPLICIT MODEL PREDICTIVE CONTROL OF THE BOOST DC-DC CONVERTER

This paper extends previous work done on the optimal control of the dc-dc converter boost circuit topology. The same control problem formulation is maintained but a simpler piecewise affine approximation of the system dynamics is derived and employed to obtain an explicit solution whose complexity renders it viable for implementation on a realistic circuit setup.

A codimension-two scenario of sliding solutions in grazing–sliding bifurcations

This paper investigates codimension-two bifurcations that involve grazing–sliding and fold scenarios. An analytical unfolding of this novel codimension-two bifurcation is presented. Using the discontinuity mapping techniques it is shown that the fold curve is one-sided and cubically tangent to the grazing curve locally to the codimension-two point. This theory is then applied to explain the dynamics of a dry-friction oscillator where such a codimension-two point has been found. In particular, the presence and the character of essential bifurcation curves that merge at the codimension-two point are confirmed. This allows us to study the dynamics away from the codimension-two point using a piecewise affine approximation of the normal form for grazing–sliding bifurcations and explain the dynamics observed in the friction system.

Averaging techniques yield reliable a posteriori finite element error control for obstacle problems

The reliability of frequently applied averaging techniques for a posteriori error control has recently been established for a series of finite element methods in the context of second-order partial differential equations. This paper establishes related reliable and efficient a posteriori error estimates for the energy-norm error of an obstacle problem on unstructured grids as a model example for variational inequalities. The surprising main result asserts that the distance of the piecewise constant discrete gradient to any continuous piecewise affine approximation is a reliable upper error bound up to known higher order terms, consistency terms, and a multiplicative constant.

Identification of the B– H curve from external measurements using complementary formulations

This paper deals with theoretical and numerical problems related with the inverse problem of the reconstruction of the first piece of the piecewise affine approximation of the B– H nonlinear characteristic starting from the knowledge of measured flux–current relationship. The reconstruction of the first piece of the characteristic is critical, as highlighted in novel approaches reconstructing the whole characteristic without relying on the common assumption that the driving system produces a uniformly distributed magnetic field inside the specimen. In this paper a proof will be given of the uniqueness of the solution of the inverse problem based on elementary analysis arguments and a numerical procedure that, by means of the use of complementary formulations, allows one to compute and control the reconstruction error due to the numerical formulation. The paper is organized as follows: a brief discussion of the problem is reported in Section 1, the numerical formulation and the error bounds are reported in Section 2, the uniqueness of the inverse problem is addressed in Section 3 and numerical examples are reported in Section 4.

Lower bounding aggregation and direct computation for an infinite horizon one-reservoir model

We present a specialized policy iteration method for the computation of optimal and approximately optimal policies for a discrete-time model of a single reservoir whose discharges generate hydroelectric power. The model is described in (Lamond et al., 1995) and (Drouin et al., 1996), where the special structure of optimal policies is given and an approximate value iteration method is presented, using piecewise affine approximations of the optimal return functions. Here, we present a finite method for computing an optimal policy in O( n 3) arithmetic operations, where n is the number of states in the associated Markov decision process, and a finite method for computing a lower bound on the optimal value function in O( m 2 n) where m is the number of nodes of the piecewise affine approximation.

Piecewise affine approximations for the control of a one-reservoir hydroelectric system

Piecewise affine approximations for the control of a one-reservoir hydroelectric system

Piecewise affine approximations for the control of a one-reservoir hydroelectric system

We analyze the computation of optimal and approximately optimal policies for a discrete-time model of a single reservoir whose discharges generate hydroelectric power. Inflows in successive periods are random variables. Revenue from hydroelectric production is represented by a piecewise linear function. We use the special structure of optimal policies, together with piecewise affine approximations of the optimal return functions at each stage of dynamic programming, to decrease the computational effort by an order of magnitude compared with ordinary value iteration. The method is then used to obtain easily computable lower and upper bounds on the value function of an optimal policy, and a policy whose value function is between the bounds.

Conformal image warping

Numerical and computer-graphic methods for conformal image mapping between two simply connected regions are described. The immediate motivation for this application is that the visual field is represented in the brain by mappings which are, at least approximately, conformal. Thus, to simulate the imaging properties of the human visual system (and perhaps other sensory systems), conformal image mapping is a necessary technique. For generating the conformal map, a method for analytic mappings and an implementation of the Symm algorithm for numerical conformal mapping are shown. The first method evaluates the inverse mapping function at each pixel of the range, with antialiasing by multiresolution texture prefiltering and bilinear interpolation. The second method is based on constructing a piecewise affine approximation of the mapping in the form of a joint triangulation, or triangulation map, in which only the nodes of the triangulation are conformally mapped. The texture is then mapped by a local affine transformation on each pixel of the range triangulation with the same antialiasing used in the first method. The algorithms are illustrated with examples of conformal mappings constructed analytically from elementary mappings, such as the linear fractional map, the complex algorithm, etc. Applications of numerically generated maps between highly irregular regions and an example of the visual field mapping that motivates this work are also shown. >

Contraction Mappings in the Theory Underlying Dynamic Programming

Contraction Mappings in the Theory Underlying Dynamic Programming