All stabilizing PID controllers for time delay systems

This paper presents a method to compute the entire set of stabilizing PID controller parameters for an arbitrary (including unstable) linear time delay system. The main contribution is to handle the infinite number of stability boundaries in the (k_d,k_i)-plane for a fixed proportional gain k_p. For retarded open loops, it is shown that the stable region in the (k_d,k_i)-plane consists of convex polygons. Concerning neutral loops, a new phenomenon is introduced. For certain systems and certain k_p, the exact stable region in the (k_d,k_i)-plane can be described by the limit of a sequence of polygons with an infinite number of vertices. This sequence may be well approximated by convex polygons. Moreover, the paper describes a necessary condition for k_p-intervals potentially having a stable region in the (k_d,k_i)-plane. Thus, the set of stabilizing controller parameters can be calculated after gridding k_p in these intervals. A Matlab tool implementing the presented method is available.     

Parametrization of all stabilizing controllers of nonlinear systems

We consider a nonlinear multiobjective optimal control problem, where a controller is to be designed such that a number of constraints are to be satisfied for all possible disturbances. We show that, under mild continuity assumptions, such a controller exists if and only if, for every weighted combination of the constraints, there exists a controller satisfying the weighted constraint.     

A class of globally stabilizing controllers for nonlinear systems

A class of globally stabilizing controllers for nonlinear systems are parametrized as fractional transformations on some input-to-state stable parameters.     

Design of stabilizing controllers of upper triangular nonlinear time-delay systems

In this paper, it is considered the state feedback controller design for a class of upper triangular nonlinear systems with simultaneous input and state delays. By using the state transformation of nonlinear systems, the problem of designing controller can be converted into that of designing a dynamic parameter, which is dynamically regulated by a dynamic equation. Then, by appraising the nonlinear terms of the given systems, a dynamic equation can be delicately constructed. At last, with the help of Lyapunov stability theorem, it is provided the stability analysis for the closed-loop system consisting of the designed controller and the given systems. Both discrete delays and continuous delays with integral form are considered here. Different from many existing control designs for upper triangular nonlinear systems, neither forwarding recursive nor saturation computation is utilized here, and thus our design procedure is simpler. A simulation example is given to demonstrate the effectiveness of the proposed design procedure.     

二阶加纯滞后过程的非脆弱PID稳定化控制器设计

根据Hermite-Biehler定理在准多项式稳定性问题上的推广，研究用PID控制器镇定二阶加纯滞后过程的问题，给出了使闭环系统稳定的PID参数区域，并以稳定的PID参数区域内切圆半径为目标函数，寻优得到非脆弱PID控制器参数，最后通过线性规划给出了非脆弱PID稳定化控制器的设计方法。     

Stabilization of uncertain dynamic systems including state delay

The authors consider a class of time-varying systems with time-varying state delay and uncertain input and parameters. They present sufficient conditions on the open loop to obtain an arbitrary-degree closed-loop stability. They construct a min-max controller from knowledge of the upper bound of the delay     

A sufficient condition for output feedback stabilization ofuncertain dynamical systems

A different proof than the one given by E. Zeheb (ibid., vol.AC-31, p.1055-7, Nov. 1986) is given for the existence of an output feedback controller which stabilizes an uncertain single-input/single-output dynamical system with a linear nominal part and matched uncertainties. However, there is no need to assume that the nominal system is stable. A simple expression is obtained for the feedback gain which is necessary for the closed-loop nominal system to become strictly positive real     

Adaptive stabilizing state feedback controllers of uncertaindynamical systems with multiple time delays

The problem of robust stabilization for a class of uncertain dynamical systems with multiple delayed state perturbations is considered. In this paper, it is assumed that each perturbation is bounded by a linear function of delayed state with unknown gains, and an adaptation law is proposed to estimate these unknown gains. Moreover, by making use of the updated values of these unknown bounds we propose a memoryless state feedback controller for such a class of uncertain time-delay systems. Based on Lyapunov stability theory and Lyapunov-Krasovskii functional, it is shown that the closed-loop dynamical system resulting from the proposed adaptive robust control schemes is globally stable in the sense of uniform ultimate boundedness. Finally, a numerical example is given to demonstrate the validity of the results     

All stabilizing controllers as frequency-shaped state estimatefeedback

It is shown that the class of all strictly proper stabilizing controllers for proper linear plants can be structured as state estimate feedback with frequency shaping in the state estimates and/or in the state estimate feedback law. The selection of where the frequency shaping takes place is at the designer's discretion. The parameterization of the controller class can be in terms of an arbitrary proper stable transfer function, with the closed-loop system transfer functions affine in the transfer function. With constant output feedback permitted in addition to the state estimate feedback, the class of all proper stabilizing controllers can be generated in a like manner     

On the order of nonlinear time-invariant stabilizing controllers

Here we consider the problem of stabilizing a finite-dimensional, stabilizable and detectable, linear time-invariant (LTI) plant in the sense that the plant state goes to zero while the controller state converges. With r the dimension of the plant output, we show that there exists a nonlinear time-invariant controller of order r+3 which stabilizes all such systems.     

Parametrization of nonlinear stabilizing controllers: theobserver-controller configuration

The paper uses the author's (1996) previous results to develop the parametrization of nonlinear stabilizing observer-controller compensators of a given nonlinear system. The configuration described permits a simple parametrization of stabilizing controllers using coprime factorisation techniques     

A parametrization of stabilizing controllers for multiratesampled-data systems

All linear multirate controllers for a given multirate sampled-data system plant are parameterized in terms of a single parameter Q(z) that is allowed to be any stable transfer matrix provided that certain causality conditions are met. The result parallels a well-known parameterization result for single-rate sampled-data plants. The parameterization suggests architectures for multirate controllers     

Exponentially stabilizing division controllers for dyadic bilinear systems

It is difficult to asymptotically stabilize a dyadic bilinear system with only multiplicative control inputs when the open-loop dynamics are unstable. The previous approach of cascading a division controller with a constant-size dead zone can only stabilize but not asymptotically stabilize the system. This note proposes a new control design which cascades a division controller with a modified dead zone whose size is proportional to the modulus of the system state. It is shown that this new division controller can globally and exponentially stabilize any open-loop unstable dyadic bilinear system as long as it is controllable.     

Lyapunov design of stabilizing controllers for cascaded systems

The design of a state feedback law for an affine nonlinear system to render a (as small as possible) compact neighborhood of the equilibrium of interest globally attractive is discussed. Following Z. Artstein's theorem (1983), the problem can be solved by designing a so-called control Lyapunov function. For systems which are in a cascade form, a Lyapunov function meeting Artstein's conditions is designed, assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem. In particular, for planar systems, this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized     

A simple derivation of all stabilizing proportional controllers for first order time‐delay systems

In this paper, a new simple derivation of all stabilizing proportional controllers, for first order linear time invariant systems with time‐delay, is presented. Although several results based on the Hermite‐Biehler theorem for finding such a set of controllers exist, the aim of this article is to present a shorter and more instructive derivation, which can be followed easily. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

An LMI approach to design extended predictor feedback controllers for dynamical systems with input delay

The stabilisation of dynamical systems with input delay by an extended predictor feedback controller is investigated. After considering a more general case of the predictor feedback controllers, the design procedure is transformed into the stabilisation problem of the system controlled by a dynamic output feedback controller. By proposing appropriate Lyapunov-Krasovskii functionals, the predictor feedback gains are then synthesised from the stability conditions which are obtained in form of linear matrix inequalities. Such design approach is more flexible and extendable in comparison to one used by the conventional predictor feedback control. As a simulation example, the designed extended predictor feedback controller is applied on an active suspension system and is used to stabilise an unstable system with input delays. The obtained results demonstrate the effectiveness of the design method.     

Self-tuning controllers for nonlinear systems

Two new self-tuning control (STC) strategies for nonlinear control problems are proposed. These strategies are applicable to a broad class of nonlinear single-input, single-output systems which can include arbitrary nonlinear functions of the output and the old inputs as well as the products of these functions and any power of the most recent input. Simulation results for a stirred-tank chemical reactor demonstrate that the new nonlinear STC strategy provides significant improvement over conventional STCs based on linear models.     

Recursive predictor design for state and output feedback controllers for linear time delay systems

This paper presents a recursive method to design state and output feedback controllers for MIMO, block-feedforward linear systems with delays in the inputs, outputs, and interconnections between the blocks. The resulting controller is of predictor-type, which means that it contains finite integrals over past state and input values. The method is a generalization of the well-known model reduction approach for systems with input delay. A recursive procedure replaces delay terms with non-delay ones step by step, from the top of the cascade structure down. Controller gains are computed for the proxy system without delays, while the construction guarantees the same closed loop poles for the delay system and the proxy one. The observer is designed by applying the duality argument and the separation principle is also shown to apply.