An overview on averaging for pulse-modulated switched systems
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paper_title: Continuous-Time Stochastic Averaging on the Infinite Interval for Locally Lipschitz Systems

paper_content:

We investigate stochastic averaging on the infinite time interval for a class of continuous-time nonlinear systems with stochastic perturbation and remove or weaken several restrictions present in existing results: global Lipschitzness of the nonlinear vector field, equilibrium preservation under the stochastic perturbation, global exponential stability of the average system, and compactness of the state space of the perturbation process. If an equilibrium of the average system is exponentially stable, we show that the original system is exponentially practically stable in probability. If, in addition, the original system has the same equilibrium as the average system, then the equilibrium of the original system is locally asymptotically stable in probability. These results extend the deterministic general averaging for aperiodic functions to the stochastic case.
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paper_title: Fast Switching Analysis of Linear Switched Systems Using Exponential Splitting

paper_content:

Stability of periodic switched linear systems with fast switching patterns is studied by combining the method of averaging with exponential splitting. This approach yields less conservative bounds on stabilizing fast switching rates than can be obtained with prior approaches. Such bounds can be useful for analysis and design of switching control laws. The method is also generalized to arbitrary (including nonperiodic nonswitched) time-varying systems. In particular, the effects of time-varying state perturbations on the stability of linear periodic switched systems are analyzed.
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paper_title: Extensions of averaging theory for power electronic systems

paper_content:

This paper extends averaging theory for power electronic systems to include feedback controlled power converters. New averaging techniques based on the integral equation description provide theoretical justification for commonly used averaging methods. The new theory provides a basis for answering fundamental questions about the averaging approximation. A ripple estimate expression is presented, along with simulation results for a feedback controlled boost power converter.<<ETX>>
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paper_title: Stability and Oscillations of Nonlinear Pulse-Modulated Systems

paper_content:

Mathematical description of pulse-modulated systems stability of equilibria - miscellaneous methods stability of equilibria - averaging method stability of processes - averaging method forced periodic oscillations - method of equations of periods oscillations of PWM systems - fixed-point method oscillations of PFM systems - fixed-point method method of harmonic balance and its justification auto-oscillations in pulse modulated systems pulse-width modulated systems of phase synchronization.
---
paper_title: Experiments on node-to-node pinning control of Chua's circuits

paper_content:

Abstract In this paper, we study the global intermittent pinning controllability of networks of coupled chaotic oscillators. We explore the feasibility of the recently presented node-to-node pinning control strategy through experiments on Chua’s circuits. We focus on the case of two peer-to-peer coupled Chua’s circuits and we build a novel test-bed platform comprised of three inductorless Chua’s oscillators. We investigate the effect of a variety of design parameters on synchronization performance, including the coupling strength between the oscillators, the control gains, and the switching frequency of node-to-node pinning control. Experimental results demonstrate the effectiveness of this novel pinning control strategy in rapidly taming chaotic oscillator dynamics onto desired reference trajectories while minimizing the overall control effort and the number of pinned network sites. From an analytical standpoint, we present sufficient conditions for global node-to-node pinning controllability and we estimate the maximum switching period for network controllability by adapting and integrating available results on Lyapunov stability theory and partial averaging techniques.
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paper_title: Input-to-State Stability and Averaging of Linear Fast Switching Systems

paper_content:

We consider the averaging method for stability of rapidly switching linear systems with disturbances. We show that the notions of strong and weak averages proposed, with partial strong average defined in this note, play an important role in the context of switched systems. Using these notions of average, we show that exponential input-to-state stability (ISS) of the strong and the partial strong average system with linear gain imply exponential ISS with linear gain of the actual system. Similarly, exponential ISS of the weak average guarantees an appropriate exponential derivative ISS (DISS) property for the actual system. Moreover, using the Lyapunov method, we show that linear ISS gains of the actual system and its average converge to each other as the switching rate is increased.
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paper_title: Global pulse synchronization of chaotic oscillators through fast-switching: theory and experiments

paper_content:

Abstract We study pulse synchronization of chaotic systems in master–slave configuration. The slave system is unidirectionally coupled to the master system through an intermittent linear error feedback coupling, whose gain matrix periodically switches among a finite set of constant matrices. Using Lyapunov-stability theory, fast-switching techniques, and the concept of matrix measure, we derive sufficient conditions for global synchronization. The derived conditions are specialized to the case of Chua’s circuits. An inductorless realization of coupled Chua’s circuits is developed to illustrate the effectiveness of the proposed approach.
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paper_title: A general unified approach to modelling switching-converter power stages

paper_content:

A method for modelling switching-converter power stages is developed, whose starting-point is the unified state-space representation of the switched notworks and whose end result is either a complete state-space description or its equivalent small-signal low-frequency linear circuit model. A new canonical circuit model is proposed, whose fixed topology contains all the essential input-output and control properties of any d.c.-to-d.c. switching converter, regardless of its detailed configuration, and by which different converters can be characterized in the form of a table conveniently stored in a computer data bank to provide a useful tool for computer-aided design and optimization. The new canonical circuit model predicts that, in general, switching action introduces both zeros and poles into the duty ratio to output transfer function in addition to those from the effective filter network.
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paper_title: Averaged modelling of switching power converters: reformulation and theoretical basis

paper_content:

Uniform averaged modeling methods for switching power converters (SPCs) are established. Averaging methods for differential equation systems with continuous right-hand sides are discussed and then generalized to discontinuous ones. It is shown that these methods are applicable to most known types of SPC, including PWM, resonant, and quasi-resonant converters. PWM converters under discontinuous conducting mode are taken as examples to further demonstrate the applicability of these methods. It is shown that the currently used state space averaging and its several extensions can be derived from these methods. >
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paper_title: Error bounds in the averaging of hybrid systems

paper_content:

The authors analyze the error introduced by the averaging of hybrid systems. These systems involve linear systems which can take a number of different realizations based on the state of an underlying finite state process. The averaging technique (based on a formula from Lie algebras known as the Backer-Campbell-Hausdorff (BCH) formula) provides a single system matrix as an approximation to the hybrid system. The two errors discussed are: (1) the error induced by the truncation of the BCH series expansion and (2) the error between the actual hybrid system and its average. A simple sufficient stability test is proposed to check the asymptotic behavior of this error. In addition, conditions are derived that allow the use of state feedback instead of averaging to arrive at a time-invariant system matrix. >
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paper_title: Time-averaging of high-speed data transfer protocols

paper_content:

We propose two modeling approaches of additive-increase/multiplicative-decrease (AIMD) congestion control mechanisms. The first separates the increase and decrease parts whereas the second describes the rate evolution of the congestion window as a continuous process governed by a differential equation. We relate these approaches and show that the second (fluid-flow) model results from deterministic time averaging of the first (discontinuous) model. A generalized class of nonlinear protocols, which includes Floyd's HighSpeed TCP, is then proposed and analyzed. Our findings are validated by simulation.
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paper_title: Sufficient conditions for fast switching synchronization in time-varying network topologies

paper_content:

In previous work [J. D. Skufca and E. Bollt, Mathematical Biosciences and Engineering, 1 (2004), pp. 347-359], empirical evidence indicated that a time-varying network could propagate sufficient in...
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paper_title: Blinking model and synchronization in small-world networks with a time-varying coupling

paper_content:

The paper proposes a new type of small-world networks of cells with chaotic behavior. This network consists of a regular lattice of cells with constant 2K-nearest neighbor couplings and time-dependent on–off couplings between any other pair of cells. In each time interval of duration τ such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times. At each moment, the coupling structure corresponds to a small-world graph, but the shortcuts change from time interval to time interval, which is a good model for many real-world dynamical networks. It is to be distinguished from networks that have randomly chosen shortcuts, fixed in time. Here, we apply the Connection Graph Stability method, developed in the companion paper (“Connection graph stability method for synchronized coupled chaotic systems”, see this issue), to the study of global synchronization in this network with the time-varying coupling structure, in the case where the on–off switching is fast with respect to the characteristic synchronization time of the network. The synchronization thresholds are explicitly linked with the average path length of the coupling graph and with the probability p of shortcut switchings in this blinking model. We prove that for the blinking model, a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length. Short interactions between cells, as in the blinking model, are important in practice. To cite prominent examples, computers networked over the Internet interact by sending packets of information, and neurons in our brain interact by sending short pulses, called spikes. The rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much. In this respect, we believe that it is more efficient than a structure of fixed random connections.
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paper_title: Tracking and formation control of multiple autonomous agents: A two-level consensus approach

paper_content:

Simultaneous tracking and formation control is addressed for a team of autonomous agents that evolve dynamically in a space containing a measurable vector field. Each agent measures the local value of the field along its trajectory and occasionally shares relevant information with other agents, in order to estimate the spatial average obtained from averaging measurements across all agents. Using shared information, agents control their trajectories in a cooperative manner, with the dual goals of driving the average field measurement to a specified value and maintaining a desired formation about the average. Two approaches to virtual leader estimation are considered. The first involves the synthesis of a common virtual leader state, whereas the second involves decentralized estimation of the virtual leader by individual agents. Under the second approach, control is posed as a two-level consensus problem, where agents reach agreement on the virtual leader state at one level and reach formation about the virtual leader at the other level. The decentralized approach is effective even when communication among agents is limited, in the sense that the associated network graph can be disconnected in frozen time.
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paper_title: Principles of power electronics

paper_content:

1. Introduction. 2. Form and Function: An Overview. 3. Introduction to Rectifier Circuits. 4. Bridge and Polyphase Rectifier Circuits. 5. Phase-Controlled Converters. 6. High-Frequency Switching dc/dc Converters. 7. Isolated High-Frequency dc/dc Converters. 8. Variable-Frequency dc/ac Converters. 9. Resonant Converters. 10. ac/ac Converters. 11. Dynamics and Control: An Overview. 12. State-Space Models. 13. Linear and Piecewise Linear Models. 14. Feedback Control Design. 15. Components: An Overview. 16. Review of Semiconductor Devices. 17. Power Diodes. 18. Power Transistors. 19. Thyristors. 20. Magnetic Components. 21. Ancillary Issues: An Overview. 22. Gate and Base Drives. 23. Thyristor Commutation Circuits. 24. Snubber Circuits and Clamps. 25. Thermal Modeling and Heat Sinking.
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paper_title: Generalized averaging method for power conversion circuits

paper_content:

A more general averaging procedure that encompasses state-space averaging and that is potentially applicable to a much broader class of circuits and systems is presented. Examples of its application in resonant and PWM power convertors are presented. The technique is shown to be effective on a number of examples. including resonant type converters. The approach offers refinements to the theory of state-space averaging, permitting a framework for analysis and design when small ripple conditions do not hold. The method may find applications in simulation and design since it is considerably easier to simulate an averaged model than a switched model. >
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paper_title: Pulse width modulated control of robotic manipulators

paper_content:

A robust discontinuous feedback control scheme of the pulse width modulation (PWM) type is proposed for the regulation of angular positions of links in robotic manipulators. The discontinuous PWM controller design is carried out on the basis of the expected linearization approach for a suitably defined average PWM-controlled continuous model. >
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paper_title: Control of hybrid systems via dehybridization

paper_content:

Presents a controller design framework for hybrid systems based on approximate continuous time models and standard continuous control techniques. The procedure has three steps: (i) designing a switching controller so that the initial hybrid system revisits one of its modes, (ii) applying a dehybridization procedure, related to singular perturbation, to eliminate all discrete states and the fast continuous states, and (iii) designing a controller for the continuous dehybridized system using standard techniques. This methodology is applied in simulation for the set point tracking of a stick-slip inertial drive.
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paper_title: Averaging of nonsmooth systems using dither

paper_content:

It was shown by Zames and Shneydor and later by Mossaheb that a high-frequency dither signal of a quite arbitrary shape can be used to narrow the effective nonlinear sector of Lipschitz continuous feedback systems. In this paper, it is shown that also discontinuous nonlinearities of feedback systems can be narrowed using dither, as long as the amplitude distribution function of the dither is absolutely continuous and has bounded derivative. The averaged system is proven to approximate the dithered system with an error of the order of dither period.
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paper_title: Justification of the averaging method for a system of functional-differential equations with variable structure and impulses

paper_content:

This paper presents a justification of the averaging method for a system of functional-differential equations with variable structure, impulses, and transformed argument, depending on the time and on the unknown function.
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paper_title: Nonlinear averaging applied to the control of pulse width modulated (PWM) pneumatic systems

paper_content:

This work presents a control methodology that enables nonlinear model based sliding mode control of PWM-controlled pneumatic servo actuators. An averaging approach is developed to describe the equivalent continuous-time dynamics of a PWM controlled nonlinear system, which renders the originally discontinuous system, possibly non-affine in the input, into an equivalent system that is in control canonical nonlinear form. A sliding mode controller is then developed for a pneumatic servo actuator based on the averaged equivalent control canonical model. The controller is implemented on an experimental system, and the effectiveness of the proposed approach validated by experimental tracking data.
---
paper_title: Stability and Oscillations of Nonlinear Pulse-Modulated Systems

paper_content:

Mathematical description of pulse-modulated systems stability of equilibria - miscellaneous methods stability of equilibria - averaging method stability of processes - averaging method forced periodic oscillations - method of equations of periods oscillations of PWM systems - fixed-point method oscillations of PFM systems - fixed-point method method of harmonic balance and its justification auto-oscillations in pulse modulated systems pulse-width modulated systems of phase synchronization.
---
paper_title: Experiments on node-to-node pinning control of Chua's circuits

paper_content:

Abstract In this paper, we study the global intermittent pinning controllability of networks of coupled chaotic oscillators. We explore the feasibility of the recently presented node-to-node pinning control strategy through experiments on Chua’s circuits. We focus on the case of two peer-to-peer coupled Chua’s circuits and we build a novel test-bed platform comprised of three inductorless Chua’s oscillators. We investigate the effect of a variety of design parameters on synchronization performance, including the coupling strength between the oscillators, the control gains, and the switching frequency of node-to-node pinning control. Experimental results demonstrate the effectiveness of this novel pinning control strategy in rapidly taming chaotic oscillator dynamics onto desired reference trajectories while minimizing the overall control effort and the number of pinned network sites. From an analytical standpoint, we present sufficient conditions for global node-to-node pinning controllability and we estimate the maximum switching period for network controllability by adapting and integrating available results on Lyapunov stability theory and partial averaging techniques.
---
paper_title: Input-to-State Stability and Averaging of Linear Fast Switching Systems

paper_content:

We consider the averaging method for stability of rapidly switching linear systems with disturbances. We show that the notions of strong and weak averages proposed, with partial strong average defined in this note, play an important role in the context of switched systems. Using these notions of average, we show that exponential input-to-state stability (ISS) of the strong and the partial strong average system with linear gain imply exponential ISS with linear gain of the actual system. Similarly, exponential ISS of the weak average guarantees an appropriate exponential derivative ISS (DISS) property for the actual system. Moreover, using the Lyapunov method, we show that linear ISS gains of the actual system and its average converge to each other as the switching rate is increased.
---
paper_title: Global pulse synchronization of chaotic oscillators through fast-switching: theory and experiments

paper_content:

Abstract We study pulse synchronization of chaotic systems in master–slave configuration. The slave system is unidirectionally coupled to the master system through an intermittent linear error feedback coupling, whose gain matrix periodically switches among a finite set of constant matrices. Using Lyapunov-stability theory, fast-switching techniques, and the concept of matrix measure, we derive sufficient conditions for global synchronization. The derived conditions are specialized to the case of Chua’s circuits. An inductorless realization of coupled Chua’s circuits is developed to illustrate the effectiveness of the proposed approach.
---
paper_title: Sufficient conditions for fast switching synchronization in time-varying network topologies

paper_content:

In previous work [J. D. Skufca and E. Bollt, Mathematical Biosciences and Engineering, 1 (2004), pp. 347-359], empirical evidence indicated that a time-varying network could propagate sufficient in...
---
paper_title: Averaging of nonsmooth systems using dither

paper_content:

It was shown by Zames and Shneydor and later by Mossaheb that a high-frequency dither signal of a quite arbitrary shape can be used to narrow the effective nonlinear sector of Lipschitz continuous feedback systems. In this paper, it is shown that also discontinuous nonlinearities of feedback systems can be narrowed using dither, as long as the amplitude distribution function of the dither is absolutely continuous and has bounded derivative. The averaged system is proven to approximate the dithered system with an error of the order of dither period.
---
paper_title: Multifrequency averaging of DC/DC converters

paper_content:

This paper presents some of the issues involved in applying frequency-selective averaging to modeling the dynamic behavior of pulsewidth-modulated (PWM) DC-DC converters. We use the boost converter as an example to show the details involved in deriving some novel extended averaged models, and we use simulations to highlight the accuracy of the models even when traditional small-ripple conditions are not satisfied.
---
paper_title: Time-averaging of high-speed data transfer protocols

paper_content:

We propose two modeling approaches of additive-increase/multiplicative-decrease (AIMD) congestion control mechanisms. The first separates the increase and decrease parts whereas the second describes the rate evolution of the congestion window as a continuous process governed by a differential equation. We relate these approaches and show that the second (fluid-flow) model results from deterministic time averaging of the first (discontinuous) model. A generalized class of nonlinear protocols, which includes Floyd's HighSpeed TCP, is then proposed and analyzed. Our findings are validated by simulation.
---
paper_title: Generalized averaging method for power conversion circuits

paper_content:

A more general averaging procedure that encompasses state-space averaging and that is potentially applicable to a much broader class of circuits and systems is presented. Examples of its application in resonant and PWM power convertors are presented. The technique is shown to be effective on a number of examples. including resonant type converters. The approach offers refinements to the theory of state-space averaging, permitting a framework for analysis and design when small ripple conditions do not hold. The method may find applications in simulation and design since it is considerably easier to simulate an averaged model than a switched model. >
---
paper_title: Nonlinear averaging applied to the control of pulse width modulated (PWM) pneumatic systems

paper_content:

This work presents a control methodology that enables nonlinear model based sliding mode control of PWM-controlled pneumatic servo actuators. An averaging approach is developed to describe the equivalent continuous-time dynamics of a PWM controlled nonlinear system, which renders the originally discontinuous system, possibly non-affine in the input, into an equivalent system that is in control canonical nonlinear form. A sliding mode controller is then developed for a pneumatic servo actuator based on the averaged equivalent control canonical model. The controller is implemented on an experimental system, and the effectiveness of the proposed approach validated by experimental tracking data.
---
paper_title: Averaging of nonsmooth systems using dither

paper_content:

It was shown by Zames and Shneydor and later by Mossaheb that a high-frequency dither signal of a quite arbitrary shape can be used to narrow the effective nonlinear sector of Lipschitz continuous feedback systems. In this paper, it is shown that also discontinuous nonlinearities of feedback systems can be narrowed using dither, as long as the amplitude distribution function of the dither is absolutely continuous and has bounded derivative. The averaged system is proven to approximate the dithered system with an error of the order of dither period.
---
paper_title: Continuous-Time Stochastic Averaging on the Infinite Interval for Locally Lipschitz Systems

paper_content:

We investigate stochastic averaging on the infinite time interval for a class of continuous-time nonlinear systems with stochastic perturbation and remove or weaken several restrictions present in existing results: global Lipschitzness of the nonlinear vector field, equilibrium preservation under the stochastic perturbation, global exponential stability of the average system, and compactness of the state space of the perturbation process. If an equilibrium of the average system is exponentially stable, we show that the original system is exponentially practically stable in probability. If, in addition, the original system has the same equilibrium as the average system, then the equilibrium of the original system is locally asymptotically stable in probability. These results extend the deterministic general averaging for aperiodic functions to the stochastic case.
---
paper_title: Blinking model and synchronization in small-world networks with a time-varying coupling

paper_content:

The paper proposes a new type of small-world networks of cells with chaotic behavior. This network consists of a regular lattice of cells with constant 2K-nearest neighbor couplings and time-dependent on–off couplings between any other pair of cells. In each time interval of duration τ such a coupling is switched on with probability p and the corresponding switching random variables are independent for different links and for different times. At each moment, the coupling structure corresponds to a small-world graph, but the shortcuts change from time interval to time interval, which is a good model for many real-world dynamical networks. It is to be distinguished from networks that have randomly chosen shortcuts, fixed in time. Here, we apply the Connection Graph Stability method, developed in the companion paper (“Connection graph stability method for synchronized coupled chaotic systems”, see this issue), to the study of global synchronization in this network with the time-varying coupling structure, in the case where the on–off switching is fast with respect to the characteristic synchronization time of the network. The synchronization thresholds are explicitly linked with the average path length of the coupling graph and with the probability p of shortcut switchings in this blinking model. We prove that for the blinking model, a few random shortcut additions significantly lower the synchronization threshold together with the effective characteristic path length. Short interactions between cells, as in the blinking model, are important in practice. To cite prominent examples, computers networked over the Internet interact by sending packets of information, and neurons in our brain interact by sending short pulses, called spikes. The rare interaction between arbitrary nodes in the network greatly facilitates synchronization without loading the network much. In this respect, we believe that it is more efficient than a structure of fixed random connections.
---
paper_title: Averaging of nonsmooth systems using dither

paper_content:

It was shown by Zames and Shneydor and later by Mossaheb that a high-frequency dither signal of a quite arbitrary shape can be used to narrow the effective nonlinear sector of Lipschitz continuous feedback systems. In this paper, it is shown that also discontinuous nonlinearities of feedback systems can be narrowed using dither, as long as the amplitude distribution function of the dither is absolutely continuous and has bounded derivative. The averaged system is proven to approximate the dithered system with an error of the order of dither period.
---
paper_title: Averaging of nonsmooth systems using dither

paper_content:

It was shown by Zames and Shneydor and later by Mossaheb that a high-frequency dither signal of a quite arbitrary shape can be used to narrow the effective nonlinear sector of Lipschitz continuous feedback systems. In this paper, it is shown that also discontinuous nonlinearities of feedback systems can be narrowed using dither, as long as the amplitude distribution function of the dither is absolutely continuous and has bounded derivative. The averaged system is proven to approximate the dithered system with an error of the order of dither period.
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Title: An Overview on Averaging for Pulse-Modulated Switched Systems

Section 1: INTRODUCTION
Description 1: Introduce the concept of switched systems, highlight the challenges in their analysis, and provide an overview of the averaging approach and its applications.

Section 2: HIGH FREQUENCY MODULATED SWITCHED SYSTEMS
Description 2: Describe the mathematical modeling of high-frequency modulated switched systems, including the definitions, assumptions, and representation of the switching function and cycle-time.

Section 3: AVERAGING OVER THE CYCLE-TIME
Description 3: Explain the averaging technique for switched systems with constant cycle-time, including the duty ratio functions, averaged system definitions, and stability results.

Section 4: Dynamic phasor model
Description 4: Introduce the dynamic phasor model as a generalization of the averaged model and discuss its implications and applications.

Section 5: AVERAGING WITH STATE-DEPENDENT SWITCHINGS
Description 5: Review the averaging results for switched systems where the switching function depends on the state and low-frequency inputs, and introduce partial averaged systems.

Section 6: Averaging through dithering
Description 6: Discuss the use of dithering in continuous pulse-width modulated systems, including the definition, behavior, and impact on system stability.

Section 7: Generalized averaging for hybrid systems
Description 7: Extend the averaging results for switched systems to a broader class of hybrid systems, highlighting solution closeness and practical stability conditions.

Section 8: CONCLUSIONS
Description 8: Summarize the findings, identify open issues and gaps in the research on averaging for nonsmooth systems, and suggest possible directions for future research.