Generalized Signal Decoupling Problem with Stability for Discrete-Time Systems
Journal of Optimization Theory and Applications
This paper deals with decoupling problems of unknown, measurable and previewed signals. First the well known solutions of unknown and measurable disturbance decoupling problems are recalled. Then new necessary and sufficient constructive conditions for the previewed signal decoupling problem are proposed. The discrete time case is considered. In this domain previewing a signal by p steps means that the k-th sample of the signal to be decoupled is known p steps in advance. The main result is to
... rove that the stability condition for all of the mentioned decoupling problems does not change, i.e. the resolving subspace to be stabilized is the same independently of the type of signal to be decoupled, being it completely unknown (disturbance), measured or previewed. The problem has been studied through self-bounded controlled invariants, thus minimizing the dimension of the resolving subspace which corresponds to the infimum of a lattice. Note that reduced dimension of the resolving controlled invariant subspace yields to reduce the order of the controller units. In this paper the decoupling control problem is approached in a more general setting. Signals which are known in advance or previewed by a given amount of time are considered. Such problem will be referred to as previewed signal decoupling problem (PSDP). The PSDP has been investigated by Willems  who first derived, in the continuous time domain, a necessary and sufficient condition to solve the PSDP with pole placement. This solution was based on the so called proportional-integral-derivative control laws consisting of a feedback of the state system and of a linear combination of signal (to be decoupled) and its time derivatives. The major drawback of these extensions of the disturbance decoupling problem in continuous time domain is that control laws include distributions, hence are not practically implementable. Independently, Imai and Shinozuka  proposed a similar necessary and sufficient condition for the PSDP with stability in both discrete and continuous time cases. In , Estrada and Malabre proposed a synthesis procedure to solve the PSDP problem with stability using the minimum number of required differentiators for the signal to be decoupled. Conditions for the PSDP to be solved, given in [12, 9, 1] , do not care about dimensionality of the resolving controlled invariant subspace. Furthermore, to the best of our knowledge, the problem of reducing the dimension of the resolving subspace for the PSDP has not been thoroughly investigated in the literature. Note that using controlled invariants of minimal dimensions yields to reduce the order of the controller units and possible state observers. In this paper a new solution for the PSDP with stability based on a subspace with reduced dimension is proposed. Such dimension optimization is obtained thanks to self-bounded controlled invariants. This is a special class of controlled invariants introduced by Basile and Marro in [7, 11] which enjoys interesting properties, the most important of which is to be a lattice instead of a semilattice, hence to admit an infimum other than a supremum. Moreover this paper provides a unique necessary and sufficient condition for signal decoupling problems with stability independently of the type of signal to be decoupled, being it completely unknown (disturbance), measured or previewed. In other terms it is shown that the proposed resolving subspace for the PSDP problems and the well known resolving subspace of the DDP problem proposed by Basile Marro and Piazzi in  are equivalent. The proofs are carried out in a geometric framework and are based on several lattices of self bounded (A, B)-controlled invariants. In this paper discrete-time systems are considered. In such domain, the solution of the PSDP is more elegant and is practically implementable. The structure of the compensator, whereby the signal decoupling of previewed signals is obtained, is discussed. It consists of a preaction and a postaction unit. A new synthesis procedure, based on geometric approach algorithms, is provided. Preliminary results of this work have been presented in [2, 3]. The following notation is used. R stands for the field of real numbers. Sets, vector spaces and subspaces are denoted by script capitals like X , I, V, etc.. Since most of the geometric theory of dynamic system herein presented is developed in the vector space R n , we reserve the symbol X for the full space, i.e., we assume X := R n . Matrices and linear maps are denoted by slanted capitals like A, B, etc., the image and the null space of the generic matrix or linear transformation A by imA and kerA, respectively, the transpose of the generic real matrix A by A T , its spectrum by σ(A) and its pseudoinverse by A # . The reminder of this paper is organized as follows. Section 2 presents the structural conditions for the general PSDP. In Section 3 new necessary and sufficient conditions for the PSDP with stability are stated, in section 4 a synthesis procedure for the decoupling compensator is reported and finally in Section 5 an illustrative example is discussed.