Alejandro J. Rojas

Output feedback stabilisation over bandwidth limited, signal to noise ratio constrained communication channels

Stabilisability of an open loop unstable plant is studied under the presence of a bandwidth limited additive coloured noise communication channel with constrained signal to noise ratio. The problem is addressed through the use of an LTI filter explicitly modelling the bandwidth limitation, and another LTI filter to model the additive coloured noise. Results in this paper show that a bandwidth limitation increases the minimum value of signal to noise ratio required for stabilisability, in comparison to the infinite bandwidth, white noise case. Examples are used to illustrate the results in the continuous and discrete framework

Feedback Stabilization Over a First Order Moving Average Gaussian Noise Channel

Recent developments in information theory by Y.-H. Kim have established the feedback capacity of a first order moving average additive Gaussian noise channel. Separate developments in control theory have examined linear time invariant feedback control stabilization under signal to noise ratio (SNR) constraints, including colored noise channels. This note considers the particular case of a minimum phase plant with relative degree one and a single unstable pole at z=phi (with |phi| > 1) over a first order moving average Gaussian channel. SNR constrained stabilization in this case is possible precisely when the feedback capacity of the channel satisfies CFB ges log2 |phi|. Furthermore, using the results of Kim we show that there exist linear encoding and decoding schemes that achieve stabilization within the SNR constraint precisely when CFB ges log2 |phi|.

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.

Control over a Bandwidth Limited Signal to Noise Ratio constrained Communication Channel

Stabilisability of a minimum phase unstable continuous plant is studied under the presence of a bandwidth limited and Signal to Noise Ratio constrained communication link. The problem is addressed in two different ways: first through the use of an LTI filter explicitly modelling the bandwidth limitation, and in second place, for the case of one real unstable pole, through the Poisson Integral Formula and design attenuation requirements on the power outside the assigned bandwidth. Results show that when a bandwidth limitation is in existence this increases the minimum value of Signal to Noise Ratio required for stabilisability. An example is used to study both approaches.

Optimal Signal to Noise Ratio in Feedback over Communication Channels with Memory

Communication channels impose a number of obstacles to feedback control, such as delay, noise, and constraints in communication data-rate. One alternate line of recent work considers the problem of feedback stabilization subject to a constraint in the signal-to-noise ratio (SNR). It has been shown for continuous-time systems that the optimal control problem arising in achieving minimal SNR can be formulated as a linear quadratic Gaussian (LQG) control problem with weights chosen as in the loop transfer recovery (LTR) technique. The present paper extends such LQG/LTR formulation to discrete-time systems with feedback over channels with memory. By using such formulation, we derive exact expressions for the LTI controller and loop sensitivity functions that achieve minimal SNR under the effect of time-delay, non minimum phase zeros and colored additive noise. For the minimum-phase case with white noise and no time delay, we show that the optimal feedback loop obtained after applying LTR has a structure equivalent to that of a communication channel with feedback from the output to the input

Signal-to-Noise Ratio Limited Output Feedback Control Subject to Channel Input Quantization

Communication channels impose a number of obstacles to feedback control, such as delay, noise, and constraints in the communication transmission data rate. One line of research considers the problem of finding a stabilizing feedback controller for a linear time invariant (LTI) plant subject to a constraint on the signal-to-noise-ratio (SNR) of a channel located in the same loop. The present technical note extends such formulation by considering logarithmic or uniform quantization at the channel input.

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 L 2 norm of a partial fraction expansion with repeated poles in the Laplace domain. We then use the squared L 2 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.

Input disturbance rejection in channel signal-to-noise ratio constrained feedback control

Communication channels impose a number of obstacles to feedback control. One recent line of work considers the problem of feedback stabilisation subject to a constraint on the channel signal-to-noise ratio (SNR). It has been shown for continuous-time systems that the optimal control problem of achieving the infimal SNR can be formulated as a linear quadratic Gaussian (LQG) control problem with weights chosen as in the loop transfer recovery (LTR) technique. The present paper extends this formulation to: discrete- time systems; communications over channels with memory; and input disturbance rejection. By using this formulation, we derive exact expressions for the linear time invariant (LTI) controller that achieves the infimal SNR under the effect of time-delay and additive coloured noise. We then quantify the infimal SNR required for both stabilisation and input disturbance rejection for a relative degree one, minimum phase plant and a memoryless Gaussian channel.

Closed-form solution for a class of discrete-time algebraic Riccati equations

In the present paper we obtain a closed-form solution for the class of discrete-time algebraic Riccati equations (ARE) with vanishing state weighting, whenever the unstable eigenvalues are distinct. The AREs in such a class solve a minimum energy control problem for a single-input single-output (SISO) system. The obtained closed-form solution gives insight on issues such as loss of controllability and it might also prove competitive in terms of numerical precision over current solving algorithms.

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.