Advances in the Theory of Control, Signals and Systems with Physical Modeling

Control problems of motion of multi-body mechanical systems under constraints and/or with redundancy in system’s degrees-of-freedom (DOF) are treated from the standpoint of Riemannian geometry. A multi-joint reaching problem with excess DOF is tackled and it is shown that a task space PD feedback with damping shaping in joints maneuvers the endpoint of the robot arm to reach a given target in the sense of exponentially asymptotic convergence. An artificial potential inducing the position feedback in task space can be regarded as a Morse-Bott function introduced in Riemannian geometry, from which the Lagrange stability theorem can be directly extended to this redundant case. The speed of convergence of both the orbit of the endpoint in task space and the trajectory of joint vector in joint space can be adjusted by damping shaping and adequately choosing a single stiffness parameter. In the case that the endpoint is constrained on a hypersurface in E ³, the original Lagrange dynamics expressed in an implicit form by introducing a Lagrange multiplier is decomposed into two partial dynamics with the aid of decomposition of the tangent space into the image of the endpoint Jacobian matrix and the kernel orthogonally complemented to the image. The stability problem of point-to-point endpoint movement on the constraint surface is reduced to the former case without constraint.

Modeling of 2-dimensional grasping and object manipulation under rolling contacts by a pair of multi-joint robot fingers with an arbitrary fingertip contour curve is discussed. Stabilization of grasping by using a control signal based on the fingers-thumb opposability is discussed from the analysis of a Morse-Bott function introduced as an artificial potential. An exentsion of modeling of 3-D grasping under rolling contact constraints is discussed under the circumstance of arbitrary shapes of the fingertips and object.

This paper presents a cascaded sliding mode control scheme for a new pneumatic linear axis. Its guided carriage is driven by a nonlinear mechanism consisting of a rocker with a pair of pneumatic muscle actuators arranged at both sides. Modelling leads to a system of four nonlinear differential equations including polynomial approximations of the volume characteristic as well as the force characteristic of the pneumatic muscles. The differential flatness of the system is exploited in combination with sliding mode techniques to stabilize the error dynamics. Furthermore, a proxy-based sliding mode controller was designed, which is a modified version of sliding mode control as well as an extension of PID control. It allows for accurate tracking during normal operation and smooth recovery from large position errors after unexpected incidents. The internal pressure of each pneumatic muscle is controlled by a fast underlying control loop, whereas in an outer control loop the carriage position and the mean internal pressure of the muscles are controlled. Remaining model uncertainties are compensated by a disturbance observer. Experimental results show an excellent control performance.

An Hamiltonian formulation with complex fluxes and currents is proposed. This formulation is derived from a recent Lagrangian formulation with complex electrical quantities. The complexification process avoids the usual separation into real and imaginary parts and notably simplifies modeling issues. Simple modifications of the magnetic energy underlying standard (α,β) models yield new (α,β) models describing machines with magnetic saturation and saliency. We prove that the usual expression of the electro-mechanical torque (wedge product of fluxes and currents) is related to a rotational invariance characterizing sinusoidal machines.

In this paper it is shown how Stochastic Approximation theory can be used to derive and analyse well-known Iterative Learning Control algorithms for linear systems. The Stochastic Approximation theory gives conditions that, when satisfied, ensure almost sure convergence of the algorithms to the optimal input in the presence of stochastic disturbances. The practical issues of monotonic convergence and robustness to model uncertainty are considered. Specific choices of the learning matrix are studied, as well as a model-free choice. Moreover, the model-free method is applied to a linear motor system, leading to greatly improved tracking.

The work presented here exploits elimination theory (solving systems of polynomial equations in several variables) [1][2] to perform nonlinear parameter identification. In particular show how this technique can be used to estimate the rotor time constant and the stator resistance values of an induction machine. Although the example here is restricted to an induction machine, parameter estimation is applicable to many practical engineering problems. In [3], L. Ljung has outlined many of the challenges of nonlinear system identification as well as its particular importance for biological systems.

This chapter provides a short review on the popular yet still very important area of controlling underactuated mechanical systems. New solutions to the simultaneous stabilization and tracking problem are proposed for nonholonomic mobile robots using state and output feedback. Some open problems are discussed with a unique objective to solicit fundamentally novel techniques for the further development of modern nonlinear control theory.

Planar tree-like structures consisting of rigid links with rotational joints are considered. These models can be used to describe the dynamics of planar biped robots, in particular during the single support phase. Another simple structure of this type is the so-called acrobot. In both cases, the base joint is unactuated while motors are available at all other joints. As a result, motion planning and control of such systems remain challenging tasks. It is shown that flatness-based methods can be helpful to their solution if time scaling is taken into account. To this end the known concept of orbital flatness has to be extended. Moreover, controlled time scaling turns out to provide a helpful additional degree of freedom.Motion planning and feedback design are briefly discussed.

This paper addresses issues related to the modelling and the control of electrostatic microelectromechanical systems (MEMS) in applications requiring high accuracy positioning, wide operation range, and high control bandwidth.A particular emphasis is put on the choice of control system architecture and its influence on potential performance in different practical operation conditions.

We survey two approaches to flatness necessary and sufficient conditions and compare them on examples.

In this review paper we consider some of the basics of nonholonomic systems, considering in particular how it is possible do derive nonholonomic equations of motion as a limit of a Lagrangian system subject to dissipation. This in then extended to show how dissipation may be induced from a Hamiltonian field with a view to quantization of the system.

We report our recent progress on the method of controlled Lagrangians. We present the following: a set of new matching conditions for controlled Lagrangian systems with external forces including velocity-independent forces; a criterion for energy shaping and exponential stabilizability by dissipation for all linear controlled Lagrangian systems; and a criterion for energy shaping and exponential stabilizability by dissipation for all controlled Lagrangian systems with one degree of underactuation.We illustrate the criteria with examples.

We present a tutorial introduction to methods for stabilization of systems with long input delays. The methods are based on techniques originally developed for boundary control of partial differential equations. We start with a consideration of linear systems, first with a known delay and then subject to a small uncertainty in the delay. Then we study linear systems with constant delays that are completely unknown, which requires an adaptive control approach. For linear systems, we also present a method for compensating arbitrarily large but known time-varying delays. Finally, we consider nonlinear control problems in the presence of arbitrarily long input delays.

An enormous wealth of knowledge and research results exists for control of systems with state delays and input delays. Problems with long input delays, for unstable plants, represent a particular challenge. In fact, they were the first challenge to be tackled, in Otto J. M. Smith’s article [1], where the compensator known as the Smith predictor was introduced five decades ago. The Smith predictor’s value is in its ability to compensate for a long input or output delay in set point regulation or constant disturbance rejection problems. However, its major limitation is that, when the plant is unstable, it fails to recover the stabilizing property of a nominal controller when delay is introduced.

A substantial modification to the Smith predictor, which removes its limitation to stable plants was developed three decades ago in the form of finite spectrum assignment (FSA) controllers [2, 3, 4]. More recent treatment of this subject can also be found in the books [5, 6]. In the FSA approach, the system

where X is the state vector, U is the control input (scalar in our consideration here), D is an arbitrarily long delay, and (A,B) is a controllable pair, is stabilized with the infinite-dimensional predictor feedback

where the gain K is chosen so that the matrix A + BK is Hurwitz. The word ‘predictor’ comes from the fact that the bracketed quantity is actually the future state X(t + D), expressed using the current state X(t) as the initial condition and using the controls U(θ) from the past time window [t − D,t]. Concerns are raised in [7] regarding the robustness of the feedback law (2) to digital implementation of the distributed delay (integral) term but are resolved with appropriate discretization schemes [8,9].

One can view the feedback law (2) as being given implicitly, since U appears both on the left and on the right, however, one should observe that the input memory U(θ), θ ∈ [t − D,t] is actually a part of the state of the overall infinite-dimensional system, so the control law is actually given by an explicit full-state feedback formula. The predictor feedback (2) actually represents a particular form of boundary control, commonly encountered in the context of control of partial differential equations.

Motivated by our recent efforts in solving boundary control problems for various classes of partial differential equations (PDEs) using the continuum version of the backstepping method [10,11], we review in this article various extensions to the predictor feedback design that we have recently developed, particularly for nonlinear and PDE systems. These extensions are the subject of our new book [12]. They include the extension of predictor feedback to nonlinear systems and PDEs with input delays, various robustness and inverse optimality results, a delay-adaptive design, an extension to time-varying delays, and observer design in the presence of sensor delays and PDE dynamics. This article is a tutorial introduction to these design tools and concludes with a brief review of some open problems and research opportunities.

One dimensional boundary value problems with lumped controls are considered. Such systems can be modeled as modules over a ring of Beurling ultradistributions with compact support. This ring appears naturally from a corresponding Cauchy problem. The heat equation with different boundary conditions serves for illustration.

A two-level quantum system model describing population transfer driven by a laser field is studied. A four-dimensional real-variable differential equation model is first obtained from the complex-valued two-level model describing the wave function of the system. Due to bilinearity in the control and the states Lie-algebraic techniques can be applied for constructing the state transition matrix of the system. The Wei-Norman technique is used in the construction. The exponential representation of the transition matrix includes three base functions, two of which serves as the parameter functions, which can be chosen freely. This corresponds to considering the overall control system as an underdetermined differential system. In this framework the initial and final states can be defined corresponding to the two levels of the original system model. Then flatness-based design is applied for explicitly calculating the parameter functions, which in turn give the desired input-output pairs. This input then drives the state of the system from the given initial state to the given final state in a finite time.

In recent years, powerful interval arithmetic tools have been developed for the computation of guaranteed enclosures of the sets of all reachable states of dynamical systems. In such simulations, uncertainties in initial conditions and parameters are taken into account by worst-case bounds. The resulting enclosures are verified in the sense that all reachable states are guaranteed to be included. This is achieved by taking into account both the influence of the above-mentioned uncertainties as well as numerical inaccuracies arising from computer implementations using finite-precision floating-point arithmetic. In this contribution, a computational framework for both offline and online applications of interval tools in control design is presented. Verified computational procedures and their applications to the solution of initial value problems for both ordinary differential equations and differential algebraic equations are summarized. These algorithms are employed for verified feedforward control design as well as state and disturbance estimation for a distributed heating system.

Let g ₁, g ₂,...,g _n be a sequence of elements of a Lie group, (knot points). Our problem is to find a smooth, parameterised curve in the group that passes through these elements at parameter values t ₁, t ₂,...,t _n . There are many variations on this basic problem. For example we could take account of velocities. Perhaps we might only require the curve to be near the knot points.

The purpose of this note is to describe a recent generalisation of the well-known Goursat normal form and explore its possible role in control theory. For instance, we give a new, straightforward, general procedure for linearising nonlinear control systems, including time-varying, fully nonlinear systems and we illustrate the method by elementary pedagogical examples. We also exhibit an apparently non-flat control system which can nevertheless be explicitly linearised and therefore posseses an infinite symmetry group.

Although TGF-β is widely known to appear to function paradoxically as a tumor suppressor in normal cells, and as a tumor promoter in cancer cells, the underlying mechanisms by which a single cytokine plays such a dual—and diametrically opposed—role are unknown. In particular, it remains a mystery why the level of TGF-β is unusually high in the primary cancer tissue and blood samples of cancer patients with the poorest prognosis, given that this cytokine is primarily a tumor suppressor. To provide a quantitative explanation of these paradoxical observations, we have developed, from a control theory perspective, a mechanistic model of TGF-β-driven regulation of cell homeostasis. Analysis of the overall system model yields quantitative insight into how the cell population is regulated, enabling us to propose a plausible explanation for the paradox: with the tumor suppressor role of TGF-β unchanged from normal to cancer cells, we demonstrate that the observed increased level of TGF-β is an effect of cancer cell characteristics (specifically, acquired TGF-β resistance), not the cause. We are thus able to explain precisely why the clinically observed correlation between elevated TGF-β levels and poor prognosis is in fact consistent with TGF-β’s original (and unchanged) role as a tumor suppressor.

AMS Classification: 92C30, 92C50.

Most cellular oscillators rely on interlocked positive and negative regulatory feedback loops. While a negative circuit is necessary and sufficient to have limit-cycle oscillations, the role of positive feedbacks is not clear. Here we investigate the possible role of positive feedbacks in the robustness of the oscillations in presence of molecular noise. We performed stochastic simulations of a minimal 3-variable model of the cell cycle. We compare the robustness of the oscillations in the 3-variable model and in a modified model which incorporates a positive feedback loop through an auto-catalytic activation. We find that the model with a positive feedback loop is more robust to molecular noise than the model without the positive feedback loop. This increase of robustness is parameter-independent and can be explained by the attractivity properties of the limit-cycle.

A persistent problem hampering our understanding of the dynamics of large-scale metabolic networks is the lack of experimentally determined kinetic parameters that are necessary to build computational models of biochemical processes. To overcome some of the limitations imposed by absent or incomplete kinetic data, structural kinetic modeling (SKM) was proposed recently as an intermediate approach between stoichiometric analysis and a full kinetic description. SKM extends stationary flux-balance analysis (FBA) by a local stability analysis utilizing an appropriate parametrization of the Jacobian matrix. To characterize the Jacobian, we utilize results from robust control theory to determine subintervals of the Jacobian’ entries that correspond to asymptotically stable metabolic states. Furthermore, we propose an efficient sampling scheme in combination with methods from computational geometry to sketch the stability region. A glycolytic pathway model comprising 12 uncertain parameters is used to assess the feasibility of the method.

We construct a model of competition of three consumers for one single biotic resource ; simulations show that the three species coexist. Using singular perturbations theory we sketch a mathematical proof for this coexistence. The main mathematical tool used is an extension of the Pontryagin-Rodygin theorem on the “slow” motion of a “slow-fast” differential system when the “fast” motion possesses a stable limit cycle. The mathematical analysis is done within the framework of Non Standard Analysis.

The purpose of this paper is to expose several recent challenging control problems for mono-dimensional fluids or reactive fluids. These problems have in common the existence of a moving interface separating two spatial zones where the dynamics are rather different. All these problems are grounded on topics of engineering interest. The aim of the author is to expose the main control issues, possible solutions and to spur an interest for other future contributors. As will appear, mobile interfaces play key roles in various problems, and truly capture main phenomena at stake in the dynamics of the considered systems.

This paper discusses the geometric formulation of the dynamics of chemical reaction networks within the port-Hamiltonian formalism [10, 9, 6]. The basic idea dates back to the innovative work of Oster, Perselson and Katchalsky [8, 7]. The main contribution concerns the formulation of a Dirac structure based on the stoichiometric matrix, which is underlying the port-Hamiltonian formulation. Interaction with the environment is modelled through the boundary metabolites and their boundary fluxes and affinities. This allows a compositional view on chemical reaction network dynamics.

From the systemic point of view protein aggregation is a compensatory mechanism allowing transition of a system (protein solution) from an initially stable equilibrium, which became unstable under a stress, to another stable equilibrium, which bifurcates from the initial one because of the stress. The simplest bifurcation of this type is Logistic bifurcation with a positive small parameter.

We realize this bifurcation as a model of protein aggregation through a large-dimensional Becker-Döring system with a one-dimensional Logistic attractor (BDL) containing two equilibria. BDL depends on the magnitude δ of stress as a small parameter. Kinetics on the attractor is transformed by the observable (which is a fewnomial, i.e., a high-degree polynomial with a number of terms that is small relative to the degree) into the observed kinetics of the experiment. This model explains Gompertzian growth, unimodality of size distribution of aggregates, and relations between Rate, Plateau and time elapsed from onset to inflection moments. The explanation is based on the existence of a nonequilibrium partition function. It exists under the assumption of formation of aggregation-competent monomer as a precursor of the aggregation.

In this work, we study the problem of computing outer bounds for the region of steady states of biochemical reaction networks modelled by ordinary differential equations, with respect to parameters that are allowed to vary within a predefined region. An improved implementation of an algorithm which we presented earlier is developed in order to increase the computational efficiency. The gain in efficiency enables the analysis of medium scale biochemical network models. The applicability of the algorithm to such networks is illustrated by studying a newly developed model for a tumor necrosis factor signalling pathway. This pathway is of major importance for the inflammatory response in mammals and therefore of high biomedical interest. The proposed uncertainty analysis algorithm is applied to the model in order to understand how variations in the parameters and co-stimulation of different receptor types may affect the signalling response in this pathway.