TY - JOUR

T1 - Computing compositional proofs of Input-to-Output Stability using SOS optimization and δ-decidability

AU - Murthy, Abhishek

AU - Islam, Md Ariful

AU - Smolka, Scott A.

AU - Grosu, Radu

N1 - Funding Information:
We would like to thank the anonymous reviewers for their valuable comments. Research supported in part by NSF grants CCF-0926190 , CCF-1018459 , CPS-1446832 and CPS-1446725 , and by AFOSR grant FA0550-09-1-0481 .
Publisher Copyright:
© 2016 Elsevier Ltd

PY - 2017/2/1

Y1 - 2017/2/1

N2 - We present BFComp, an automated framework based on Sum-Of-Squares (SOS) optimization and δ-decidability over the reals, to compute Bisimulation Functions (BFs) that characterize Input-to-Output Stability (IOS) of dynamical systems. BFs are Lyapunov-like functions that decay along the trajectories of a given pair of systems, and can be used to establish the stability of the outputs with respect to bounded input deviations. In addition to establishing IOS, BFComp is designed to provide tight bounds on the squared output errors between systems whenever possible. For this purpose, two SOS optimization formulations are employed: SOSP 1, which enforces the decay requirements on a discretized grid over the input space, and SOSP 2, which covers the input space exhaustively. SOSP 2 is attempted first, and if the resulting error bounds are not satisfactory, SOSP 1 is used to compute a Candidate BF (CBF). The decay requirement for the BFs is then encoded as a δ-decidable formula and validated over a level set of the CBF using the dReal tool. If dReal produces a counterexample containing the states and inputs where the decay requirement is violated, this pair of vectors is used to refine the input-space grid and SOSP 1 is iterated. By computing BFs that appeal to a small-gain theorem, the BFComp framework can be used to show that a subsystem of a feedback-composed system can be replaced–with bounded error–by an approximately equivalent abstraction, thereby enabling approximate model-order reduction of dynamical systems. The BFs can then be used to obtain bounds on the error between the outputs of the original system and its reduced approximation. To this end, we illustrate the utility of BFComp on a canonical cardiac-cell model, showing that the four-variable Markovian model for the slowly activating Potassium current IKs can be safely replaced by a one-variable Hodgkin–Huxley-type approximation. In addition to a detailed performance evaluation of BFComp, our case study also presents workarounds for systems with non-polynomial vector fields, which are not amenable to standard SOS optimizers.

AB - We present BFComp, an automated framework based on Sum-Of-Squares (SOS) optimization and δ-decidability over the reals, to compute Bisimulation Functions (BFs) that characterize Input-to-Output Stability (IOS) of dynamical systems. BFs are Lyapunov-like functions that decay along the trajectories of a given pair of systems, and can be used to establish the stability of the outputs with respect to bounded input deviations. In addition to establishing IOS, BFComp is designed to provide tight bounds on the squared output errors between systems whenever possible. For this purpose, two SOS optimization formulations are employed: SOSP 1, which enforces the decay requirements on a discretized grid over the input space, and SOSP 2, which covers the input space exhaustively. SOSP 2 is attempted first, and if the resulting error bounds are not satisfactory, SOSP 1 is used to compute a Candidate BF (CBF). The decay requirement for the BFs is then encoded as a δ-decidable formula and validated over a level set of the CBF using the dReal tool. If dReal produces a counterexample containing the states and inputs where the decay requirement is violated, this pair of vectors is used to refine the input-space grid and SOSP 1 is iterated. By computing BFs that appeal to a small-gain theorem, the BFComp framework can be used to show that a subsystem of a feedback-composed system can be replaced–with bounded error–by an approximately equivalent abstraction, thereby enabling approximate model-order reduction of dynamical systems. The BFs can then be used to obtain bounds on the error between the outputs of the original system and its reduced approximation. To this end, we illustrate the utility of BFComp on a canonical cardiac-cell model, showing that the four-variable Markovian model for the slowly activating Potassium current IKs can be safely replaced by a one-variable Hodgkin–Huxley-type approximation. In addition to a detailed performance evaluation of BFComp, our case study also presents workarounds for systems with non-polynomial vector fields, which are not amenable to standard SOS optimizers.

KW - Approximate bisimulation

KW - Cardiac cell model

KW - Ionic channel

KW - Model-order reduction

UR - http://www.scopus.com/inward/record.url?scp=84967017610&partnerID=8YFLogxK

U2 - 10.1016/j.nahs.2016.03.008

DO - 10.1016/j.nahs.2016.03.008

M3 - Article

AN - SCOPUS:84967017610

SN - 1751-570X

VL - 23

SP - 272

EP - 286

JO - Nonlinear Analysis: Hybrid Systems

JF - Nonlinear Analysis: Hybrid Systems

ER -