Formal Non-Fragile Stability Verification of Digital Control Systems with Uncertainty

Abstract

A verification methodology is described and evaluated to formally determine uncertain linear systems stability in digital controllers with considerations to the implementation aspects. In particular, this methodology is combined with the digital-system verifier (DSVerifier), which is a verification tool that employs Bounded Model Checking based on Satisfiability Modulo Theories to check the stability of digital control systems with uncertainty. DSVerifier determines the control system stability, considering all the plant interval variation set, together with the Finite Word-length (FWL) effects in the digital controller implementation; DSVerifier checks the robust non-fragile stability of a given closed-loop system. The proposed methodology and respective tool are evaluated considering non-fragile control examples from literature. Experimental results show that the approach used in this study is able to foresee fragility problems in robust controllers, which could be overlooked by other existing approaches due to underestimating of FWL effects.

Sound and Automated Synthesis of Digital Stabilizing Controllers for Continuous Plants

“…The uncertain model may be rewritten as a vector of coefficients in the z-domain using equation (8) as Ĝ = G + ∆p G. The parametric uncertainties in the plant are assumed to have the same order as the plant model, since errors of higher order can move the closed-loop poles by large amounts, thus preventing any given controller from stabilizing such a setup. This is a reasonable assumption since most tolerances do not change the architecture of the plant.…”

“…Proposition 1. [8,15] Consider a feedback closed-loop control system as given in Figure 1 with a FWL implementation of the digital controllerC(z) = FM C ,N C I, F (C(z)) and uncertain discrete model of the plant from (6), (7)…”

“…Next, we describe the specific property that we pass to the program synthesizer as the specification σ. There are a number of algorithms in our verification engine that can be used for stability analysis [7,8]. Here we choose Jury's criterion [4] in view of its efficiency and ease of integration within DSSynth: we employ this method to check the stability in the z-domain for the characteristic polynomial S(z) defined in (16).…”

DSSynth: An automated digital controller synthesis tool for physical plants

“…Our evaluation consists of 18 SISO control system benchmarks extracted from the literature [7], [8], [14], [15], [20]- [28], which include physical plants for an unmanned aerial Step Response Time (seconds) Amplitude Fig. 9:…”

Verifying digital systems with MATLAB

“…In order to detect the mentioned errors in digital systems, a model-checking procedure based on Boolean Satisfiability (SAT) and Satisfiability Modulo Theories (SMT) has been proposed, named as Digital-System Verifier (DSVerifier) [6]. DSVerifier checks specific properties related to overflow, limit cycle, stability, and minimum-phase, in digitalsystem implementations [2], and also supports the verification of robust stability, considering parametric uncertainties for closed-loop systems represented by transfer functions [7]. Recently, DSVerifier was extended to support state-space systems, considering single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems [8], in order to verify violations in stability, controllability, observability, and quantization-error properties.…”