Deadbeat control with disturbance rejection

A deadbeat control problem with disturbance rejection is considered for a SISO discrete time plant. Disturbances are supposed to enter into the input to the plant and the output from the plant. The two-degree-of-freedom controllers are employed to internally stabilize the feedback control system, to make the output of the plant track a reference signal and to reject the disturbances in the sense of the deadbeat response. Necessary and sufficient conditions for the problem to have a solution are shown. And the set of all controllers meeting the design requirements are represented using two free polynomials.     

Parametrization of all ripple-free deadbeat controllers

This paper is concerned with deadbeat control in sampled-data systems. Deadbeat control achieves finite-time settling (deadbeat settling) at sampling instants, but there may exist error called ripple “between” sampling instants even after the response is settled “at” sampling instants. The objective of this paper is to give a parametrization of all ripple-free deadbeat controllers (controllers which achieve deadbeat settling without ripple) in sampled-data systems. It is also shown that the following holds in general: minimum-time deadbeat control causes ripple when the pulse transfer function to be controlled has stable zeros.     

Dead-beat control of simple Hammerstein models

Dead-beat controllers for simple Hammerstein systems are investigated. Several designs for nonminimum-time state dead-beat controllers are given for certain classes of simple Hammerstein systems. A general minimum-time state dead-beat controller is presented for a class of simple Hammerstein systems. A design for a family of minimum-time control laws is provided. This enables, to a certain extent, shaping of transient response via choosing an appropriate control law. Finally, the authors design an output dead-beat controller for a class of Hammerstein systems that are not necessarily state dead-beat controllable     

Explicit parameterization of state deadbeat controllers

A general state deadbeat control problem, not restricted to the controllability indexes, is posed and solved. A result for the explicit parameterization of deadbeat controllers is obtained. With the parameterization, only the genuine independent free parameters appear in the deadbeat controllers     

Organization of the non-uniqueness of a canonical structure of linear multivariable systems

This paper is concerned with organizing the non-uniqueness of the canonical structure of a class of linear time-invariant multivariable systems in such a way as to provide an efficient tool in control system design. A phase-variable block form of state equations is used to describe the system class in the transformed coordinates. The transformation procedure is characterized by an arbitrary parameter (in the form of a submatrix) which adds an attractive degree of freedom in the solution of practical problems. As a demonstration, the procedure is applied to synthesize a minimum-time deadbeat controller for linear discrete-time systems. A family of controllers, not identifiable by earlier approaches, is developed and that controller which additionally minimizes a control energy criterion is determined. A numerical example is provided to illustrate the different aspects of this work.     

On the general solution of the state deadbeat control problem

In this note, the state deadbeat control problem is considered. It is shown that, after appropriate change of basis of input and state spaces, the general solution of the state deadbeat control problem can be expressed completely by the rows of the powers of system matrix. This result yields a very simple procedure for the calculation of a state feedback deadbeat control gain. It also provides the number of free parameters which could be used for further design purposes. The results are illustrated by an example at the end of the note     

Output deadbeat controllers with stability for discrete-time multivariable linear systems

This short paper Treats the problem of designing output deadbeat controllers having the property that the control input to the system converges to zero as time goes to infinity, for discrete-time multivariable linear systems. Two configurations of controllers are considered: one is of state feedback; the other is a dynamic controller using an observer. The existence of such controllers is examined, and the methods are presented for designing such controllers when they exist. The controller using a state feedback obtained in this paper is optimal in the sense that the controller settles the output in zero for any initial state in the minimum number of steps. On the other hand, the dynamic controller is not optimal in that sense, but it minimizest, wheretis defined as an integer such that the controller drives the output to zero in no more thantsteps for any set of initial conditions of the system and the observer.     

Continuous-time deadbeat control for sampled-data systems

In this paper, the continuous-time deadbeat control problem for the sampled-data systems is considered. We derived the class of all controllers that achieve the continuous-time deadbeat control     

一类离散时间切换线性系统的无差拍控制

研究一类带多控制器和多传感器离散时间线性系统的无差拍控制.对能控系统,通过适当的状态坐标变换获得系统矩阵的块三角结构,再设计状态反馈和周期切换策略使得状态反馈矩阵在有限周期内为零,从而保证闭环系统的无差拍稳定.进一步,对能观系统,设计具有有限时间精确估计的动态输出反馈,通过适当的周期切换策略实现闭环系统的无差拍稳定.最后,给出一个例子以验证所提设计方法的有效性.     

A note on parameterization of the class of deadbeat controllers

The problem of parameterizing the class of deadbeat controllers for a given discrete-time system through the minimum number of parameters was solved by Schlegel [1]. This note shows how to utilize the above solution to study some problems in designing deadbeat controllers. First, an algorithm is developed to compute a controller which minimizes-in an average sense-a given objective function. Second, a necessary and sufficient condition is given for the existence of an output deadbeat controller. Finally, the problem of parameterizing the set of deadbeat controllers for those systems transformable to the phase-variable block-canonical form is reconsidered.     

Deadbeat control of linear multivariable generalized state-spacesystems

The classic idea of deadbeat control is extended to linear multivariable discrete-time generalized state-space systems using algebraic methods. The asymptotic properties of the linear quadratic regulator theory are used to obtain the classes of deadbeat controllers using stabilizing full semistate feedback. The solution is constructed from a `cheap control' problem. Both semistate and output deadbeat control laws are considered. The main design criteria are to drive the semistate and/or outputs of the system to zero in minimum time and that the closed-loop system be internally stable. Unique properties of these types of control laws are discussed. For semistate deadbeat control, all the (dynamic) poles including the ones at infinity are moved to the origin, whereas for output deadbeat, some of the finite transmission zeros are canceled. Numerically reliable algorithms are developed to solve both problems     

Minimum-gain minimum-time deadbeat controllers

This paper considers the design of minimum-gain minimum-time deadbeat controllers (MGMTDC) for linear discrete-time systems. A new direct method for constructing the MGMTDC of minimum Frobenius norm is developed. Compared with many existing techniques, the proposed method gives analytic expressions for the constructed MGMTDC, requires less computational effort, and does not require that the transition matrix be nonsingular.     

Parameterization of the class of deadbeat controllers via thetheory of decoupling

A novel approach is presented for parameterizing the class of minimum-time deadbeat controllers (MTDC) through the minimum number of parameters. The approach is based on the theory of decoupling and the properties of square decouplable systems. The main result is a compact parametric form for the class of MTDC of a discrete-time system. Contrary to many existing techniques, no special assumption on the invertibility of the transition matrix is required     

Multivariable closed-loop deadbeat control: a polynomial-matrix approach

The closed-loop deadbeat servo problem (CDSP), considered in this paper, consists of the synthesis of a linear, output feedback controller such that the control signal and tracking error both vanish, after a finite period of time, for every reference sequence from a prespecified class and for every initial state of a plant and the controller. The closed-loop structure is determined by studying necessary and sufficient conditions for deadbeat tracking performance. A new theorem asserts that if an open-loop deadbeat control strategy exists for every initial state of the plant and every reference function from a given class, then CDSP is solvable and all desired control laws are found in an explicit parametric form by solving simple, unilateral, linear equations in polynomial matrices. On the basis of this theorem a design algorithm is developed. Asymptotic stability of the closed-loop system exhibiting deadbeat properties is demonstrated. A numerical example is given to illustrate the usefulness and computational efficiency of the new design algorithm presented.     

Ripple-free deadbeat control of SISO discrete systems

The ripple-free deadbeat control problem for arbitrary (not necessarily stable) SISO linear discrete plants and reference signals is treated. It is established that a causal, stabilizing ripple-free deadbeat controller exists if and only if the zeros of the plant and the poles of the reference signal are disjoint, and a complete characterization of all such controllers is obtained. Solutions to two illustrative problems are presented.     

Double objective optimal multivariable ripple-free deadbeat control

This paper presents novel results on optimal multivariable deadbeat control. Given a discrete-time, stable, linear, time invariant plant model, we give a simple parameterization of all stabilizing ripple-free deadbeat controllers of a given order. The free parameter is then optimized in the sense that a quadratic index is kept minimal. The optimality criterion has the advantage of accounting for both tracking performance and magnitude of the control effort. The proposed design procedure is simple to use and allows the tuning of the controller with a scalar weighting factor. Simulation results are included to illustrate the effectiveness of the proposed design algorithm.     

Theoretical developments in discrete-time control

Significant theoretical developments in discrete-time control over the past 10–15 years are reviewed. Topics reviewed include optimal control, Riccati equations, controllability, stability, robustness, deadbeat control, minimum-time systems, sampling, quantization, microprocessor implementation, and stochastic systems. Emphasis is placed on topics that are peculiar to discrete-time systems and do not generalize from continuous-time theory.     

Comparison and application of DDC algorithms for a heat exchanger

In recent years, several algorithms for Direct Digital Control (DDC) have been proposed in literature. Although some of these, such as PID or cascade controllers, are very commonly used in industrial applications, the more recent ones like optimal state feedback controllers using an observer or parameter adaptive controllers have rarely been applied to a real plant. The primary difficulty behind this application has been perhaps the lack of testing such algorithms on a pilot plant. Moreover, there has been no serious attempt to make a comparative study of the merits of such algorithms for an existing plant under actual operating conditions. In this paper, seven DDC algorithms are applied to the temperature control of a heat exchanger. These algorithms are: PID, cascade, compensation (pole assignment), deadbeat, half-proportional, adaptive and optimal state feedback controller using an observer. The system performance and sensitivity with respect to changes of the plant parameters, disturbances and set point variations are investigated for the heat exchanger using these algorithms. The results indicate that the more sophisticated algorithms, e.g. optimal state feedback, compensation and adaptive controllers, requiring more computer time and memory, yield relatively less improvement when applied to a low-order plant than do the simple algorithms such as PID or cascade. It was deduced that the PID controller with anti-windup is the most suitable one.     

Pole assignment by state transition graph

The pole assignment problem is considered, using the graph representation of a matrix. The parametrization of controllers which have a specified characteristic polynomial is given. A simple algorithm based on graphs is presented and two examples are given     

Comments on “ Extension of dual coordination to a class of non-linear systems ”

The problem of constructing an observer for a discrete linear system so as to minimize a quadratic function of the state estimation error is considered. The choice of a particular canonical representation for the system is shown to lead to a simple design procedure. The resulting observer is shown also to possess the minimum-time state-estimation property.