Simple Algebraic Solution Research Articles

Time-optimal output regulators for linear multivariate discrete-time systems Part 2. All classes of right-invertible systems

In Part 1 of this paper (Hassan and Amin 1987) a simple algebraic solution for the time-optimal output regulator problem was presented. This solution, which consists of a state feedback control law, has been obtained for all classes of right-invertible decouplable systems S(A, B, C, E). The results of Part 1 are here extended to all classes of right-invertible systems S(A, B, C, E). A set of optimal output deadbeat indices (called the ‘optimal set’) is defined and related to the observability indices of the optimal closed-loop system. The time-optimal output regulator problem for a right-invertible non-decouplable system S(A, B, C, E) is resolved by transforming S into a decouplable system Sc(A, B, Cc, Ec) having the optimal output deadbeat index σ* of S. First, an algorithm is presented to construct iteratively, in a well-defined optimal sense, a unimodular left compensator L(z) and a compensated decouplable system Sc(A, B, Cc, Ec) from the state-space parameters of S. Then, a family of optimal state fee...

Time-optimal output regulators for linear multivariable discrete-time systems Part 1. Right invertible decouplable systems

Corsetti and Houpis (1985) presented a simple algebraic solution for the timeoptimal output regulator problem. This solution has been obtained for a special class of right invertible decouplable systems S(A, B, C) satisfying the condition that CB is of full rank. In this paper a general method is presented to extend the results of Corsetti and Houpis to all classes of right invertible decouplable systems S(A, B, C, E). It is shown that a simple state feedback control law drives each output of S(A, B, C, E) in a minimum number of steps equal to the order of the associated infinite zero. This state feedback represents the optimal solution for the considered time-optimal output regulator problem. A class of optimal solutions, parametrized by a free parameter matrix K, is obtained for non-square systems. The properties of the resulting optimal closed-loop system is given and a numerical example is worked out to illustrate the generality and feasibility of the proposed method.

An exact algebraic solution for the recursive path model with manifest variables

It is shown that the classical recursive path “model” By = Ax + Uz, where Cov(x,z) = 0 and B is lower triangular, has a very simple algebraic solution that is essentially given by the choleski factor of the partial variance-covariance matrix, Var (y|x). This solution always exists, and it is unique provided only that Var(x) and Var(y|x) are nonsingular. It may be useful for exploratory work and also as a starting configuration to speed up convergence of iterative estimation schemes.

On Direct Algebraic Solutions of Linear Differential Equations using Walsh Transforms

Numerical integral and differential operators are developed using Walsh transforms. These are matrices with coefficients for integrating Walsh functions and their inverses. Their application to linear differential equations yields a simple algebraic solution. It is clear that this solution is the mean value over each subinterval of the true solution to the defined differential equation.

Use of the numerical phase-difference filter for the digital channel bank

The use of the numerical phase-difference filter in the single-sideband modulation scheme for frequency-division multiplexing is investigated and compared with the Weaver-modulator scheme proposed by Freeny et al. A simple algebraic solution to the approximation problem for arbitrary specifications in the stopband is given. The sensitivity of the stopband attenuation is the main problem that precludes the use of RC active structures; it is shown that the digital version can be implemented with a reasonable coefficients wordlength. The structure of the filter is optimised for round-off noise performance. Finally, the two modulation schemes are compared.

An algebraic solution for a class of subjective metrics models

It is shown that an obvious generalization of the subjective metrics model by Bloxom, Horan, Carroll and Chang has a very simple algebraic solution which was previously considered by Meredith in a different context. This solution is readily adapted to the special case treated by Bloxom, Horan, Carroll and Chang. In addition to being very simple, this algebraic solution also permits testing the constraints of these models explicitly. A numerical example is given.

Constant-Temperature Magneto-Gasdynamic Channel Flow

In the course of investigating boundary-layer flow in continuous plasma accelerators with crossed electric and magnetic fields, it was found advantageous to have at hand simple closed-form solutions for the magneto-gas dynamic flow in the duct which could serve as free-stream conditions for the boundary layers. Nontrivial solutions of this sort are not available at present. and in fact, as in the work of Resler and Sears, the variation of conditions along the flow axis must be obtained through numerical integration. Consequently, some simple solutions of magneto-gasdynamic channel flow were sought, possessing sufficient algebraic simplicity to serve as free-stream boundary conditions for analytic investigations of the boundary layer in a physically reasonable accelerator. In particular, since the cooling of the accelerator tube is likely to be an important physical problem because of the high gas temperatures required to provide sufficient gaseous conductivity, channel flow with constant temperature appears interesting. Some simple algebraic solutions for the case of a constant temperature plasma are developed in the following paragraphs.

How Old Is Ann?

Although it is not at all a difficult problem, it occurred to me that perhaps the readers of The Mathematics Teacher might be interested to see the simple algebraic solution to Dr. A. A. Bennett's “How Old Is Ann?” printed at the end of his article in the November (1937) issue.