Ernesto Kofman and Julio H. Braslavsky, Level Crossing Sampling in Feedback Stabilization Technical Report EE06010, School of Electrical Engineering and Computer Science, The University of Newcastle, Australia,
Abstract: This paper introduces a novel event-driven sampled-data feedback scheme based on hysteretic quantization. In the proposed sampling scheme, the plant output samples are triggered by the crossings---with hysteresis---of the signal through its quantization levels. The plant and controller communicate over binary channels that operate asynchronously and are assumed to be error and delay--free. The paper proposes two systematic output feedback control design strategies. The first strategy consists in the digital emulation of a previously designed analog controller. If such analog controller achieves closed-loop asymptotic stability, the proposed emulation design guarantees closed-loop practical stability of the resulting asynchronous sampled-data system. The second design strategy is a simple direct design that drives the plant state to the origin in finite time after a total transmission of $2n+2$ bits, where $n$ is the order of the plant. These results exhibit the potential of the proposed scheme for the development of general tools for the analysis and design of practical event-driven sampled-data control systems.
J.H. Braslavsky, R.H. Middleton and J.S. Freudenberg Effects of Time Delay on Feedback Stabilization over Signal-to-Noise Ratio Constrained Channels. Technical Report EE04019, ARC Centre for Complex Dynamic Systems and Control, The University of Newcastle, Australia, September 2004.
Abstract: The expanding integration of control and communication networks in recent years has generated an increasing interest in control problems with feedback over a communication channel. Significant research activity has concentrated on stabilisation in face of channel effects such as quantisation and data-rate limits. In a recent paper, the authors have studied the problem of feedback stabilisation over a communication channel with a constraint on the admissible signal-to-noise ratio (SNR). It has been shown therein that for a delay-free, linear time-invariant feedback loop, a SNR constraint in the feedback channel imposes fundamental limitations in the ability to achieve closed-loop stability. The present paper extends these results by introducing a time-delay in the loop, and shows that the lowest SNR required for closed-loop stability increases by a factor that may grow exponentially with the time-delay and the unstable open loop poles of the system. This result contributes to the quantification of performance tradeoffs in integrated control and communication environments.
R.H. Middleton, J.H. Braslavsky and J.S. Freudenberg Stabilization of Non-Minimum Phase Plants over Signal-to-Noise Ratio Constrained Channels. Proceedings of the 5th Asian Control Conference, Melbourne, Australia, 20-23 July 2004.
Abstract: We recently considered feedback stabilization over a Signal to Noise Ratio (SNR) constrained channel. The results were examined for the state feedback and minimum phase cases, with links to bit-rate limited control. In this paper, we extend this analysis to Non-Minimum Phase (NMP) plants, and show that for Linear Time Invariant (LTI) control, NMP zeros further constrain the ability to stabilize over an SNR limited channel. This differs from the situation of bit-rate limited stabilization where NMP zeros do not play a role. We show that by considering linear time varying feedback the effect of NMP zeros in SNR limited stabilization may be eliminated.
J.H. Braslavsky, R.H. Middleton and J.S. Freudenberg Feedback Stabilization over Signal-to-Noise Ratio Constrained Channels. To appear at the 2004 American Control Conference.
Abstract: There has recently been significant interest in feedback stabilization problems over communication channels, including several with bit rate limited feedback. Motivated by considering one source of such bit rate limits, we study the problem of stabilization over a signal-to-noise ratio (SNR) constrained channel. We discuss both continuous and discrete time cases, and show that for either state feedback; or for output feedback delay-free, minimum phase plants, there are limitations on the ability to stabilize an unstable plant over an SNR constrained channel. These limitations in fact match precisely those that might have been inferred by considering the associated ideal Shannon capacity bit rate over the same channel.
K. Lau, R. Middleton and J.H. Braslavsky. Undershoot and settling time trade-offs for nonlinear non-minimum phase systems. To appear in the IEEE Transactions on Automatic Control.
Abstract: In the case of linear systems, it has been known for some time that non-minimum phase zeros may imply undershoot in the step response. Bounds on such undershoot depend on the settling time demanded and the zero locations. In this paper we review such constraints for the linear time invariant case, including providing stronger bounds. We also extend these results to the nonlinear case. In particular, using concepts of constrained reachability, we show that scalar separable unstable zero dynamics imply undershoot in the step response. Furthermore, this undershoot cannot be small if a rapid settling time is required and the zero dynamics are slow.
Abstract: Two seemingly independent streams of control systems research have examined logarithmic sensitivity integrals and limiting linear quadratic optimal control problems. These apparently diverse problems yield some results with an identical right hand side. The main contribution of this paper is to directly explain the commonality between these streams. This explanation involves the use of Parseval's theorem to derive tight inequality bounds between frequency domain logarithmic sensitivity integrals, and the achievable quadratic performance of a linear time invariant system.
Abstract: For strict-feedback nonlinear systems, this paper shows that it is impossible to reduce to zero the optimal cost in the regulation of more states than the number of control inputs in the system, even using unrestricted control effort. By constructing a near optimal cheap control law, we characterise the infimum value of the optimal regulation cost as the optimal value of a reduced-order regulator problem where the states with lower relative degree drive those with higher relative degree. We illustrate our results with two examples of practical interest: the optimal regulation of the rotational motion of a free rigid body, and the optimal control of a magnetic suspension system.
Abstract: In this paper it is shown that Iterative Feedback Tuning can be applied to multi-input multi-output linear sampled-data systems where the sensed outputs may be sampled at different rates and where the controllers operate synchronously at one sampling rate which is a multiple of the sampling rates of the controlled variables. It is also shown that generalized sampled-data hold functions (GSDHF) can be tuned. By way of an example, it is shown that one case where GSDHF's may improve the performance significantly is when the continuous time system is minimum phase but the sampled counterpart is non-minimum phase.
Abstract: This paper shows that there exists a structural limitation in reducing the integral of the squared regulation error in strict-feedback nonlinear systems with more outputs (that is, performance variables) than inputs. This limitation arises from the fact that, in these systems, some outputs are controlled only indirectly through other outputs with lower relative degree. It is then impossible to uniformly reduce the overall regulation cost to zero, the infimum value of which is characterised by a reduced-order optimal regulation problem. Although essentially nonlinear, the results are similar in nature and have direct connections with their linear counterparts.
Abstract: In Technical Report EE98024, the authors present a closed-form expression for the quantification of model error in transfer function estimation by characterising undermodelling by a random walk process in the frequency domain. The present note shows that, for systems with resonant modes, less conservative model error quantifications may be obtained by alternatively characterising undermodelling by an integrated random walk process in the frequency domain.
Abstract: This paper presents a consistent framework for the quantification of noise and undermodelling errors in transfer function model estimation. We use the, so-called, ``stochastic embedding'' approach, in which both noise and undermodelling errors are treated as stochastic processes. In contrast to previous applications of stochastic embedding, in this paper we represent the undermodeling as a multiplicative error characterised by random walk processes in the frequency domain. The benefit of the present formulation is that it significantly simplifies the estimation of the parameters of the embedded process yielding a closed-form expression for the model error quantification. An example illustrates how the random walk effectively captures typical cases of undermodelling found in practice.
Abstract: We study the lowest achievable mean square estimation error in two limiting optimal linear filtering problems. First, when the intensity of the process noise tends to zero, the lowest achievable mean square estimation error is a function of the unstable poles of the system. Second, when the intensity of the measurement noise tends to zero, the lowest achievable mean square estimation error is a function of the nonminimum phase zeros of the system. We link these results with Bode integral characterizations of performance limitations in linear filtering.
Abstract: Feedback limitations of nonlinear systems are investigated using the cheap control approach. The main result is that in the limit, when the control effort is free, the smallest achievable L_2 norm of the output is equal to the least amount of control energy (L_2 norm) needed to stabilize the unstable zero dynamics. This nonlinear result is structurally similar to an earlier linear result by Qiu & Davison (1993), which, in turn, is connected with a Bodetype integral derived by Middleton (1991).