Stochastic hybrid systems: A modeling and stability theory tutorial

Teel, Andrew R.; Hespanha, João P.

doi:10.1109/cdc.2015.7402688

Cited by 23 publications

(28 citation statements)

References 37 publications

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“…where Π is the unique solution of (5) and M is given by equation 10, is a robust reduced order model of system (5) at (L, S) for any fixed G such that σ(S) ∩ σ(S − GL) = ∅ and the pair (S − GL, CΠ) is observable. Proof: For model (6), F , G, H and M are free parameters which must be used to solve the conditions (7), (8), (9).…”

Section: Moment Matching With Additive Disturbancesmentioning

confidence: 99%

“…They are also useful to model systems with uncertain right-hand side, such as systems in which some of the parameters are known to belong to a certain set but cannot be determined precisely [2]. Recently, differential inclusions have also being used in the literature of hybrid systems and stochastic hybrid systems to model non-unique solutions [4], [5]. Computing the solutions of differential inclusions is numerically challenging since each initial condition can produce a different set of dense solutions.…”

Section: Introductionmentioning

confidence: 99%

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Model reduction for linear differential inclusions: moment-set and time-variance

Scarciotti

Teel

Astolfi

2017

2017 American Control Conference (ACC)

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This paper deals with the problem of model reduction by moment matching for linear differential inclusions. The problem is formally formulated and the notions of momentset, perturbed moment trajectory, approximate reduced order model and robust reduced order model are introduced. Two sets of results are presented. The first part of the paper deals with robustness of the reduced order models with respect to input perturbations. Exploiting this result an enhanced model reduction scheme for linear differential equations is presented. In the second part of the paper we focus on the problem of model reduction by moment matching for time-varying systems driven by time-varying signal generators. Finally, these two sets of results are used to solve the problem of model reduction for linear differential inclusions. The results are illustrated by means of numerical examples.

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Section: Moment Matching With Additive Disturbancesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Model reduction for linear differential inclusions: moment-set and time-variance

Scarciotti

Teel

Astolfi

2017

2017 American Control Conference (ACC)

Self Cite

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“…Randomly switching dynamical systems have generated much interest in the probability literature recently [2-4, 7-9, 21, 29, 30, 48] and are involved in a growing number of applications to biology [11,12,20,52], physics [6,18], engineering [25,31,53], and finance [57]. Due to the diversity of the groups studying these switching systems, they have been given several names, including stochastic hybrid systems, piecewise deterministic Markov processes, dichotomous Markov noise processes, velocity jump processes, and random evolutions.…”

Section: Introductionmentioning

confidence: 99%

Blowup from randomly switching between stable boundary conditions for the heat equation

Lawley

2018

Communications in Mathematical Sciences

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We find a pair of boundary conditions for the heat equation such that the solution goes to zero for either boundary condition, but if the boundary condition randomly switches, then the average solution grows exponentially in time. Specifically, we prove that the mean of the random solution grows exponentially under certain mild assumptions, and we use formal asymptotic methods to argue that the random solution grows exponentially almost surely. To our knowledge, this could be the first PDE example showing that randomly switching between two globally asymptotically stable systems can produce a blowup. We devise several methods to analyze this random PDE. First, we use the method of lines to approximate the switching PDE by a large number of switching ODEs and then apply recent results to determine if they grow or decay in the limit of fast switching. We then use perturbation theory to obtain more detailed information on the switching PDE in this fast switching limit. To understand the case of finite switching rates, we characterize the parameter regimes in which the first and second moments of the random PDE grow or decay. This moment analysis reveals rich dynamical behavior, including a region of parameter space in which the mean of the random PDE oscillates with ever increasing amplitude for slow switching rates, grows exponentially for fast switching rates, but decays to zero for intermediate switching rates. We also highlight cases in which the second moment is necessary to understand the switching system's qualitative behavior, rather than just the mean. Finally, we give a PDE example in which randomly switching between two unstable systems produces a stable system. All of our analysis is accompanied by numerical simulation.

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“…A controlled hybrid automaton, a subset of Branicky's model, is a hybrid automaton, whose underlying continuous dynamics are modelled by inhomogeneous ordinary differential equations. More advanced hybrid systems were introduced in recent research work, such as stochastic hybrid systems [13].…”

Section: Introductionmentioning

confidence: 99%

Biped Robot Modeling and Control Using Controlled Hybrid Automata

Shang¹

2017

IJSSAM

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Hybrid systems are dynamical systems consisting of interacting discrete event and continuous state subsystems. A controlled hybrid automaton is a hybrid automaton whose continuous-state dynamics are described by inhomogeneous differential equations. This paper presents a sufficient condition for the existence of global non-terminating solutions in controlled hybrid automata. The condition is based on a recursive algorithm that can always terminate after a finite number of iterations to a limit set of states, i.e. the fixed point of the recursion. If the fixed point is non-empty, then there exists a measurable control under which the hybrid automaton generates a global non-terminating solution. The more important is that this result can also be used to infer the existence of global solutions to compositions of controlled hybrid automata, thereby providing a foundation for the analysis of large scale hybrid systems. The controlled hybrid automata model can be used for robotics system modeling and control. By solving the global non-terminating solution to controlled hybrid automata, the biped robots can be guaranteed to keep the walking gait without falling down.

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Stochastic hybrid systems: A modeling and stability theory tutorial

Cited by 23 publications

References 37 publications

Model reduction for linear differential inclusions: moment-set and time-variance

Model reduction for linear differential inclusions: moment-set and time-variance

Blowup from randomly switching between stable boundary conditions for the heat equation

Biped Robot Modeling and Control Using Controlled Hybrid Automata

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