[Paper Review] Mean-square boundedness of stochastic networked control systems with bounded control inputs

Debasish Chatterjee, Saurabh Amin+3 more|arXiv (Cornell University)|Apr 6, 2010

Stability and Control of Uncertain Systems12 references4 citations

TL;DR

This paper proposes a control policy for marginally stable linear systems with bounded control inputs in networked control settings where control signals are transmitted over a noisy channel. Under mild noise assumptions and sufficient control authority, the authors prove that the closed-loop system's state remains mean-square bounded, extending prior results to stochastic, noisy communication channels with hard input constraints.

ABSTRACT

We consider the problem of controlling marginally stable linear systems using bounded control inputs for networked control settings in which the communication channel between the remote controller and the system is unreliable. We assume that the states are perfectly observed, but the control inputs are transmitted over a noisy communication channel. Under mild hypotheses on the noise introduced by the control communication channel and large enough control authority, we construct a control policy that renders the state of the closed-loop system mean-square bounded.

Motivation & Objective

• To address the challenge of stabilizing marginally stable linear systems under bounded control inputs in networked control systems with unreliable communication channels.
• To extend existing results on mean-square boundedness to scenarios where control signals are corrupted by multiplicative noise in the communication channel.
• To ensure hard bounds on control inputs while maintaining stochastic stability under noisy transmission.
• To generalize prior deterministic results to a stochastic setting with noisy control channels and bounded control authority.

Proposed method

• The authors model the control channel as a noisy channel with multiplicative noise affecting the control input, while sensor-to-controller communication is assumed noiseless.
• They employ a saturation-based control policy that limits control inputs to a predefined bound, ensuring hard constraints are satisfied.
• The analysis uses a subsampled system approach, focusing on the state evolution at periodic intervals to simplify the stochastic dynamics.
• A Lyapunov-type argument is applied to the subsampled process, leveraging the properties of the system's transition matrices and noise bounds.
• The proof relies on verifying conditions for mean-square boundedness using the expected norm growth of the state over time steps.
• The control policy is designed such that the expected norm of the state remains bounded, even under noisy channel effects.

Experimental results

Research questions

• RQ1 Can mean-square boundedness be achieved in linear networked control systems with bounded control inputs when control signals are transmitted over a noisy channel?
• RQ2 How does multiplicative noise in the control channel affect the stability of a system with hard control input constraints?
• RQ3 Under what conditions on system dynamics and control authority can the state remain stochastically bounded despite noisy transmissions?
• RQ4 Can the separation principle from LQG control be adapted to systems with bounded inputs and noisy channels?

Key findings

• The closed-loop system's state is mean-square bounded under the proposed control policy, provided the control authority is sufficiently large.
• The authors establish that mean-square boundedness is achievable even when control inputs are constrained and the control channel introduces multiplicative noise.
• The bound on the expected state norm depends on the system's spectral properties, noise characteristics, and the control input saturation level.
• The result generalizes prior deterministic boundedness results to a stochastic setting with noisy control channels.
• The proof relies on analyzing a subsampled version of the system and verifying two key conditions for boundedness using the expected norm growth.
• The existence of a finite upper bound on the second moment of the state is guaranteed for all initial conditions, under mild assumptions on the noise and system parameters.