State-feedback control of non-linear systems†

A design method for state-feedback controllers for single-input non-linear systems is proposed. The method makes use of the transformations of the non-linear system into ‘controllable-like’ canonical forms. The resulting non-linear state feedback is designed in such a way that the eigenvalues of the linearized closed-loop model are invariant with respect to any constant operating point. The method constitutes an alternative approach to the design methodology recently proposed by Baumann and Rugh. Also a review of different transformation methods for non-linear systems is presented. An example and simulation results of different control strategies are provided to illustrate the design technique.     

Dual-input describing series and their use for forced non-linear system analysis†

The concept of the non-linearity specific vector is defined. It is shown that this vector depends exclusively on the non-linearity parameters. The expressions for the describing series for dual-input are determined. These are valid for both single-valued and double-valued non-linearities. A theorem referring to the correlation of the non-linearity specific vectors with the approximated dual-input describing series is stated and proved. The equivalent gain characteristics obtained for two non-linear elements with dual-input are also compared with those obtained by means of the conventional method of DIDF. The conditions which allow the dual-input describing series in matrix form to be much more general than the conventional means are examined and a stability criterion for forced systems is established.     

Space delta method for analysis and synthesis of non-linear lumped systems†

The paper reviews introduction and applications of the so-called Delta Method of investigating phase-space trajectories and further indicates new possibilities occurred recently in applying an inversion of this method to problems of Synthesis, Control and Identification.     

Computer generation of sensitivity functions for sampled-data control systems†

Methods have been presented previously (Suryanarayanan and Soudack 1970) for the generation of sensitivity functions for general non-linear sampled-data systems. In this paper, it is shown that economy of simulator components (for hybrid computer generation) or of computational time (for digital computer generation) is achieved for simultaneously generating a number of sensitivity functions in a class of linear sampled-data control systems. Extensive use of signal flow graph algebra is made for this purpose and comprehensive examples are presented.     

Diagonalization and inverses for non-linear systems†

The problem of decoupling and inverting dynamic systems is considered. Attention is focused on the problems that arise as available linear techniques ore applied to nonlinear systems. The decoupling of non-linear multi-variate differential equations is illustrated in a series of three examples.     

Closed-loop control of a class of non-linear systems†

A technique is described for the closed-loop control of a class of non-linear systems iii a potentially large neighbourhood of a nominal trajectory. By the introduction of extra states it is shown how to reduce non-linear differential equations to a related canonical linear form. Since the linear form is not readily amenable to standard design techniques a new synthesis procedure is proposed for the construction of a closed-loop control. The proposed technique relies heavily on computer computation. The computation for the closed-loop control of a lifting body, a system not completely controllable, is given in detail to illustrate the applicability of the method.     

Sensitivity in the decoupling of non-linear control systems†

The problem of decoupling the inputs and outputs of non-linear control systems is considered for the situation where the system parameters are subject to variations or disturbances. The exact decoupling is obtained by an adaptive state feedback controller which requires full knowledge of the instantaneous values of the system parameters. By utilizing the parameter sensitivity concept an approximate adaptive decoupling controller is synthesized in the present paper which is shown to decouple the system to first (or higher) order in the variations of the system parameters from their known nominal values. The sensitivity equations which generate the state sensitivity function required for the synthesis are derived. The results of the paper actually provide the tools for reducing the system sensitivity to parameter variations when the final goal is the improvement of decoupling.     

Graphical analysis of a class of non-linear feedback control systems with step input†

The class of systems considered in this investigation is a cascade combination of a linear memory system and a non-linear no-memory system in the forward path of a unity feedback control system. The output of the non-linear no-memory system is assumed to be a polynomial function of the input. Regardless of the exact nature of the non-linearity, the objective of this method of analysis is to predict the behaviour of higher-order non-linear systems with different initial conditions for step inputs.

Two different cascade combinations of linear and non-linear blocks in the forward path are considered. For both configurations a similar non-linear differential equation is obtained for some variable in the system. The non-linear differential equation is further reduced to a first-order equation, explicitly independent of the independent variable, time t. Treating all other coefficients as parameters and eliminating each in turn, finally the required phase-plane trajectory is obtained.     

Limit cycle analysis for control systems of high order†

A generalized method for the limit cycle analysis of a class of nth-order systems is proposed. The method utilizes describing function technique and is applicable to dynamical systems with both memoryless and with memory non-linearities. The critical gain for self-sustained oscillations is found without explicitly solving the system characteristic equation. Relatively simple expressions are derived for the evaluation of limit cycle frequency. The results reduce to the conventional Routh equation for memoryless non-linearities. Application of the proposed method is demonstrated by several examples.     

Suboptimal adaptive control of a class of non-linear systems†

The problem of controlling a class of nonlinear systems with uncertain parameters and imprecisely defined system and measurement noise terms is considered. A sub-optimal adaptive control algorithm is developed as a solution to coupled linear modelling and control optimization problems. Features of the algorithm are the forward-recursive generation of the modelling and control processes and necessity for past and present values of states and parameters alone in order to implement the algorithm (i.e. no precomputed nominal trajectory is required). These features make the algorithm ideally suited to real-time control applications

Real-time control of a non-trivial fifth-order system, (a model of a plasma torch furnace) is considered and the results of an experimental study are presented.     

A method of determining the crosscorrelation functions for a class of non-linear systems†

A use is made of the method of finding output autocorrelation functions of continuous time-invariant nonlinear systems, when they are represented by their functional expansion. This enables the input-output crosscorrelation functions to be calculated by the use of multi-dimensional Laplace transforms.

A simple example for a pseudo random binary sequence input is presented.     

Modified maximum principle for non-linear open-loop PFM control systems†

The problem of finding the optimum control function for pulse frequency modulated (PFM) control systems is considered in this paper. The modified maximum principle (MMP) is extended to cover non-linear open-loop systems. A method for obtaining the optimum control function is presented and illustrated by an example.     

Stability criteria for control systems with one non-linear element†

Liapunov functions are used to develop frequency domain criteria for a class of systems in which the given non–linear characteristic is bounded in both sector and slope. The criteria include the special case when order of numerator and denominator is equal in the linear transfer functions of the class of systems.     

High-gain feedback in non-linear control systems†

High-gain state and output feedback are investigated for non-linear control systems with a single additive input by using singular perturbation techniques.

Classical approximation results (Tihonov-like theorems) in singular perturbation theory are extended to non-linear control systems by defining a composite additive control strategy, a control-dependent fast equilibrium manifold and non-linear change of coordinates.

Those tools and an appropriate change of coordinates show that high-gain state feedback and variable structure control systems can be equivalently used for approximate non-linearity compensation in feedback-linearizable systems.

Next the effect of high-gain output feedback is shown to be related to the strong invertibility property and the relative order of invertibility. For strongly invertible systems the slow reduced subsystem coincides with the dynamics of the inverse system when zero input is applied and with the unobservable dynamics when a certain input-output feedback-linearizable transformation is applied.     

Stabilization of non-linear systems†

The problem of stabilizing a discrete-time non-linear system is considered. For a rather large class of common stabilizable non-linear systems, a procedure leading to the stabilization of a given non-linear system Σ belonging to that class is derived. In this procedure, a pair of compensators is constructed, consisting of a precompensator and an output feedback compensator, which, when connected in closed loop around the system Σ, yield a closed-loop system that is internally stable for bounded input sequences. The procedure allows the construction of infinitely many different pairs of such compensators, thus facilitating the choice of a convenient one.     

Time-averaged non-linear systems: steady-state analysis†

A non-linear system with time averaging can be regarded as one composed of a time varying system with a variable parameter. A non-linear operation, such as squaring is performed on the signal of the system and followed by low pass filtering. This filtered signal controls, in turn, the variable parameter. This paper considers the steady state analysis of time averaged non-linear systems. The analysis covers both passive (with input) and oscillatory (no input) systems, In the passive case it is shown that the response is similar to that of a passive system with the poles shifted. The harmonic content and distortion are computed for both the oscillatory and passive case. An evaluation of the frequency is carried out for the oscillatory case. An interesting result is that the significant filter characteristic is the attenuation at twice the frequency.     

Computer control of non-linear systems with varying performance criteria†

A non-linear system, or process whose parameters are adjustable, is considered.

The system is supposed to work under varying performance specifications. Stability is to be maintained at all times, of course. An algorithm combining Popov's stability criterion and a non-linear programming technique is proposed. Using this algorithm the system's parameters may be adjusted using a time-sharing console connected to a central computer and to the control system.

Examples of non-linear third- and fourth-order systems are solved in detail.

It is clear that the algorithm may be easily extended to higher-order systems.     

Asymptotic observers for non-linear systems†

Observer design for non-linear systems is discussed. Some recent approaches are based on state and output change of coordinates to transform a non-linear system into a particular observer form, from which an asymptotic observer can be designed ensuring the asymptotic stability of the error dynamics in the new coordinates. In this paper, the stability properties of the error dynamics are studied in the original coordinates. With some examples, it is shown how the asymptotic stability in the new coordinates does not imply, in general, the asymptotic stability in the original ones. Some general results are stated and proved to guarantee the asymptotic stability of the error dynamics in the original coordinates.     

Network models for co-state equations of linear and non-linear systems†

The paper presents a topological correspondence between a network and its adjoint which, in the final result, renders it possible to proceed to the adjoint network without going through the intermediate step of deriving the co-state equations. Algorithms are developed for arriving at the topology of adjoint networks for the terminal control problem of a linear network with a linear criterion as well as with a quadratic criterion. The results are also extended to non-linear networks. The topological interpretation of adjoint networks facilitates inclusion of the results into presently available programmes on computer-aided design, which, in fact, provides the primary motivation for the work.