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.     

Approximation of discrete linear systems†

Given a multi–input, multi–output linear time–invariant discrete–time system or its weighting matrices, equations are derived to characterize a system of lower dimension which optimally approximates the given system with respect to the sum of squared output errors of the impulse response. These equations are shown to imply that in principle a modified B. L. Ho algorithm can be used to find the approximation. The connection with projection methods of approximation is demonstrated, Some comments on computational difficulties are included.     

A discrete maximum principle for systems with pure delays†

This paper deals with necessary arid sufficient conditions for the optimization of nonlinear multivariate discrete systems containing pure delays in dynamics, in performance index, and in constraints. Using non-linear programming, it is shown that a discrete maximum principle similar to the one derived by Pearson and Sridhar (1966) is valid for a subclass of these problems, that is systems with linear dynamics, concave inequality constraints, and convex performance index. As an application, the optimal piceewise constant control with optimal non-uniform sampling of systems described by differential equations is derived.     

Minimal-time control of linear-distributed systems with a composite-norm constraint†

This paper deals with the time-optimal control of a class of linear-distributed systems with a composite-norm constraint. The method of i-theory of moments is used to obtain the solution. This gives rise to two auxiliary problems: (i) an infinite set of Fredholm integral equations, and (ii)a minimization problem in the dual space.     

Synthesis of linear digital controllers using dead-beat observers for systems with inaccessible states†

A procedure is given to synthesize linear digital controllers for uniformly observable linear time-varying discrete data systems where some of the state variables are not available for feedback. Dead-beat observer theory is used and the controller generates the exact optimal control law in at most q?1 sampling periods, where q is the observability index of the system.     

An approach for improved modal control of multi-input multi-output discrete time systems†

It is shown that the state feedback gains for a given multi-input multi-output discrete time system can be computed by matching a finite model output sequence members to prespecified ones. Relationship of the pole positioning control schemes to such matching is first examined and it is shown that better control is possible by adopting the method proposed in this paper. The results are illustrated through a numerical example.     

Large-scale discrete systems : a two-level optimization scheme†

Design of controllers for large-scale systems often requires methods based on decomposition and decentralization to overcome the twin problems of high dimensionality and constraints on the information structure. In this paper a two-level control scheme that effectively utilizes these methods for the optimization of large-scale discrete systems is presented. Decentralized controllers that optimize individual decoupled subsystems while ignoring the interactions constitute a set of local controllers at the first level, while a global controller at a higher hierarchical level is used to minimize the suboptimality of performance resulting due to the presence of the interactions. Such a control scheme is robust, simple to implement and guarantees global asymptotic stability of the overall system.     

Linear functional observers with guaranteed ε-convergence for discrete time-delay systems with input/output disturbances

The problem of designing linear functional observers for discrete time-delay systems with unknown-but-bounded disturbances in both the plant and the output is considered for the first time in this paper. A novel approach to design a minimum-order observer is proposed to guarantee that the observer error is ε-convergent, which means that the estimate converges robustly within an ε-bound of the true state. Conditions for the existence of this observer are first derived. Then, by utilising an extended Lyapunov–Krasovskii functional and the free-weighting matrix technique, a sufficient condition for ε-convergence of the observer error system is given. This condition is presented in terms of linear matrix inequalities with two parameters needed to be tuned, so that it can be efficiently solved by incorporating a two-dimensional search method into convex optimisation algorithms to obtain the smallest possible value for ε. Three numerical examples, including the well-known single-link flexible joint robotic system, are given to illustrate the feasibility and effectiveness of our results.     

Two-dimensional disturbance decoupled observers†

A procedure is developed for the design of a minimal-order observer capable of estimating the states of a two-dimensional system in the presence of unknown inputs. This is done using singular-value decomposition. The assignment of eigenvalues to the observer is facilitated by choosing the observer structure.     

Multilevel discrete time system identification in large scale systems†

Hierarchical decomposition is considered to be one of the most powerful and offective tools to deal with complexity. Hierarchical system theory, which deals with system decomposition and coordination, can be used to decentralize and reduce the computational efforts requirements for many large-scale problems. This is achieved by decomposing the original system problem into several lower order easier to handle sub-problems, which are then coordinated such that the overall system objectives are met. In this work a hierarchical system theory approach to the discrete-time system identification problem is considered for stochastic large-scale system applications. A set of sequential discrete-time hierarchical identification algorithms, suitable for known and unknown system noise moments, are first obtained using a maximum a posteriori (MAP) approach with covariance matching and maximum likelihood (ML) methods. This is conducted in a two level hierarchical structure with two principles of coordination. Next, the hierarchical system identification algorithms are extended to multilevel hierarchical structures based on system characteristics of priority of action, spatial structure and time behaviour. This results in multilevel and composite multilevel coordinated system identification procedures, where each subsystem unit can be treated independently in reaching the overall system optimality. Application of these algorithms for the purpose of decentralization and reduction of computational requirements as well as adaptation to structural changes in growth and merger are considered.     

On smoothing in linear discrete systems with time delays†

An algorithm is presented for optimal linear fixed-point and fixed-lag smoothing in non-stationary linear discrete systems with multiple time delays. Relations derived previously in the problem of filtering are used directly to obtain the fixed-point and fixed-lag smoothing filters. The smoothing error covariance matrix is obtained via a recursive matrix equation. For the particular case of systems without delay terms, the algorithm defines new smoothing procedures.     

Sensitivity-state models for linear systems†

In this paper time-domain models for sensitivity analyses of linear systems have been developed. The procedures for developing those models are quite similar to the formulation of state models which are based on graph-theoretic concepts. Sensitivity analyses with respect to variations in component values, the initial conditions and the driving functions appear as three different facets of the same analytical procedure. These models admit a very convenient interpretation in terms of networks which properly can be exploited for digital computer studies. Additionally, attention is focused on developing sensitivity models for three specific types of interconnection, viz. parallel, cascade and feedback. On the basis of these models, the effect of system structure on sensitivity is studied.     

Phase-variable canonical forms for linear,multi-input,multi-output systems†

Three types of system equivalent relations are introduced to unify existing definitions of system equivalence. Called α-, β- and γ-equivalence, their characteristics are examined mathematically and physically. Based on those definitions, two phase-variable canonical forma are developed for linear, multi-input, multi-output systems, resulting in a canonical form for control and a canonical form for observation. It is shown that a characteristic quantity called a stage sequence can be defined for a given system to uniquely determine its equivalent system in the canonical forms. The compact structures of these canonical forms are convenient representations of the mathematical and physical compositions of systems and for subsequent system analyses.     

Optimal control of distributed parameter systems with discrete constrained inputs†

The optimal control of linear distributed parameter systems, which are represent able by a linear vector integral equation, is investigated. Restricting the control action to be discrete in time, the problem of minimizing the mean-square error, between specified desired final state functions and the actual state functions at a prescribed final time, subject to an energy constraint on the controlling functions, is treated. A necessary and sufficient condition from functional analysis is used to derive an equation whose solution yields the optimal control vector. Two convergence properties for the discrete problem are established which can be used to determine a good approximate solution to the corresponding measurable optimal control problem. An illustrative example is given.     

Stability of discrete systems over a finite interval of time†

In many cases of practical interest there is concern with the behaviour of dynamic systems only over a finite time interval. This concern may arise in one of two ways: In one case the system under consideration is defined over a fixed and finite interval of time, while in the second case the system in question is defined for all time; however, the behaviour of the system is of interest only over a finite time interval.

Recently, Weiss and Infante (1965, 1967) treated the problem of system stability over a finite time interval for the ease of continuous systems. In this paper a theory is developed which concerns itself with the stability of discrete systems over a finite interval of time. The dynamic systems which are considered are general enough so as to include unforced systems, systems under the influence of perturbing forces, linear systems, non-linear systems, time invariant systems, time-varying systems, simple systems and composite systems.

In the present development various definitions of stability are considered and corresponding stability theorems are stated and proved. These theorems yield sufficient conditions for stability and in general involve the existence of Lyapunov-like functions which do not possess the usual definiteness requirements on V and ΔV.     

Analytical and experimental studies in signal stabilization of discrete systems†

This paper describes the definition, development and application of a dual input discrete describing function (D.I.D.D.F.) for the analytical study of non-linear sampled-data systems. An analytical procedure is described which aims at deriving the actual D.I.D.D.F. for an ideal relay. The limit cycles that are ordinarily found in non-linear sampled-data systems can be completely quenched by injecting a high-frequency dither signal at the input to the non-linear element. By applying the D.I.D.D.F. technique, the magnitude of the dither signal required for complete suppression of the limit cycles can be obtained analytically. This technique can be applied to the study of a problem utilizing sinusoidal, triangular or square-wave dither signals. The latter half of the paper presents the results obtained from a comprehensive experimental study of the problem on a model built in our laboratory.     

Minimal controllability for systems with delays†

In this paper we consider the problem of minimal controllability for a general class of linear spectrally controllable systems. We show that the smallest number of controls that are necessary in order that spectral controllability may hold depends only upon the matrices of the uncontrolled system and that it is equal to the smallest number of outputs necessary for the dual problem of ‘spectral observability’; moreover, any spectrally controllable system can be made minimally controllable by acting upon the control matrix only, and any spectrally observable system can be made minimally observable by acting upon the output matrix only. In the sequel this result is explained in the cases of the delayed systems and of some kind of generalized systems.     

A sufficiency condition for discrete time optimal processes†

Sufficient conditions for optimality of a class of linear discrete time processes are presented. In distinction to previously obtained necessary conditions for optimality, these sufficient conditions hold under very weak hypotheses. An example, which might arise in a sampled data system, is presented. For the example the sufficient conditions yield the optimal control, but the necessary conditions are not applicable since the required hypotheses are not satisfied.     

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.     

Design of multifunctional reduced order observers†

This paper presents a method for the design of an observer capable of reconstucting several linear functionals of the states of a linear, finite-dimensional system. The goal of this method is the design of an observer having minimum order subject to the restriction that the observer eigenvalues be freely assignable. The method is based on the reduction of a state observer formulated from an observable canonic form for the system when the functionals are treated as if they were additional outputs.