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.     

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     

Self-tuning deadbeat controllers

A deterministic self-tuning deadbeat control scheme is presented which is extremely simple to implement in the single-variable and multivariable cases. Its convergence properties are analysed by straightforward application of de Larminat's convergence results.     

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.     

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.     

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.     

Multipurpose deadbeat controllers for multivariable systems

A simple and direct deadbeat control design algorithm for discrete-time multivariable systems is introduced. Our brief was to synthesize a realizable controller to achieve the following objectives: (1) input-output decoupling, (2) deadbeat tracking of any prespecified class of changeable deterministic reference signals and (3) changeable deterministic disturbances rejection where the changeable reference signals and disturbances can be different at each channel. Since the internal stability requirement is satisfied, our design algorithm can easily handle both unstable and non-minimum phase systems. Some constraints on the transfer function are also derived to make sure that the derived controller is realizable. Since all solutions can be described in parametric form, some of the performance criteria can be combined.     

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,     

A convex parameterization of robustly stabilizing controllers

This paper treats synthesis of robust controllers for linear time-invariant systems. Uncertain real parameters are assumed to appear linearly in the closed loop characteristic polynomial. The main contribution is to give a convex parameterization of all controllers that simultaneously stabilize the system for all possible parameter combinations. With the new parameterization, certain robust performance problems can be stated in terms of quasi-convex optimization     

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     

A parameterization of reduced stable models and controllers

This note addresses the problem of instability in both model and controller reduction methods based on similarity transformations. For a given model (controller) a general framework is proposed that parameterizes a large set of reduced models (controllers) that preserve the stability of the original model (closed loop system). In addition, a sufficient condition for the existence of such a framework is derived. As an application of the main results, it is shown how different reduction methods can be modified, if they fail to maintain stability.     

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.     

Decentralized stable factors and a parameterization ofdecentralized controllers

Decentralized stable factors (DSF) are used to extend the stable factorization approach to the design of decentralized controllers. The DSF allow a parameterization of all the stabilizing decentralized controllers in the form of a Youla parameterization. A decentralized controller serves as the central controller, and the Youla parameter has a particular structure and must satisfy an interaction constraint. This parameterization encompasses several other parameterization results on decentralized control systems     

A direct computation of state deadbeat feedback gains

A method for computing a feedback gain that achieves state deadbeat control is given. From systems given in the staircase form, this method derives the deadbeat gain in a numerically reliable way. It is shown that the gain turns out to be LQ optimal for some weightings     

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     

Rough controllers with state feedback

This paper addresses a new approach to design rule-based controllers using concepts of rough sets and techniques of state feedback. The goal is to obtain rule-based models that allow the construction of control loops, ensuring stable conditions and suitable dynamic characteristics for nonlinear systems. The study performs a comparison of the procedure proposed with results obtained with a conventional control, where a system with nonlinear behavior is used. Numerical examples derived from computer simulations and real applications are shown. Experiments in a level plant were performed, checking the potential of the rule-based controllers.     

A partial parameterization of nonlinear output feedback controllers for saturated linear systems

This paper addresses the stabilization problem of linear systems subject to input saturation. The major purpose is to introduce nonlinearity into control laws so as to expand the design freedom for performance enhancement. To this end, a new approach is developed which involves a partial differential matrix inequality (PDMI). A class of stabilizing feedback laws is explicitly obtained by solving this PDMI analytically and the feedback laws are parameterized by a nonlinear function. Further, it is revealed that any linear observer can be used to realize the output feedback stabilization. Numerical examples, including the seek control of a hard disk drive, show that the introduced nonlinearity does contribute to the improvement of system performance. The application to integral control is also discussed.     

Damping of harmonic disturbances in sampled-data systems—parameterization of all optimal controllers

Optimal damping of harmonic disturbances of known frequencies is studied for sampled-data systems. A sampled-data output feedback controller is designed to minimize the intersample variations of the controlled variable. The set of all stabilizing optimal controllers is obtained in terms of the Youla parameterization and a set of interpolation conditions at the disturbance frequencies, which ensure that the stationary cost is minimized.     

An algorithm to compute state feedback matrices for multi-input deadbeat control

An algorithm for the computation of a state feedback controller shifting all eigenvalues of the closed loop system to the origin is presented. The available freedom in multi-input systems is used to reduce the norm of the feedback matrix. The algorithm places one or more eigenvalues in each step using partial feedback laws of minimal norm. This results in an overall feedback matrix whose norm is close to its minimum value in most cases.