J. S. Freudenberg

Inherent design limitations for linear sampled-data feedback systems

There is a well-developed theory describing inherent design limitations for linear time invariant feedback systems consisting of an analogue plant and analogue controller. This theory describes limitations on achievable performance present when the plant has non-minimum phase zeros, unstable poles, and/or time delays. The parallel theory for linear time invariant discrete time systems is less interesting because it describes system behaviour only at sampling instants. This paper develops a theory of design limitations for sampled-data feedback systems wherein the response of the analogue system output is considered. This is done using the fact that the steady-state response of a hybrid feedback system to a sinusoidal input consists of a fundamental component at the frequency of the input together with infinitely many harmonics at frequencies spaced integer multiples of the sampling frequency away from the fundamental. This fact allows fundamental sensitivity and complementary sensitivity functions that re...

Fundamental design limitations of the general control configuration

The theory of fundamental design limitations is well understood for the case that the performance variable is measured for feedback. In the present paper, we extend the theory to systems for which the performance variable is not measured. We consider only the special case for which the performance and measured outputs and the control and exogenous inputs are all scalar signals. The results of the paper depend on the control architecture, specifically, on the location of the sensor relative to the performance output, and the actuator relative to the exogenous input. We show that there may exist a tradeoff between disturbance attenuation and stability robustness that is in addition to the tradeoffs that exist when the performance output is measured. We also develop a set of interpolation constraints that must be satisfied by the disturbance response at certain closed right half plane poles and zeros, and translate these constraints into generalizations of the Bode and Poisson sensitivity integrals. In the absence of problematic interpolation constraints we show that there exists a stabilizing control law that achieves arbitrarily small disturbance response. Depending on the system architecture, this control law will either be high gain feedback or a finite gain controller that depends explicitly on the plant model. We illustrate the results of this paper with the problem of active noise control in an acoustic duct.

A survey of inherent design limitations

The theory of inherent design limitations provides the basic scientific background for the field of linear feedback control, and all control engineers need to know the basic results from this area. Knowledge of inherent limitations is useful in designing a control law that achieves a reasonable tradeoff between conflicting system goals. More significantly, such knowledge is useful much earlier in the engineering design cycle, in order to configure the plant and specify sensors and actuators so that the resulting feedback control problem is tractable. Our purpose in the paper is to provide an overview of available results on design limitations, showing how they may be used to assess the potential difficulty of a feedback control design. We discuss results applicable to single input single output feedback systems in detail, and describe some extensions to multivariable feedback systems.

Control-Oriented Model of a Dual Equal Variable Cam Timing Spark Ignition Engine

A control-oriented engine model is developed to represent a spark ignited engine equipped with a variable cam timing mechanism over a wide range of operating conditions. Based upon laboratory measurements a continuous, nonlinear, low-frequency phenomenological engine model is developed. With respect to a fixed-cam timing engine, the VCT mechanism alters the mass air flow into the cylinders, the torque response, and the emissions of the engine. The developed model reflects all of these modifications and includes a representation of the breathing process, torque and emission generation, and sensor/actuator dynamics. The model has been validated with engine-dynamometer experimental data and can be used in powertrain controller design and development.

Non-pathological sampling for generalized sampled-data hold functions

Abstract Given a generalized sampled-data hold function (GSHF), we define a frequency dependent response function having many properties analogous to those of a transfer function. In particular, the zeros of the response function have transmission blocking properties. We then study the problem of non-pathological sampling with a GSHF. Sampling is said to be pathological if the discretized version of a stabilizable/detectable continuous time plant is not itself stabilizable and detectable. Sufficient conditions for nonpathological sampling with a zero order hold have long been known. We extend these to the case of a GSHF, and describe the role in non-pathological sampling played by right half plane zeros of the response function. The results are presented for square multivariable linear systems and include a generalization to allow for a time delay.

The minimal signal-to-noise ratio required to stabilize over a noisy channel

We consider the problem of stabilizing a discrete-time plant over a noisy channel that is subject to a signal-to-noise ratio (SNR) constraint. It has previously been shown that a necessary condition for stabilization with linear time-invariant state feedback is that the SNR satisfy a lower bound whose value depends solely upon the unstable plant poles. The same lower bound is also necessary for stabilization using linear time invariant output feedback for a minimum phase plant with relative degree one. Following Nair and Evans, we use the concept of entropy power to show that the same bound is also necessary when nonlinear time-varying control laws are applied. Another previous result is that the aforementioned bound on the SNR is also sufficient for stabilization. We apply the concept of entropy rate to provide an alternate proof of this result that also demonstrates an intimate connection between the discrete Bode sensitivity integral and the problem of disturbance-free estimation

Sensor systems for real‐time feedback control of reactive ion etching

Previous efforts from our group have shown encouraging initial results in stabilizing etch rates versus time during a run by using real‐time, multivariable feedback control (RTC) in an Applied 8300 reactive ion etcher. That work indicated the need for improvements in our sensor systems, both the sensors currently used in feedback control and those monitoring the effects of the control on the wafers being etched. In this article we report on our efforts in the development and improvement of two such sensor systems. The first is an optical emission spectroscopy system which simultaneously measures two emission line intensities for use in actinometry. The second sensor system uses spectral reflectometry data to determine the in situ film thickness, from which we calculate the etch rate. We show examples of RTC using the actinometry sensor system during fluorine‐based polycrystalline silicon etching. The results of using these sensors for RTC are presented by comparing open loop signals with those from real‐t...

Control over Signal-to-Noise Ratio Constrained Channels: Stabilization and Performance

Previous papers have considered the problem of using linear time invariant control to stabilize an unstable plant over a signal-to-noise ratio constrained communication channel, and have shown that this problem reduces to one of minimizing the H2norm of the complementary sensitivity function. Different techniques were used to derive the state and output feedback results, and it is not straightforward to characterize the performance that is achieved when output feedback is used with a nonminimum phase plant. In the present paper, a unified treatment of the state and output feedback cases is obtained by posing the problem as one of LQG optimal control. Doing so allows us both to analyze the achieved performance, in terms of sensitivity reduction, as well as to incorporate performance into the problem statement. When performance is considered, the results have interpretations in terms of Wiener filtering.

Plant directionality, coupling and multivariable loop-shaping

We study the multivariable loop-shaping problem when the plant is ill-conditioned and feedback properties at both plant input and output are of interest. Directionality and coupling are described, using the singular value decomposition and principal angles between high- and low-gain subspaces. Our first results are new first-order approximations to feedback properties at one loop-breaking point in terms of the open loop transfer function at that point. Secondly, we study the relation between feedback properties at different loop-breaking points in the event that the plant is ill-conditioned. The nominal relation is studied first. Then robustness is studied, using a first-order approximation to feedback properties at one point with respect to uncertainty at the other. This analysis yields conditions that must be satisfied by feedback properties at one point to prevent poor feedback properties at the other. Finally, these two sets of results are combined to show how feedback properties at both points may be...

Relations between properties of multivariable feedback systems at different loop-breaking points: Part I

The purpose of this paper is to investigate the relation between feedback properties at the plant output and those at the plant input. It is shown that these may differ greatly when the plant has a high condition number. The structure of a highly conditioned plant is explored to provide insights useful in design.