An algorithm for constructing a minimal partial realization in the multivariable case

It is known that the Berlekamp-Massey algorithm from information theory can be used to compute scalar minimal partial realizations. Recently, it has been interpreted in terms of exact modeling of behaviors. In this paper, we extend these results to the multivariable case. We put the behavioral theory of exact modeling to work in deriving an extension of the Berlekamp-Massey algorithm for computing multivariable minimal partial realizations.     

The set of all minimal partial realizations

This paper is concerned with the uniqueness of minimal partial realizations. Earlier papers are typically concerned with the problem of the existence and the determination of a matrix triple(tilde{A},tilde{B},tilde{C})which is a minimal partial realization of a sequence{Y_{1},...,Y_{m}}of Markov parameters. In this paper a parameterized realization(A(y),B, C)(yis a parameter vector) is defined which characterizes the set of all minimal partial realizations of the sequence{ Y_{1},. . . , Y_{m}}. An example is provided and the utility of the parameterization is discussed.     

Nonisomorphic classes of inhomogeneous bilinear realizations: Realizability and existence

This paper is concerned with the nonisomorphic classes of the inhomogeneos bilinear realizations with unknown initial state.A necessary and sufficient condition is given for the factorizability of the generalized Hankel matrix into the specified form. And then a new realizability condition for inhomogeneous bilinear systems is obtained, which is of practical use.A necessary and sufficient condition for the existence of nonisomorphic (in the sense of linear transformation) spancanonical realizations is given. And the parameters determining the classes of nonisomorphic realizations are obtained in explicit form. As a result, it is shown that if nonisomorphic realizations exist, the cardinality of the family of nonisomorphic classes is equal to that of the real number field.     

Least order,stable solution of the exact model matching problem

The set of all solutions of the minimal design problem (MDP) is presented in parametric form. This result is obtained by first deriving a parametric representation of all minimal bases of a particular vector space. From this the set of all solutions of the MDP are obtained. Then conditions are placed on the parameters, which conditions define the set of stable solutions of the MDP.Solutions to the nonminimal model matching problem are also presented in parametric form, and it is shown how in principle solutions of successively higher degree can be searched for a stable solution, should the MDP have no stable solution, so that a solution to the model matching problem can be found which has the lowest order consistent with a stability constraint.     

Realization by inspection

We investigate which first-order representations can be obtained from high-order representations of linear systems “by inspection”, that is, just by rearrangement of the data. Under quite weak conditions it is possible to obtain minimal realizations in the so-called pencil form; under stronger conditions one can obtain minimal realizations in standard state-space form by inspection. The development is based on a reformulation of the realization problem as a problem of finding a complete set of basis vectors for the nullspace of a given constant matrix. Since no numerical computation is needed, the realization method in particular is suitable for situations in which some of the coefficients are symbolic rather than numerical     

Decoupling linear multiinput multioutput plants by dynamic output feedback: An algebraic theory

This paper presents an algebraic theory for the design of a decoupling compensator for linear time-invariant multiinput multioutput systems. The design method uses a two-input one-output compensator, which gives a convenient parametrization of all diagonal input-output (I/ O) maps and all disturbance-to-output (D/O) maps achievable by a stabilizing compensator for a given plant. It is shown that this method has two degrees of freedom: any achievable diagonal I/O map and any achievable D/O map can be realized simultaneously by a choice of an appropriate compensator. The difference between all achievable diagonal and nondiagonal I/O maps and the "cost" of decoupling is discussed for some particular algebraic settings.     

Nudelman interpolation, parametrizations of lossless functions and balanced realizations

We investigate the parametrization issue for discrete-time stable all-pass multivariable systems by means of a Schur algorithm involving a Nudelman interpolation condition. A recursive construction of balanced realizations is associated with it, that possesses a very good numerical behavior. Several atlases of charts or families of local parametrizations are presented and for each atlas a chart selection strategy is proposed. The last one can be viewed as a nice mutual encoding property of lossless functions and turns out to be very efficient. These parametrizations allow for solving optimization problems within the fields of system identification and optimal control.     

Convergence properties of constrained linear system under MPC control law using affine disturbance feedback

This paper shows new convergence properties of constrained linear discrete time system with bounded disturbances under Model Predictive Control (MPC) law. The MPC control law is obtained using an affine disturbance feedback parametrization with an additional linear state feedback term. This parametrization has the same representative ability as some recent disturbance feedback parametrization, but its choice together with an appropriate cost function results in a different closed-loop convergence property. More exactly, the state of the closed-loop system converges to a minimal invariant set with probability one. Deterministic convergence to the same minimal invariant set is also possible if a less intuitive cost function is used. Numerical experiments are provided that validate the results.     

Input/Output Decoupling of Square Linear Systems by Dynamic Two‐Parameter Stabilizing Control

Analytical expressions of the free parameters of a two‐parameter stabilizing control (TPSC), solving an input/output (I/O) decoupling problem, are presented, and stability conditions are given. Multi‐input‐multi‐output (MIMO), proper, lumped and linear time invariant (LTI) systems are considered. These systems have stabilizable and detectable realizations. The separation principle is applied to design a dynamic output control in a controller‐observer feedback configuration. The I/O relation of the overall system is equivalent to a subsystem, in which the I/O decoupling problem has a solution. Also, if the state dimension of the plant is even, and is double the input dimension of the plant, then coprime factorizations of the plant used for the stabilizing controllers are proposed. The results are illustrated through an example.     

On the realization of periodic functions

In this article, canonical and spectrally minimal infinite-dimensional state space realizations for periodic functions are considered. It is shown that the periodic functions having ℓ₁ Fourier coefficients are precisely those realizable by a Riesz spectral system (RSS) in which the system operator generates a periodic strongly continuous semigroup and the observation operator is bounded. This realization can easily be converted to a canonical and spectrally minimal form. It is shown how the use of RSS and Cesáro sums of Fourier series allows the construction of a state space realization for a given periodic function merely integrable over its period. Simple finite-dimensional approximations with error bounds are derived for the RSS realization. Regular well-posed linear systems (WPLS) are used to construct a Fuhrmann-type realization for a given periodic function integrable over its period. It is shown that the RSS realizations and WPLS realizations are precisely equally good at coping with the possible ill behavior of a given bounded periodic function integrable over its period, but the WPLS realization is not always spectrally minimal or canonical.     

Parametric Jordan Form Assignment Revisited

The Jordan form assignment problem (complete and partial) for linear multivariable control system is formulated and completely solved by the unified Sylvester matrix equation approach. A new explicit parametrization of all state feedbacks assigning an admissible Jordan form through the minimum number of parameters is provided. The main advantage of this parametrization, unlike previous ones, is that it is explicit; thus, it can be used, not only for numeric, but also symbolic computations. This fact is demonstrated on the problem of parametrization of all output feedbacks assigning an admissible Jordan form, where Gröbner basis method is applied. Developed algorithms for Jordan form assignment by state and output feedback are implemented in the freely available Maple package.     

Generalized dynamic covers for linear systems with applications to deterministic identification and realization problems

Several deterministic identification problems, with partial realization being a special case, are unified in the framework of the mathematical problem "generalized dynamic covers." An algorithm to find such a minimal dynamic cover as well as a uniqueness criterion is given, which yields several identifiability results. This problem also includes the "observer" and the "exact model matching" problems, as well as the problem of finding "minimal inverses for linear systems with arbitrary initial states." It is shown that the problem of finding minimal realizations from a transfer function matrix can also be solved in this framework.     

A method of constructing minimal approximate realizations of linear input-output behavior

An algorithm for constructing minimal realizations of stationary linear dynamical systems is presented. The ‘input’ to this algorithm is the data set representation. In the case of the minimal partial realization problem this algorithm provides for several different approximations to prescribed input-output behavior.     

Minimal parametrizations of single-input linear dynamical systems

The parametrization of the set of all controllable single-input time-invariant linear dynamical systems is defined as a map π from a parameter set M_π onto a given set of canonical forms 𝒸_π. A parametrization is called ‘minimal’ if it induces a continuous canonical map and the number of parameters is minimal. Some of its general properties, necessary and sufficient conditions, and several new concrete forms suitable for identification are also given.     

Minimal coverings for incompletely specified sequential machines

Summary In the literature the problem of finding minimal realizations for incompletely specified machines has been treated in a number of different ways. Solutions to this problem depend on the precise definition of what minimal realization means in this case. If the behaviour of a state is definied as its related partial input-output function then the behaviour of one machine A can cover the behaviour of another machine B if it contains better definied I/O functions for all states of B. Finding a minimal covering in this case is known to be NP-complete. We develop an algebraic treatment of the problem and give a homomorphic characterization of the covering-relation. The construction of state-splitting is also characterized as a special morphism. Then a heuristic method is proposed for finding minimal coverings.     

Continuity properties of balanced realizations

In this paper topological and geometrical properties of pre-balanced and balanced realizations are considered. It is shown that analytic pre-balancing coordinate transformations do exist and that the set of pre-balanced realizations forms an analytic submanifold. Explicit formulas for the tangent spaces and their dimension are derived. Continuity properties of balanced realizations are studied. For systems with more than two inputs and two outputs explicit points of discontinuity of balancing coordinate transformations are described. A certain class of balancing coordinate transformations is shown to be discontinuous, even for distinct and fixed singular values.     

Searching a minimal semantically-equivalent subset of a set of partial values

Imprecise data exist in databases due to their unavailability or to data/ schema incompatibilities in a multidatabase system. Partial values have been used to represent imprecise data. Manipulation of partial values is therefore necessary to process queries involving imprecise data. In this article, we study the problem of eliminating redundant partial values that result from a projection on an attribute with partial values. The redundancy of partial values is defined through the interpretation of a set of partial values. This problem is equivalent to searching a minimal semantically-equivalent subset of a set of partial values. A semantically-equivalent subset contains exactly the same information as the original set. We derive a set of useful properties and apply a graph matching technique to develop an efficient algorithm for searching such a minimal subset and therefore eliminating redundant partial values. By this process, we not only provide a concise answer to the user, but also reduce the communication cost when partial values are requested to be transmitted from one site to another site in a distributed environment. Moreover, further manipulation of the partial values can be simplified. This work is also extended to the case of multi-attribute projections.     

Balancing related methods for minimal realization of periodic systems

We propose balancing related numerically reliable methods to compute minimal realizations of linear periodic systems with time-varying dimensions. The first method belongs to the family of square-root methods with guaranteed enhanced computational accuracy and can be used to compute balanced minimal order realizations. An alternative balancing-free square-root method has the advantage of a potentially better numerical accuracy in case of poorly scaled original systems. The key numerical computation in both methods is the solution of nonnegative periodic Lyapunov equations directly for the Cholesky factors of the solutions. For this purpose, a numerically reliable computational algorithm is proposed to solve nonnegative periodic Lyapunov equations with time-varying dimensions.