The design of deadbeat controllers by state transition graph

Two kinds of deadbeat control problems are considered. One is the state deadbeat control problem and the other is the pointwise minimum-time deadbeat control problem. A simple graph called the state transition graph of a matrix is introduced, and simple algorithms based on it giving deadbeat controllers are presented. The set of pointwise minimum-time deadbeat controllers is characterized. The set of output feedback deadbeat controller is also considered     

Stabilizing and deadbeat properties of receding-horizon controllers

It is shown that receding-horizon controllers with horizon length N ≥ v (v being the controllability index of the system) stabilize a given discrete-time linear multi-variable system. Necessary and sufficient conditions for a receding-horizon controller to be a deadbeat controller are also given. It is further shown that by modifying a receding-horizon controller m of the poles of the closed-loop system (where m is the dimension of the input space) can be assigned to zero with simultaneous stabilization. The deadbeat properties of such modified receding-horizon controllers are also investigated,     

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.     

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.     

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.     

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     

Optimal pole placement design for SISO discrete-time systems

In this paper, we incorporate the placement of one real closed-loop pole into a compensator design framework based upon the Youla parameterization, duality theory, and linear programming. This framework has been used to design discrete-time compensators to solve the l₁ controller design problem as well as other related time-domain optimization problems. Previous work on these problems has focused on deadbeat systems. It is known that these can require high-order controllers. Part of the motivation for this work is to improve the tradeoff between controller order and performance over that, using deadbeat control     

On deadbeat controllers

The possibility of achieving a deadbeat regulation and tracking in linear multivariable systems independently of their initial conditions is investigated. Necessary and sufficient conditions for the existence of various deadbeat controllers are established in a constructive way. A detailed analysis of the control sequence generated by any deadbeat controller is then presented.     

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     

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.     

Deadbeat control system with time-varying uncertainty: Minimization of worst case steady-state tracking error

This paper poses and solves a new problem of the synthesis of a controller that minimizes the worst case steady-state controlled error in the presence of time-varying unstructured uncertainty. We present an easy, straightforward design method for obtaining a controller that minimizes the worst case steady-state controlled error under the assumption that the plant is strictly proper. The detailed contributions are as follows. First, we derive a new, simple expression for calculating the worst steady-state error, which gives useful and interesting suggestions about the controller design. We then show that the synthesis problem is reduced to a simple and tractable l1 -norm minimization problem. Therefore, this reduction provides a feasible method for solving the design problem of a controller that minimizes the worst case steadystate error. Second, a deadbeat tracking control problem is considered and a straightforward design method is presented for obtaining a deadbeat controller with given settling steps both to guarantee robust stability and to minimize the worst case steady-state error. The proposed controller is easily obtained by solving a simple linear programming problem. Finally, we show that the proposed controller minimizes the maximum error bound of the controlled output to persistent bounded disturbance.     

Deadbeat algorithms for output controllers in discrete-time multivariable systems

This paper studies the problem of designing output deadbeat controllers to force the state of discrete-time multivariable systems to zero in a finite number of samples. Two algorithms are considered. The first is based on the fact that the closed-loop eigenstructure assignable by output feedback is constrained by the requirement that the left and right eigenvectors must be in certain subspaces. In the second algorithm, the output gain matrix is computed through the optimization of certain parameters of the controller, while maintaining its structural constraints. Computer programs have been developed to realize the two algorithms and examples are given to illustrate the feasibility of the techniques.     

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.     

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.     

Algebraic method for the design of output deadbeat controller

Based on the deadbeat model approach and the technique of output spectra matching, an algebraic method for the design of an approximate deadbeat controller with a specified configuration has been proposed. The chosen controller is determined by solving a set of simultaneous algebraic equations. One example is given to show the characteristic features of this approach.     

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     

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.     

Design of linear time-invariant controllers for multirate systems

In this paper we discuss a frequency domain approach to model multirate single-input single-output (SISO) systems which facilitates design of linear time-invariant (LTI) controllers operating at the fast rate. To illustrate the approach we consider a dual-rate system with slow output measurements and fast control actions. We obtain necessary and sufficient conditions for the existence of stabilizing linear time-invariant (LTI) controllers for which model matching is also achieved at the fast rate with a desired single-rate system. Moreover, a solution to the problem of parameterizing the set of such LTI controllers is also given.     

A technique for choosing zero locations for minimal overshoot

An approach is presented to the specification of optimal overshoot controllers when the controller order is fixed and the closed-loop system poles are at a fixed location. The essential effect of the technique is to pick the best possible locations of the free zeros of the system in terms of minimizing the overshoot. The solution is obtained by solving an affine minimax optimization problem. An example illustrates how this technique can be used as a design aid when it is necessary to trade off conflicting design requirements, such as overshoot and settling-time specifications. It is shown that the method is not restricted to the deadbeat case, but can also be used for any prescribed set of closed-loop poles     

A numerical optimization approach for tuning fuzzy logiccontrollers

This paper develops a method to tune fuzzy controllers using numerical optimization. The main attribute of this approach is that it allows fuzzy logic controllers to be tuned to achieve global performance requirements. Furthermore, this approach allows design constraints to be implemented during the tuning process. The method tunes the controller by parameterizing the membership functions for error, change-in-error, and control output. The resulting parameters form a design vector which is iteratively changed to minimize an objective function. The minimal objective function results in an optimal performance of the system. A spacecraft mounted science payload line-of-sight pointing control is used to demonstrate results.