Mathematical Control Theory I

Frontmatter

Chapter 1. A Port-Hamiltonian Formulation of a Wireless Communication System

Abstract

In this chapter we model the traffic dynamics in a wireless communication system (characterized by a set of routers exchanging data with each other) in the port-Hamiltonian framework. Communication systems are characterized by elements which produce significant time delays (by design) in their response, unlike (say) an idealized circuit or mechanical element. Furthermore, the communication between two routers (compositionality) involves losses due to the characteristics of radio signal propagation. In this paper we study the type of Dirac structure used to model a communication element (a router), we analyze the stability properties of a router and finally we study the compositionality properties that evolve under lossy interconnections.

Viswanath Talasila, Ramkrishna Pasumarthy

Chapter 2. Dirac Structures and Control by Interconnection for Distributed Port-Hamiltonian Systems

Abstract

The aim of this work is to show how the Dirac structure properties can be exploited in the development of energy-based boundary control laws for distributed port-Hamiltonian systems. Stabilisation of non-zero equilibria has been achieved by looking at, or generating, a set of structural invariants, namely Casimir functions, in closed-loop, and geometric conditions for the problem to be solved are determined. However, it is well known that this method fails when an infinite amount of energy is required at the equilibrium (dissipation obstacle). So, a novel approach that enlarges the class of stabilising controllers within the control by interconnection paradigm is also discussed. In this respect, it is shown how to determine a different control port that is instrumental for removing the intrinsic constraints imposed by the dissipative structure of the system. The general theory is illustrated with the help of two related examples, namely the boundary stabilisation of the shallow water equation with and without distributed dissipation.

Alessandro Macchelli

Chapter 3. Energy-Aware Robotics

Abstract

This chapter has a tutorial nature in introducing a number of useful concepts which resulted by reasoning with power ports rather than with signals, as people usually do in control. Arjan is one of the Godfathers in this way of thinking and he has been a pioneer in bringing these concepts to a new level, introducing proper geometry, a sound system theoretic basis and divulgating these issues. This chapter shows how, by using these concepts, it is possible to address or solve certain problems in robotics, control and passivity in a simple and straightforward way. It also presents a formal proof of a claim which is often used as a conjecture and which gives theoretical arguments to counteract the statement which is often used against passivity and saying that passivity is too restrictive and stability is what should be looked for. Many of the concepts reported in the chapter have been the results of discussions with Arjan or are still issues that I am working on with Arjan. It is a great pleasure and honour to have the opportunity to contribute in this way to a recognition of the incredible career of a college and friend for which I have incredible respect from an intellectual and personal point of view.

Stefano Stramigioli

Chapter 4. Time-Varying Phasors and Their Application to Power Analysis

Abstract

The classical complex phasor representation of sinusoidal voltages and currents is generalized to arbitrary waveforms. The method relies on the so-called analytic signal using the Hilbert transform. This naturally leads to the notion of a time-varying power triangle and its associated instantaneous power factor. Additionally, it is shown for linear systems that Budeanu’s reactive power can be related to energy oscillations, but only in an average sense. Furthermore, Budeanu’s distortion power is decomposed into a part representing a measure of the fluctuation of power around the active power and a part that represents the fluctuation of power around Budeanu’s reactive power. The results are presented for single-phase systems.

Dimitri Jeltsema

Chapter 5. Handling Biological Complexity Using Kron Reduction

Abstract

We revisit a model reduction method for detailed-balanced chemical reaction networks based on Kron reduction on the graph of complexes. The resulting reduced model preserves a number of important properties of the original model, such as, the kinetics law and identity of the chemical species. For determining the set of chemical complexes for the deletion, we propose two alternative methods to the computation of error integral which requires numerical integration of the state equations. The first one is based on the spectral clustering method and the second one is based on the eigenvalue interlacing property of Kron reduction on the graph. The efficacy of the proposed methods is evaluated on two biological models.

Bayu Jayawardhana, Shodhan Rao, Ward Sikkema, Barbara M. Bakker

Chapter 6. Distributed Line Search for Multiagent Convex Optimization

Abstract

This note considers multiagent systems seeking to optimize a convex aggregate function. We assume that the gradient of this function is distributed, meaning that each agent can compute its corresponding partial derivative with information about its neighbors and itself only. In such scenarios, the discrete-time implementation of the gradient descent method poses the basic challenge of determining appropriate agent stepsizes that guarantee the monotonic evolution of the objective function. We provide a distributed algorithmic solution to this problem based on the aggregation of agent stepsizes via adaptive convex combinations. Simulations illustrate our results.

Jorge Cortés, Sonia Martínez

Chapter 7. Optimal Management with Hybrid Dynamics—The Shallow Lake Problem

Abstract

In this article we analyze an optimal management problem that arises in ecological economics using hybrid systems modeling. First, we introduce a discounted autonomous infinite horizon hybrid optimal control problem and develop few tools to analyze the necessary conditions for optimality. Next, using these tools we study the classical shallow lake problem where the nonlinear lake dynamics is described by hybrid dynamics. We show that our results agree with earlier studies on the problem, that is, variation of system parameters induce bifurcations in the optimal solution.

P. V. Reddy, J. M. Schumacher, J. C. Engwerda

Chapter 8. Modeling Perspectives of Hybrid Systems and Network Systems

Abstract

This article presents two topics, i.e., well-posedness of piecewise affine systems, and model reduction of network systems. The well-posedness problem, i.e., the problem of existence and uniqueness of solutions, of hybrid systems is one of the fundamental research topics, which the first author has collaborated with Prof. Arjan van der Schaft in 1998. Some results are revisited by focusing on the class of bimodal piecewise affine systems. The latter discusses the most recent topic that both Arjan and the first author have common interest in. In particular, the clustering-based \(H_\infty -\) and \(H_2\)-model reduction approaches of large-scale network systems, which have been independently developed by the authors, are represented in a unified way.

Jun-ichi Imura, Takayuki Ishizaki

Chapter 9. Control of HVDC Transmission Systems: From Theory to Practice and Back

Abstract

The problem of modeling and control of multi-terminal high-voltage direct-current transmission systems is addressed in this chapter, which contains three main contributions. First, to propose a unified, physically motivated, modeling framework—based on port-Hamiltonian systems representations—of the various network topologies used in this application. Second, to prove that the system can be globally asymptotically stabilized with a decentralized PI control that exploits its passivity properties. Close connections between the proposed PI and the popular Akagi’s PQ instantaneous power method are also established. Third, to reveal the transient performance limitations of the proposed controller that, interestingly, is shown to be intrinsic to PI passivity-based control. The performances of the controller are verified via simulations on a three-terminal benchmark example.

Daniele Zonetti, Romeo Ortega

Chapter 10. A Complement on Elimination and Realization in Rational Representations

Abstract

In this paper we study a number of problems in the context of rational representations of behaviors. In that context, a given proper real rational matrix can represent three behaviors. In the first place it can represent an input–output behavior. Second, it can represent the kernel behavior of the rational ‘differential operator’ associated with the rational matrix. Third, it can represent the image behavior asociated with the rational matrix. On the other hand, every proper real rational matrix admits a realization as a finite-dimensional linear state-space system. Such realization can represent three system behaviors: an input-state-output behavior, an output nulling behavior, or a driving variable behavior. In this paper we will study the relation between the three external behaviors of these state representations, and the behaviors given by the three rational representations associated with the underlying rational matrix. Preliminary results from [5] will be complemented to obtain necessary and sufficient conditions such that the respective external behaviors are equal.

Harry L. Trentelman, Tjerk W. Stegink, Sasanka V. Gottimukkala

Chapter 11. Modeling and Analysis of Energy Distribution Networks Using Switched Differential Systems

Abstract

It is a pleasure to dedicate this contribution to Prof. Arjan van der Schaft on the occasion of his 60th birthday. We study the dynamics of energy distribution networks consisting of switching power converters and multiple (dis-)connectable modules. We use parsimonious models that deal effectively with the variant complexity of the network and the inherent switching phenomena induced by power converters. We also present the solution to instability problems caused by devices with negative impedance characteristics such as constant power loads. Elements of the behavioral system theory such as linear differential behaviors and quadratic differential forms are crucial in our analysis.

Jonathan C. Mayo-Maldonado, Paolo Rapisarda

Chapter 12. Nonlinear Controller Design Based on Invariant Manifold Theory

Abstract

The role of invariant manifold in nonlinear control theory is reviewed. First, stable, center-stable and center manifold algorithms to compute flows on these manifolds are presented. Next, application results of the computational methods are illustrated for optimal stabilization, optimal output regulation and periodic orbit design problems.

Noboru Sakamoto

Chapter 13. On Geometric Properties of Triangularizations for Nonlinear Control Systems

Abstract

We consider triangular decompositions for nonlinear control systems. For systems that are exactly linearizable by static feedback it is well known that a triangular structure exists in adapted coordinates using the Frobenius theorem to straighten out a nested sequence of involutive distributions. This triangular form is based on explicit ordinary differential equations from which it can be easily seen that exactly linearizable systems are also flat. We will analyze this triangularization also from a dual perspective using a Pfaffian system representation. This point of view allows the introduction of a triangular form corresponding to implicit ordinary differential equations. For systems that are flat but not exactly linearizable by static feedback, this modified triangular form turns out to be useful in setting up a constructive algorithm to compute so-called 1-flat outputs.

Markus Schöberl, Kurt Schlacher

Chapter 14. Online Frequency Estimation of Periodic Signals

Abstract

The problem of estimating online the unknown period of a periodic signal is considered, with no a priori information on the period: this is a crucial problem in the design of learning and synchronizing controls, in fault detection, and for the attenuation of periodic disturbances. Given a measurable continuous, bounded periodic signal, with nonzero first harmonic in its Fourier series expansion, a dynamic algorithm is proposed which provides an online globally exponentially convergent estimate of the unknown period. The period estimate converges from any initial condition to a neighborhood of the true period whose size is explicitly characterized in terms of the higher order harmonics contained in the signal. The accuracy of the frequency estimation can be arbitrarily improved by increasing the order of a prefilter which is incorporated in the estimation algorithm, at the expense of reducing the rate of the exponential convergence. This online frequency estimation algorithm can be used to design hybrid disturbance attenuation controllers for periodic disturbances with unknown period.

Riccardo Marino, Patrizio Tomei

Chapter 15. Power-Based Methods for Infinite-Dimensional Systems

Abstract

In this chapter we aim to extend the Brayton Moser (BM) framework for modeling infinite-dimensional systems. Starting with an infinite-dimensional port-Hamiltonian system we derive a BM equivalent which can be defined with respect to a non-canonical Dirac structure. Based on this model we derive stability and new passivity properties for the system. The state variables in this case are the “effort” variables and the storage function is a “power-like” function called the mixed potential. The new property is derived by “differentiating” one of the port variables. We present our results with the Maxwell’s equations, and the transmission line with non-zero boundary conditions as examples.

Krishna Chaitanya Kosaraju, Ramkrishna Pasumarthy

Chapter 16. On Stabilization of Mixed Dimensional Parameter Port Hamiltonian Systems Via Energy Shaping

Abstract

For systems described by Port-Hamiltonian (PH) equations, the Control by Interconnection method, based on the existence of Casimir functions, provides a simple and elegant procedure for stabilization of nonlinear systems with finite dissipation. This work explores the possibility of extending this technique to the case where the plant contains an infinite-dimensional subsystem. Conditions for the existence of Casimir functions reveal the constraints for the application of the design procedure. A simple example of an RLC circuit coupled with an infinite-dimensional transmission line illustrates the main ideas of this paper.

H. Rodríguez-Cortés

Chapter 17. Network Topology and Synchronization of Systems with Linear Time-Delayed Coupling

Abstract

We consider networks of square input–output systems that interact via linear, time-delayed coupling functions. For given system dynamics, we give conditions for the construction of a (local, global) synchronization diagram. We show that a condition for (local, global) synchronization is that the coupling strength and time-delay are contained in the intersection of scaled copies of the (local, global) synchronization diagram, where the scaling factors are the nonzero eigenvalues of the symmetric Laplacian matrix.

Erik Steur, Henk Nijmeijer

Chapter 18. Examples on Stability for Infinite-Dimensional Systems

Abstract

By means of examples, we study stability of infinite-dimensional linear and nonlinear systems. First we show that having a (strict) Lyapunov function does not imply asymptotic stability, even not for linear systems. Second, we show that to conclude (local) exponential stability from the linearization, care must be taken how the linearization is obtained.

Hans Zwart

Chapter 19. Model Reduction by Generalized Differential Balancing

Abstract

In this chapter, we give a generalization of differential balancing method for model reduction of nonlinear systems in the direction to computation. We generalize concepts of differential controllability and observability functions, then use them for model reduction. We show some stability properties are preserved under the model reduction and estimate the error bound by the model reduction.

Yu Kawano, Jacquelien M.A. Scherpen

Chapter 20. Trajectory-Based Theory for Hybrid Systems

Abstract

This chapter presents a trajectory-based perspective in solving safety/ reachability analysis and synthesis problems and fault diagnosability analysis in hybrid systems. The main tool used in obtaining the results presented in this chapter is the concept of trajectory robustness, which is derived from the theory of approximate bisimulation. Trajectory robustness essentially provides a guarantee on how far the system’s state trajectories can deviate (in \(L_{\infty }\) norm) as a result of initial state variations. It further leads to the possibility of approximating the set of the system’s trajectories, which is infinite, with a finite set of trajectories. This fact, in turns, allows us to pose the above problems as finitely many finite problems that can be practically solved. In addition, these finite problems can be solved in parallel.

A. Agung Julius

Chapter 21. Controllability and Stabilizability of Discontinuous Bimodal Piecewise Linear Systems

Abstract

Characterizing controllability like properties of bimodal piecewise linear systems, i.e., piecewise linear systems with two modes, is known to be a notoriously hard. In this chapter, we focus on discontinuous bimodal systems that are well-posed in the sense of existence and uniqueness of solutions. The main results of the chapter are Popov–Belevitch–Hautus-type necessary and sufficient conditions for controllability and stabilizability of such systems.

Le Quang Thuan, Kanat Camlibel