In this note, we address the problem of stabilization of port-Hamiltonian systems via the ubiquitous proportional-integral-derivative (PID) controller. The design is based on passivity theory, hence the first step is to identify all passive outputs of the system, which is the first contribution of the paper. Adding a PID around this signal ensures that the closed-loop system is L 2 -stable for all positive PID gains. Global stability (and/or global attractivity) of a desired constant equilibrium is also guaranteed for a new class of systems for which a Lyapunov function can be constructed. A second contribution is to prove that this class-that is identified via some easily verifiable integrability conditions-is strictly larger than the ones previously reported in the literature. Comparisons of the proposed PID controller with control-by-interconnection passivity-based control are also discussed.

A review of the basic approaches to the artificial hand design is given. The importance of new ideas in this field using progress in automatic control theory is stressed. The most importemt feature of the new hand is the two level control. The movements of the hand can be controlled by signals produced by man as well as by external stimuli. For this purpose simple pressure sensitive transducers are placed in the hand to provide the control signals for reflexive-type movements. An adaptive control circuit is present for automatic weight adjustment. All these features are incorporated into the hand using simple and standard servocomponents. Use of appropriate feedback loops eliminates the need for complicated mechanical parts.

This paper deals with further advances of the author's recent work on optimal design of control systems and Wiener filters. Specifically, we consider the problem of designing a system to control a plant when (1) not all state variables are measurable, (2) the measured state variables are contaminated with noise, and (3) there are random disturbances. An explicit design procedure (well adapted to digital computation) is presented. In addition, some fundamental new concepts (controllability, observability, etc.) are introduced. A general theory of control systems is outlined which answers many basic questions (what is controllable? why? how?) and gives a highly efficient method of computation. This paper is to be published in the Proceedings of the First IFAC Moscow Congress by Butterworth Scientific Publications in 1960.

Sampled-data control systems generally have fixed sampling frequencies which must be set high enough to give satisfactory performance for all anticipated conditions. A study is made here of an adaptive system which varies the sampling frequency by measuring a system parameter. It is shown that a sampler followed by a zero-order hold whose sampling period is controlled by the absolute value of the first derivative of the error signal will be a more "efficient" sampler than a fixed-frequency sampler. That is, over a given time interval, fewer samples are needed with the variable-frequency system than with a fixed-frequency system while maintaining essentially the same response characteristics. Analog computer studies of simple Type I and Type II sampled-data servo systems with error sampling and unity feedback verified the method. Standard analog computer components were used to set up a simulated servo system, a rate detector, absolute-value detector, a voltage-controlled oscillator, and a sampler and zero-order hold. The system described reduced the number of samples required for response to a step input to about three-quarters that required in a fixed-sampling-frequency system. Over a long period of time savings in the number of samples required can be expected to be between 25 and 50 per cent. In many applications the savings produced by reducing the over-all number of samples required may outweigh the added complexity of the adaptive-sampling-frequency system.

The increasing applications of numerical control of processes have created a need for new methods of synthesis of control equipment. The method presented is applicable to systems commanded by discretely valued inputs, and processes whose outputs may be similarly quantized. Periodic sampling is not required. The most suitable sampling is by transmission of only significant data, as the new value obtained when the data are changed by a given increment. In certain cases, transmission of data by this means can be used to increase channel capacity. When the data are so quantized, the error signal is constrained to a finite number of discrete values, each of which may be associated with an area in the phase plane. Within each such area, the trajectories of any process subject to phase plane representation are a family of parallel curves. Thus, analytic synthesis may be simplified in the case of certain nonlinear processes. Graphical design is facilitated without requiring deduction of a mathematical representation of the process. The method is illustrated by synthesis of several systems involving a simple linear process.

In digital computations, errors resulting from sampling and amplitude quantization (round off) are unavoidable. This work evaluates the mean-square error caused by sampling and quantization at the output of a linear network which contains a single quantizer. A detailed answer is given to the question, "Given quantized samples of a signal which is a sample function of a random process, what is the optimum linear filter for recovering the signal from its samples?" This filter is determined and its characteristics are summarized graphically for a specific example. A comparison with the conventional hold circuit shows that the optimum filter is much better if high accuracy is required and quantization is coarse. The difference in performance between the two filters is small when the accuracy requirement is low and the quantization is fine. Also included as Appendix V is a survey of the general quantization errors problem, as it appears in the areas of digital computation and numerical analysis, and a study of multiquantizer networks. It is found that extension of the method to networks which contain more than one quantizer is impractical, if not impossible.

The various criteria upon which self-optimizing systems have been based are reviewed, and the operation of each type of system is discussed. A new system which is self-optimizing with respect to a measure of impulse response is described, and experimental results are presented.

A modification of the original Lyapunov stability criterion is given, which includes intermediate conditions of stability, as well as "stability in the small" and "stability in the large." A means is developed for applydng it to practical control problems that eliminates much of the guesswork usually required with Lyapunov methods of investigation. The process is based upon an integration of matrices which solves the linearized problem exactly. It gives sufficient conditions for the stability of nonlinear systems that are always correct for small disturbances and may be exact or conservative for large deviations from equilibrium. The formal procedure admits to enough variation that a wide range of nonlinear problems can be treated. Examples from both continuous and discontinuous feedback systems are given to illustrate its use.

Consider a sampled-data control system with the following sequence of components in the forward path: a sampler with period T , a zero-order hold circuit, a linear amplifier with saturation limits ±1, and a plant with transfer function G(s)=\frac{1}{\Pi\min{i=1}\max{n} (s-\lambda_{i})}. It is assumed that the poles \lambda_{1}, \lambda_{2}, ... , \lambda_{n} of G(s) are real, distinct, and non-positive (a single integral is permissible). The sampler, zero-order hold, and saturating amplifier constrain f(t) , the forcing function of G(s) , to be piecewise constant with values between -1 and +1. The forcing function f(t) is completely defined, for t>0 , by the sequence of numbers f_{1}, f_{2} , ... , where f i is the value of f(t) during the i'th sampling period. The minimal time regulator problem for the above system can then be stated as follows: Given G(s) with an arbitrary set of initial conditions [i.e., the state vector \overrightarrow{c(0)} defined by its components c(0), \dot{c}(0), ... , c^{n-1}(0) ]; find the forcing function f(t) [specified by f_{1}, f_{2}, ... and satisfying |f_{i}| \leq 1] , and the corresponding computer in the feedback loop which will bring the system to equilibrium in the minimum number of sampling periods. Any such forcing function will be called an optimal control. The first step is to consider R_{N}' the set of all initial states \overrightarrow{c(0)} from which the origin can be reached in N sampling periods or less. From this definition all such states are characterized algebraically and geometrically: R_{N}' is shown to be a convex polyhedron with 2 \sum\min{k=1}\max{n} ({N-1}\over{k-1}) vertices. Let R N be the set of all initial states \overrightarrow{c(0)} from which the origin can be reached in N sampling periods and no less. Each point of R N is shown to have a unique canonical representation. The coefficients appearing in the canonical representation suggest an optimal control. To obtain this particular optimal control we define a surface in state space called the critical surface. It is shown that this optimal control will be generated by the following procedure: at the beginning of each sampling period the distance φ from the state of the system to the critical surface is measured along a fixed specified direction; if \phi \geq 1 (or ≤ -1) then the forcing function for that sampling period is +1 (or -1); if |\phi| < 1 , then the forcing function is φ. For a third-order plant it is shown that the critical surface has certain properties which lead to a simple analog computer simulation.

PWM systems contain inherent nonlinearities which arise from their modulation scheme. Thus, for a legitimate study of stability, such systems must be treated as nonlinear sampled-data systems without initially resorting to linear approximations. For a nonlinear system whose dynamic behavior is described by a set of first-order difference equations, one of the theorems in the second method of Lyapunov gives, as a sufficient condition for asymptotic stability in the large, the existence in the whole space of a positive-definite Lyapunov's function V , whose difference \DeltaV is negative definite. Hence, by choosing a positive-definite quadratic form as V , the sufficient condition is reduced to the negative-definiteness in the whole space of \DeltaV . Upon this basis, a systematic procedure of obtaining analytically a sufficient condition for asymptotic stability in the large is developed for various types of PWM systems; the condition is stated as the negativeness of all the eigenvalues of three matrices associated with the PWM system.

The second method of Lyapunov is used to validate Aizerman's conjecture for the class of third-order nonlinear control systems described by the following differential equation: \tdot{e} + a_{2}\ddot{e} + a_{1}\dot{e} + a_{0}e + f(e)=0 In this case, the stability of the nonlinear system may be inferred by considering an associated linear system in which the nonlinear function f(e) is replaced by ke . If the linear system is asymptotically stable for k_{1} < k < k_{2} , then the nonlinear system will be asymptotically stable in-the-large for any f(e) for which k_{1} < \frac{f(e)}{e} < k_{2}. The Lyapunov function used to prove this result is determined in a straightforward manner by considering the physical behavior of the system at the extreme points of the allowable range of k .

A single-loop design is basic for a two-degree-of-freedom plant, and it is theoretically able to achieve any desired insensitivity to plant variations or rejection of disturbances, if the plant is minimum-phase. In exacting feedback problems where the parameter variations or disturbances are large, the resulting single-loop transmission may require a larger gain and bandwidth than that of the plant. The added feedback compensation networks then have rising frequency characteristics and make the system very sensitive to HF noise in the feedback path. It is shown how a multiple-loop design permits the attainment of the same benefits of feedback, but with considerably less sensitivity to the HF noise. The basic problem is how to divide up the feedback burden most efficiently among the various loop transmission functions. Detailed procedures for this purpose are presented in the paper. The treatment is restricted to cascade-type plants.

The second method of Liapunov is used to study asymptotic stability of feedback control systems with single non-linear elements. The paper aims at a systematic development of Liapunov functions in terms of canonic transformations of state variables. This approach yields various simplified stability criteria; a rather complete table of such criteria is included. Several insights believed novel concerning the failure of the method in certain practical cases are presented. The root locus of the linear portion of the system is used to predict the applicability of the method. The proposed pole- and zero-shifting techniques extend the applicability of the method to many practical systems in which the method would fail without these techniques.

It is shown that simple saturable servo systems can have two modes of response to a given input signal one a linear mode as predicted by linear feedback theory, and the other a saturated mode predictable from a large signal analysis presented here. This dual mode of response theory provides a reasonable explanation for premature saturation, hysteresis in the input-output characteristics, jump resonance and similar anomalous effects that so often degrade the steady state performance of saturable servo systems. Furthermore, systems which exhibit a large degree of anomalous steady state behavior can be expected to exhibit a correspondingly poor transient response under large signal conditions such as obtained when the system is coming out of saturation. Demonstration of the existence of this dual mode of response and prediction of the range of signal amplitude and frequencies for which it exists are both built around the "saturated" transfer characteristics of the control loop. This analysis is quantitatively useful only in certain simple cases where the necessary saturated transfer characteristics can readily be found. Its chief value lies in the insight it provides to the cause of anomalous behavior. In this respect it is particularly useful to the designer because it relates this behavior to the loop gain and phase characteristics of the saturated transfer characteristics. The analysis is applied here to two simple examples, and the results are verified by an analog computer study on simulations of these systems.

In designing a feedback control system involving a variable (or incompletely known) plant the prime consideration, besides obtaining a satisfactory transfer function, is to specify a system which is insensitive to plant variations. Some procedures for obtaining insensitive designs have been described in the literature, but they all result in systems having large open loop bandwidths. In the presence of instrument noise such systems would tend to produce an excessive noise output. In this paper a minimization is carried out, where the conflicting requirements of small sensitivity to plant variations and insensitivity to instrument noise are satisfied simultaneously. The solution is approximate, but can be justified in most physical situations. A method for iterating the solution is also described.

The stability of a system which alternates between stable and unstable configurations at random times may be investigated conveniently using Kronecker products. The system is stable with probability one if, and only if, all the eigenvalues of a specified matrix lie within the unit circle. If stability in the presence of a parameter adjustment is to be investigated, a root-locus plot of the eigenvalues is convenient.

This paper deals with the application of z -transform techniques to the analysis of sampled-data systems in which signals appear in pulse duration modulated form. The characteristics of pulse duration sampled data are first briefly described. An analytical means for studying the transient response and stability of systems which use this kind of data is then presented. Finally, a comparison is shown of analytical results with tests on several representative experimental systems. A short discussion of the relative advantages and disadvantages of pulse duration modulation in control systems is also included.

The limitations in classical feedback that might justify the more complex plant or process adaptive systems are studied. Some of the limitations cited in the adaptive literature apply only to the classical single-degree-of-freedom configuration and not to the classical two-degree-of-freedom structures. Model and conditional feedback configurations are not superior to ordinary two-degree-of-freedom configurations. Time invariant (classical) compensation is adequate for coping with the sensitivity and disturbance problem in lightly damped and drifting plant poles. Two significant limitations in ordinary linear feedback systems that may justify the adaptive approach are: a) Their susceptibility to feedback transducer noise when the plant by itself does not have the loop gain area required for the desired sensitivity properties of the system; b) The limited sensitivity reduction achievable in nonminimum phase and unstable plants. The first of these may be eased by a multiple-loop design and the second by a parallel plant design. However, it has not been shown how the adaptive systems overcome the limitations of ordinary feedback systems.