Formal Analysis of Linear Control Systems using Theorem Proving

Control systems are an integral part of almost every engineering and physical system and thus their accurate analysis is of utmost importance. Traditionally, control systems are analyzed using paper-and-pencil proof and computer simulation methods, however, both of these methods cannot provide accurate analysis due to their inherent limitations. Model checking has been widely used to analyze control systems but the continuous nature of their environment and physical components cannot be truly captured by a state-transition system in this technique. To overcome these limitations, we propose to use higher-order-logic theorem proving for analyzing linear control systems based on a formalized theory of the Laplace transform method. For this purpose, we have formalized the foundations of linear control system analysis in higher-order logic so that a linear control system can be readily modeled and analyzed. The paper presents a new formalization of the Laplace transform and the formal verification of its properties that are frequently used in the transfer function based analysis to judge the frequency response, gain margin and phase margin, and stability of a linear control system. We also formalize the active realizations of various controllers, like Proportional-Integral-Derivative (PID), Proportional-Integral (PI), Proportional-Derivative (PD), and various active and passive compensators, like lead, lag and lag-lead. For illustration, we present a formal analysis of an unmanned free-swimming submersible vehicle using the HOL Light theorem prover.Comment: International Conference on Formal Engineering Method

Synthesis of Minimal Error Control Software

Software implementations of controllers for physical systems are at the core of many embedded systems. The design of controllers uses the theory of dynamical systems to construct a mathematical control law that ensures that the controlled system has certain properties, such as asymptotic convergence to an equilibrium point, while optimizing some performance criteria. However, owing to quantization errors arising from the use of fixed-point arithmetic, the implementation of this control law can only guarantee practical stability: under the actions of the implementation, the trajectories of the controlled system converge to a bounded set around the equilibrium point, and the size of the bounded set is proportional to the error in the implementation. The problem of verifying whether a controller implementation achieves practical stability for a given bounded set has been studied before. In this paper, we change the emphasis from verification to automatic synthesis. Using synthesis, the need for formal verification can be considerably reduced thereby reducing the design time as well as design cost of embedded control software. We give a methodology and a tool to synthesize embedded control software that is Pareto optimal w.r.t. both performance criteria and practical stability regions. Our technique is a combination of static analysis to estimate quantization errors for specific controller implementations and stochastic local search over the space of possible controllers using particle swarm optimization. The effectiveness of our technique is illustrated using examples of various standard control systems: in most examples, we achieve controllers with close LQR-LQG performance but with implementation errors, hence regions of practical stability, several times as small.Comment: 18 pages, 2 figure

Analysis of control system stability under algorithmic uncertainty

Stability of control systems is one of the central subjects in control theory. The classical asymptotic stability theorem states that the norm of the residual between the state trajectory and the equilibrium is zero in limit. Unfortunately, it does not in general allow computing a concrete rate of convergence particularly due to algorithmic uncertainty which is related to numerical imperfections of floating-point arithmetic. This work proposes to revisit the asymptotic stability theory with the aim of computation of convergence rates using constructive analysis which is a mathematical tool that realizes equivalence between certain theorems and computation algorithms. Consequently, it also offers a framework which allows controlling numerical imperfections in a coherent and formal way. The overall goal of the current study also matches with the trend of introducing formal verification tools into the control theory. Besides existing approaches, constructive analysis, suggested within this work, can also be considered for formal verification of control systems. A computational example is provided that demonstrates extraction of a convergence certificate for example dynamical systems

On the Safety of Connected Cruise Control: Analysis and Synthesis with Control Barrier Functions

Connected automated vehicles have shown great potential to improve the efficiency of transportation systems in terms of passenger comfort, fuel economy, stability of driving behavior and mitigation of traffic congestions. Yet, to deploy these vehicles and leverage their benefits, the underlying algorithms must ensure their safe operation. In this paper, we address the safety of connected cruise control strategies for longitudinal car following using control barrier function (CBF) theory. In particular, we consider various safety measures such as minimum distance, time headway and time to conflict, and provide a formal analysis of these measures through the lens of CBFs. Additionally, motivated by how stability charts facilitate stable controller design, we derive safety charts for existing connected cruise controllers to identify safe choices of controller parameters. Finally, we combine the analysis of safety measures and the corresponding stability charts to synthesize safety-critical connected cruise controllers using CBFs. We verify our theoretical results by numerical simulations.Comment: Accepted to the 62nd IEEE Conference on Decision and Control. 6 pages, 5 figure

A graphical environment to express the semantics of control systems

We present the concept of a unified graphical environment for expressing the semantics of control systems. The graphical control system design environment in Simulink already allows engineers to insert a variety of assertions aimed the verification and validation of the control software. We propose extensions to a Simulink-like environment's annotation capabilities to include formal control system stability, performance properties and their proofs. We provide a conceptual description of a tool, that takes in a Simulink-like diagram of the control system as the input, and generates a graphically annotated control system diagram as the output. The annotations can either be inserted by the user or generated automatically by a third party control analysis software such as IQC$\beta$ or $\mu$-tool. We finally describe how the graphical representation of the system and its properties can be translated to annotated programs in a programming language used in verification and validation such as Lustre or C