Infimal Signal-to-noise Ratio Research Articles

Tracking Performance Limitations of Networked Control Systems With Repeated Zeros and Poles

In this article, the performance limitations problem are investigated for a class of signal-input, signal-output networked control systems with repeated zeros and poles. The additive white noise is adopted in the communication channel. The tracking performance limitations are investigated based on a two-degree of freedom (2DOF) controller. The results demonstrate that the multiplicity of nonminimum phase (NMP) zeros and unstable poles of the plant can affect the tracking performance limitations. The explicit quantitative relationship is characterized for the tracking performance limitations, which is derived based on NMP zeros, unstable poles of the plant with corresponding multiplicity, as well as the statistical characteristics of the reference noise and communication noise. The admissible infimal signal-to-noise ratio (SNR) is also obtained for the tracking system by using a 2DOF controller structure, where the admissible infimal SNR can satisfy the stability conditions while achieving tracking performance limitations. Finally, an illustrative example is presented to validate the effectiveness of our proposed scheme.

Signal-to-noise ratio constrained feedback control: Robust stability analysis

The explicit consideration of a communication channel model in a feedback control loop is known to be constrained by a fundamental limitation for stabilizability on the channel signal-to-noise ratio (SNR) when the linear time invariant (LTI) plant model is unstable. The LTI modelling approach for real, usually nonlinear, processes compromises accuracy versus complexity of the resulting model. This in turn introduces a gap between the proposed model and the real process, which is known as the model uncertainty. In this paper we then study SNR limitations by considering the continuous-time scenario and the case of an additive coloured Gaussian noise (ACGN) channel with bandwidth limitation, for which we then quantify the infimal SNR subject to the simultaneous presence of plant, channel and noise model uncertainties. We observe, for the special case of memoryless additive white Gaussian noise (AWGN) channels, that the obtained SNR limitation subject to plant model uncertainty can be redefined as a channel capacity limitation for stabilization.

Step reference tracking in signal-to-noise ratio constrained feedback control

In this paper we address the problem of step reference tracking in the context of signal-to-noise ratio (SNR) constrained feedback control. We study the presence of a channel model in the feedback loop either over the measurement path, between the plant and the controller, or the control path, between the controller and the plant. We start the analysis by considering the memoryless additive white Gaussian noise (AWGN) channel model, and follow-up with the additive coloured Gaussian noise (ACGN) channel with memory model. We observe that the standard SNR constrained feedback loop configuration, when tracking step references, will result in an infinite channel SNR. We show that this situation can be ameliorated by extending the feedback loop configuration with an encoder and decoder. The proposed configurations result in a finite SNR that converges back to the infimal SNR for regulation when the amplitude of the reference signal tends to zero.

Stabilizability of nonminimum phase unstable plants with arbitrary multiplicity over AWGN channels

In the present paper we obtain the infimal signal-to-noise ratio (SNR) required for the stabilizability of a linear output feedback loop over an additive white Gaussian noise (AWGN) channel in closed-form. The focus on AWGN channels allow us to then define the minimal channel capacity required for stabilizability. Finally, the infimal SNR for stabilizability also allow us to identify in closed-form the related stabilizing Hermitian positive semidefinite solution to the continuous-time algebraic Riccati equation of LQ control with vanishing state weight and repeated eigenvalues.

Signal-to-noise ratio fundamental limitations in the discrete-time domain

Fundamental limitations in feedback control are a well established area of research. In recent years it has been extended to the study of limitations imposed by the consideration of a communication channel in the control loop. Previous results characterised these limitations in terms of a minimal data transmission rate necessary for stabilisation. In this paper a signal-to-noise ratio (SNR) approach is used to obtain a tight condition for the linear time invariant output feedback stabilisation of a discrete-time, single-input-single-output (SISO) unstable, non-minimum phase (NMP) plant with arbitrary relative degree over an additive Gaussian coloured noise (ACGN) communication channel with memory. The obtained result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros and relative degree as well as the channel NMP zeros, bandwidth, and noise colouring. We then characterise the output feedback sensitivity function for the infimal SNR solution and follow up by quantifying the extra SNR imposed by suboptimal solutions (for example due to plant modelling errors).

Digital Output Feedback Control over Signal-to-Noise Ratio Limited Communication Channels

AbstractIn this paper we address the problem of finding a stabilizing feedback controller for a linear time invariant (LTI) plant model when measurement is performed over a communication channel model subject to channel input logarithmic quantization and a signal-to-noise ratio (SNR) constraint. The location of the communication channel is therefore on the measurement path, that is between the plant and the controller, nevertheless similar results can be obtained for the alternative location over the control path.In our first approach the communication channel model itself is selected to be an additive white Gaussian noise (AWGN) channel. We then describe the spectral power density of the channel density as a function of the power spectral density of the channel additive white Gaussian noise and the uniform distributed process model arising from a logarithmic quantization. The infimal channel SNR is then obtained through the definition for the channel input power, together with an interpretation of a bound on the logarithmic quantization relative error as a multiplicative modeling error.In our second approach we extend the proposed analysis to the class of additive colored Gaussian noise (ACGN) channels with memory. The infimal SNR is then obtained again through the definition for the channel input power. Finally, an extension to the case of sampled-data systems is proposed.

Infimal SNR for Output Disturbance Rejection

AbstractIn the present paper we address the problem of plant output disturbance rejection in the context of feedback control over a signal-to-noise ratio (SNR) constrained communication channel. We study separately the case of the communication channel when located between the controller and the plant (control path) and when located between the plant and the controller (measurement path). We also study separately the case of a memoryless additive white Gaussian noise channel model and the case of an additive colored Gaussian noise channel with memory. When possible we express the resulting infimal SNR for performance in closed-form, whilst where not possible we use a numerical approach such as linear quadratic Gaussian (LQG) optimization with loop transfer recovery (LTR).

Signal-to-Noise Ratio Fundamental Limitations in Continuous-Time Linear Output Feedback Control

In the present technical note we study the fundamental limitation on stability that arise when an additive coloured Gaussian noise (ACGN) channel is explicitly considered over either the control or measurement paths of a linear time invariant (LTI) feedback loop. By considering a linear setting we can naturally express the fundamental limitation as a lower bound on the channel signal-to-noise ratio (SNR) required for stabilisability. We start by first obtaining a closed-form expression for the squared <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm of a partial fraction expansion with repeated poles in the Laplace domain. We then use the squared <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">L</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> norm result to obtain the closed-form expression for the infimal SNR required for stabilisability. The proposed closed-form includes the case of repeated unstable plant poles and non minimum phase (NMP) zeros.

Comments on "Feedback stabilization over signal-to-noise ratio constrained channels

The above-named paper obtains in closed-form the infimal signal-to-noise ratio (SNR) required for stabilization of finite-dimensional linear time invariant (LTI) feedback loops, in continuous and discrete-time, for state feedback and output feedback control. The objective of the present note is to prove that the discrete-time output feedback infimal SNR result for stabilization reported in [1, Theorem III.2] is missing a term in delta, the component due to the relative degree of the plant model.

Signal-to-noise ratio performance limitations for input disturbance rejection in output feedback control

Communication channels impose a number of obstacles to feedback control. One recent line of research considers the problem of feedback stabilization subject to a constraint on the channel signal-to-noise ratio (SNR). We use the spectral factorization induced by the optimal solution and quantify in closed-form the infimal SNR required for both stabilization and input disturbance rejection for a minimum phase plant with relative degree one and memoryless additive white Gaussian noise (AWGN) channel. Finally we conclude by presenting a closed-form expression of the difference between the infimal AWGN channel capacity for input disturbance rejection and the infimal AWGN channel capacity required only for stabilizability.

Fundamental limitations in control over a communication channel

Fundamental limitations in feedback control is a well established area of research. In recent years it has been extended to the study of limitations imposed by the consideration of a communication channel in the control loop. Previous results characterised these limitations in terms of a minimal data transmission rate necessary for stabilisation. In this paper a signal-to-noise ratio (SNR) approach is used to obtain a tight condition for the linear time invariant output feedback stabilisation of a continuous-time, unstable, non minimum phase (NMP) plant with time-delay over an additive Gaussian coloured noise communication channel. By working on a linear setting the infimal SNR for stabilisability is defined as the infimal achievable H2 norm between the channel noise input and the channel signal input. The result gives a guideline in estimating the severity of the fundamental SNR limitation imposed by the plant unstable poles, NMP zeros, time-delay as well as the channel NMP zeros, bandwidth, and channel noise colouring.

Infimal Feedback Capacity for a Class of Additive Coloured Gaussian Noise Channels

In this paper we consider the infimal signal-to-noise ratio (SNR) required for stabilisation of a linear time invariant (LTI) scalar unstable plant over a class of additive coloured Gaussian noise channels. We apply recent results in the literature to obtain the feedback capacity of such a class of channels. We prove that the infimal SNR constrained LTI solution, when dealing with a scalar unstable plant, does achieve a channel feedback capacity equal to the infimal rate of transmission required for stability. The optimality of such channel feedback capacity is a non trivial result since we consider additive 1st order moving average (MA) and autoregressive moving average (ARMA) coloured noise.