Isaac Horowitz

Synthesis of feedback systems with large plant ignorance for prescribed time-domain tolerances†

There is given a minimum-phase plant transfer function, with prescribed bounds on its parameter values. The plant is imbedded in a two-degree-of-freedom feedback system, which is to be designed such that the system time response to a deterministic input lies within specified boundaries. Subject to the above, the design should be such as to minimize the effect of sensor white noise at the input to the plant. This report presents a design procedure for this purpose, based on frequency response concepts. The time-domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop-transmission function L(jω), in the form of continuous curves on the Nichols chart. Properties of L(jω) which satisfy these bounds with minimum effect of sensor white noise are derived. The design procedure is quite transparent, providing the designer with the insight to make necessary tradeoffs, at every step in the design process. The same design philosophy may be used to attenuate...There is given a minimum-phase plant transfer function, with prescribed bounds on its parameter values. The plant is imbedded in a two-degree-of-freedom feedback system, which is to be designed such that the system time response to a deterministic input lies within specified boundaries. Subject to the above, the design should be such as to minimize the effect of sensor white noise at the input to the plant. This report presents a design procedure for this purpose, based on frequency response concepts. The time-domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop-transmission function L(jω), in the form of continuous curves on the Nichols chart. Properties of L(jω) which satisfy these bounds with minimum effect of sensor white noise are derived. The design procedure is quite transparent, providing the designer with the insight to make necessary tradeoffs, at every step in the design process. The same design philosophy may be used to attenuate...

Non-linear design for cost of feedback reduction in systems with large parameter uncertainty †

Feedback systems containing linear, minimum-phase plants with large parameter uncertainty may be designed to achieve specified performance tolerances over the entire range of parameter uncertainty. The principal ‘cost of feedback’ is in the feedback loop bandwidth, which is generally much larger than that of the system as a whole. This makes the system very sensitive to sensor noise and high-frequency parasitics. It is shown how a non-linear ‘first-order reset element’ (FORE) may be used to drastically decrease the feedback loop transmission bandwidth. One is logically led to FORE by simple, linear feedback frequency response concepts. The paper assumes that the primary design problem is to satisfy quantitative response tolerances to command inputs. However, disturbances at the plant are not neglected, but the specification on such disturbances is in the damping of the step response. An important feature of the non-linear design is that the system response to command inputs is almost exactly that of a lin...

Quantitative Synthesis of Uncertain Multiple Input-Output Feedback System,

There is given an n input, n output plant with a specified range of parameter uncertainty and specified tolerances on the n2 system response to command functions and the n2 response to disturbance functions. It is shown how Schauders fixed point theorem may be used to generate a variety of synthesis techniques, for a largo class of such plants. The design guarantees the specifications are satisfied over the range of parameter uncertainty. An attractive property is that design execution is that of successive single-loop designs, with no interaction between them and no iteration necessary. Stability over the range of parameter uncertainty is automatically included. By an additional use of Schauders theorem, these same synthesis techniques can be rigorously used for quantitative design in the same sense as above, for n × n uncertain non-linear plants, even non-linear time-varying plants, in response to a finite number of inputs.

Synthesis of a non-linear feedback system with significant plant-ignorance for prescribed system tolerances†

In the design of a linear feedback system to achieve prescribed response tolerances despite significant plant uncertainty, the principal price paid is in the amplification of sensor noise, which tends to saturate the plant elements. This noise amplification is due to the fixed relation between gain and phase of an analytic function, which forces relatively slow reduction of the loop transmission magnitude, as a function of frequency. A nonlinear element, the Clegg Integrator (C.I.), is used to alleviate this relation, permitting faster reduction of the loop transmission magnitude. The major difficulty is in finding a description of C.I. usable for synthesis. This is done by considering the class of step inputs, and locating the C.I. such that, from the inputs and the system output specifications, the nature of the inputs to the C.I. is known, permitting an equivalent linear characterization. A quantitative design procedure is then available to precisely design to achieve specified tolerances. A design exa...

Optimization of the loop transfer function

Abstract In quantitative feedback synthesis, the objective is to satisfy assigned performance tolerances over given ranges of plant uncertainty and external disturbances. In such linear and non-linear problems, whether single, multiple-loop or multivariable, the synthesis techniques result in frequency-domain bounds ψi(ω) in the complex plane, on the loop transmission functions Li(jω). This paper presents a simple proof that an optimum Li(jω) lies on its ψi(ω) for each ω∊[0, ∞). Also, a numerical technique is presented for deriving any desired approximation to the optimum.

Optimum synthesis of non-minimum phase feedback systems with plant uncertainty†

Linear time-invariant feedback systems in which the constrained plant transfer function has right half-plane zeros, are perforce non-minimum-phase, and their attainable benefits of feedback are inherently restricted. This paper presents criteria for determining whether a given set of performance specifications are achievable and, if so, a synthesis procedure is included for deriving the optimum design, defined as that with an effectively minimum loop transmission bandwidth. The properties of the optimum design are derived and its uniqueness proven, for both the minimum and non-minimum-phase feedback systems.

Optimum loop transfer function in single-loop minimum-phase feedback systems†

Abstract In the single-loop, linear, time-invariant feedback control system with parameter ignorance and/or unwanted disturbances, the sensitivity specification may be formulated in terms of bounds on the acceptable value of the loop transfer function L(s=σ+jω) on the imaginary axis jω, for each ω. Due to the ever-present noise in the feedback return path, it is highly important to satisfy these bounds with an L(jω) whose magnitude is as small as possible at large id. Such an optimum exists, is unique and lies on the associated boundary at each value of ω

Superiority of transfer function over state-variable methods in linear time-invariant feedback system design

The objectives and achievements of state-variable methods in linear time-invariant feedback system synthesis are examined. It is argued that the philosophy and objectives associated with eigenvalue realization by state feedback, with or without observers, are highly naive and incomplete in the practical context of control systems. Furthermore, even the objectives undertaken have not really been attained by the state-variable techniques which have been developed. The extremely important factors of sensor noise and loop bandwidths are obscured by the state-variable formulation and have been ignored in the state-variable literature. The basic fundamental problem of sensitivity in the face of significant plant parameter uncertainty has hardly received any attention. Instead, the literature has concentrated primarily on differential sensitivity functions and even those results are so highly obscured in the state-variable formation as to lead to incorrect conclusions. In contrast, the important practical considerations and constraints have been clearly revealed and considered in the transfer function formulation. Differential sensitivity results are simple and transparent. For single input-output systems, there exists an exact design technique for achieving quantitative sensitivity specifications in the face of significant parameter uncertainty, which is optimum in an important practical sense. This problem is much more difficult and has not been completely solved for multivariable systems, but it has at least been realistically attacked by some transfer function methods. Finally, the concepts of controllability and observability so much emphasized in the state-variable literature are examined. It is argued that their importance in this problem class has been greatly exaggerated. On the one hand, transfer function methods can be used to check for their existence. On the other hand, nothing is lost when they are ignored, if the synthesis problem is treated as one with parameter uncertainty by transfer function methods.

Fundamental theory of automatic linear feedback control systems

The reasons for using feedback are reviewed. The beneficial aspects of feedback are quantitatively expressed in sensitivity functions and noise transmission functions. The physical constraints on the controlled process (or plant) determine the maximum number of independent functions realizable. Any configuration with the same number of degrees of freedom may be used. With this approach the study of conditional feedback, model feedback, combined positive and negative feedback, etc. is of secondary interest. The benefits of feedback are paid for in gain-bandwidth of active elements over and above what is needed to physically do the job. The minimum price that must be paid is independent of configuration. The system with two degrees of freedom is studied in detail. Two methods are presented for the precise design of a system that will be as insensitive as may be desired to large parameter variations. One method uses root-locus techniques and is suitable for systems with a small number of dominant poles and zeros. The second method is based on frequency response and can be used for systems of any complexity. Numerical examples are given.

Improved design technique for uncertain multiple-input-multiple-output feedback systems†

This paper presents a synthesis technique for linear time invariant n×n multiple-input-multiple-output (MIMO) feedback systems with constrained ‘ plant’ P, and output feedback. Due to uncertainty, P is known only to be a member of a set &𝒫 = {P}. It is required for all Pep, the n2 system transfer functions tuv be members of specified sets of acceptable outputs a uv;; u. v+ 1, [tdot] n. The problem is rigorously converted into a number of single-input-single output (SISO) uncertainty problems, whose solutions are guaranteed to solve the original MIMO problem. The technique has several advantages over previous ones : (1) fixed point theory is not needed to rigorously justify the theory—the justification is very simple ; (2) there is significantly less overdesign inherent in the method ; and (3) if arbitrary small sensitivity is desired over arbitrary large bandwidth, then the set p must satisfy certain constraints as 8→∞. It is shown that these constraints are inherent in all linear time invariant compensat...