Pole assignment using matrix generalized inverses

The problem of pole assignment in a completely controllable linear time-invariant system dx/t = Ax + Bu, y = Cx is considered. A method using matrix generalized inverses is developed for the computation of a matrix K such that the matrix A + BK has prescribed eigenvalues which need satisfy only the condition that a certain number of them are distinct and real; then a feedback law of the form u = r + Kx can be used to achieve the desired pole-placement. The method does not require solution of sets of non-linear equations or manipulation of polynomial matrices, and no knowledge of eigenvalues and/or eigenvectors of A is necessary. If the computed matrix K and the given matrix C satisfy a consistency condition, a matrix Kν such that KνC = K can be directly obtained from K and the desired pole-placement can be realized by an output feedback law u = r + Kνy.     

General eigenvalue placement in linear control systems by output feedback

Consider the time-invariant system E[xdot] = Ax + Bu, y = Cx where E is a square matrix that may be singular. The problem is to find constant matrices K and L, such that the feedback law u = Ky+L[ydot] yields x = exp (λt)v_i (where v_i is some constant vector) for some preassigned λ_i (i=l, 2, [tdot], r). This problem is equivalent to that of finding K and L which makes a preassigned λ _i an eigenvalue corresponding to the general eigenvalue problem {λ(E ? BLC) ? (A + BKC)}v=0. Using matrix generalized inverses, a method is developed for the construction of a linear system of equations from which the elements of K and L may be computed.     

Non-classicalH 1 projection and Galerkin methods for non-linear parabolic integro-differential equations

In this paper the Galerkin method is analyzed for the following nonlinear integro-differential equation of parabolic type: $$c(u)u_t = \nabla \cdot \{ a(u)\nabla u + \int_0^t {b(x, t, r, u(x, r))} \nabla u(x, r) dr\} + f (u)$$ Optimal L ² error estimates for Crank-Nicolson and extrapolated Crank-Nicolson approximations are derived by using a non-classicalH ¹ projection associated with the above equation. Both schemes result in procedures which are second order correct in time, but the latter requires the solution of a linear algebraic system only once per time step.     

Smallest destabilizing perturbations for linear systems

Necessary and sufficient conditions on a perturbation £ that will guarantee the asymptotic stability of the linear system [xdot] = (A + E)x given that the nominal system [xdot] = Ax is asymptotically stable are derived. The class of all perturbations E of minimal norm that will destabilize the nominal system is characterized in terms of the norm of the resolvent matrix (iωI — A)_?1 for an appropriate ω ? R These results are specialized to the euclidean norm and expressed in terms of singular values. Analogous results are also obtained for the difference equation x(k + 1) = Ax(k) (k = 0,1,2,.)     

Dyadic output feedback laws generated as Kronecker products : optimal and suboptimal solutions

The problem of eigenvalue assignment in the system dx/dt equals; Ax + Bu, y = Dx, using the dyadic output feedback law u = u₀ + q · p^Ty is considered via a formulation developed earlier by the author, in which p and q occur in the Kronecker product vector p?q. The equations governing the values of p and q which give an optimum approximation to a prescribed spectrum of eigenvalues are derived, and a special case is solved. Various facets of the problem of generating suboptimal solutions are discussed.     

Applications of complete discrimination system for polynomial for classifications of traveling wave solutions to nonlinear differential equations

If a partial differential equation is reduced to an ordinary differential equation in the form u^′(ξ)=G(u,θ₁,…,θm) under the traveling wave transformation, where θ₁,…,θm are parameters, its solutions can be written as an integral form . Therefore, the key steps are to determine the parameters' scopes and to solve the corresponding integral. When G is related to a polynomial, a mathematical tool named complete discrimination system for polynomial is applied to this problem so that the parameter's scopes can be determined easily. The complete discrimination system for polynomial is a natural generalization of the discrimination △=b²−4ac of the second degree polynomial ax²+bx+c. For example, the complete discrimination system for the third degree polynomial F(w)=w³+d₂w²+d₁w+d₀ is given by and . In the paper, we give some new applications of the complete discrimination system for polynomial, that is, we give the classifications of traveling wave solutions to some nonlinear differential equations through solving the corresponding integrals. In finally, as a result, we give a partial answer to a problem on Fan's expansion method.     

广义系统的能量受限的输出调节

本文在文[1]的工作基础上讨论了广义系统通过MPD反馈的能量受限的输出调节问题,得到了相应的充分条件及算法。     

Identification and adaptive control of SISO singular systems

A general adaptive control scheme is presented for an unknown time invariant singular system of the form Ex(t + 1) = Ax(t) + Bu(t); y(t) = Cx(t). Owing to the non-causality of this kind of system, the identification of unknown parameters is re-considered and a new residual signal is constructed and used in the recursive calculations. A general design procedure is obtained that uses the identified parameters and includes two steps: (i) preliminary output feedback gain design in order to make the original system causal; (ii) adaptive control design for the causal system. It has been shown that any adaptive control algorithm can be combined with this scheme to obtain a globally stable closed-loop system. The design procedure is shown to perform well on a simulation of a third-order singular system     

An Unconditionally Stable ADI Method for the Linear Hyperbolic Equation in Three Space Dimensions

An unconditionally stable alternating direction implicit (ADI) method of O(k 2 +h 2 ) of Lees type for solving the three space dimensional linear hyperbolic equation u tt +2 f u t + g 2 u = u xx + u yy + u zz + f ( x , y , z , t ), 0<x, y, z<1, t>0 subject to appropriate initial and Dirichlet boundary conditions is proposed, where f >0 and g S 0 are real numbers. For this method, we use a single computational cell. The resulting system of algebraic equations is solved by three step split method. The new method is demonstrated by a suitable numerical example.     

Determination of zeros and zero directions of linear time-invariant systems by matrix generalized inverses

Consider the linear control system whose state-space equations are dx/dl = Ax + Bu, y = Cx + Du where x, u and y are, respectively, the state, input and output vectors, and A, B, C, D are constant matrices. Problem : find constant scalars r and constant vectors v, w such that an input of the form v exp (rt) 1(t) will yield a state of the form x = w exp (rt) and output y ≡ 0, t ≥ 0. The method developed is this: the output equation is solved for x and the solution is substituted into the state equation. The resulting equation must satisfy a consistency condition involving r, in order for a solution of the problem to exist. Matrix generalized inverses are employed both to derive consistency conditions and to obtain solutions. Several examples illustrate a variety of conditions.     

On the solution of a matrix equation in the theory of Luenberger observers

An explicit solution is presented for a matrix equation involved in the design of a Luenberger observer to realize an acceptable approximation to a desired state-variable feedback lawu = u_{0} + Kxfor the systemdx/dt = Ax + Bu, y = Dx.     

An improved algorithm for optimal state regulation of generalized systems

The system considered is a generalized system E = Ax + Bu, Y = Cx with matrix E singular. When the impulsive modes of the system are both controllable and observable, it is shown in this paper that these modes can be eliminated by almost any constant output feedback. For a strongly controllable and observable generalized system, after its impulsive modes are eliminated, the poles of the system can be assigned arbitrarily by state feedback. Even if the states are not available, the paper adopts an efficient method to design a compensator for the system.     

Algorithms of constructing a stabilizing solution of the algebraic Riccati equation based on its linear reduction in the problems of analytical construction of controllers

In the problem of the stabilizing solution of the algebraic Riccati equation, the resolvent Θ(s) = (s I _2n ? H)^?1 of the Hamilton 2n × 2n-matrix H of the algebraic Riccati equation allows us to reduce the problem to a linear matrix equation. In [1], the constructions necessary for this and the theorem of existence and representation of the stabilized solutions to an algebraic Riccati equation was proposed. In this paper, the methods of constructing the resolvent and the linear reduction matrix defined by it necessary for the application of the theorem, and in addition, the algorithms of constructing stabilizing solution of the algebraic Riccati equation are proposed.