Patent Document ID: 5379210
Application ID: 07919764
Patent Flag: 1

Claim One:
1. An apparatus for physically forcing a physical vector output y of a physical system to converge to a selected physical output vector y.sub.d in accordance with a selected vector tracking property T; in which the output vector y is affected by the internal physical dynamics of the system, external physical disturbances, and the physical responses of physical actuators in the system to an external physical control vector u; and in which the internal dynamics of the system are not necessarily known, and in which the values and effects of the external disturbances are not necessarily known; comprising: (a) means for measuring the difference vector e between the selected output vector y.sub.d and the output vector y; (b) means for determining the vector tracking property T as a selected function of the difference vector e; (c) means for determining an internal control vector w(t), where w(t) is expressible in a form substantially equivalent to EQU w(t)=.mu.w(t-.epsilon.)+VT or to EQU w(t)=kVT .mu. is the internal control gain, where 0&lt;&lt;.mu..ltoreq.1; k=(1-.mu.).sup.-1, whereby k is a high gain; t is the time; .epsilon. is a small, non-negative interval of time; w(t) is a function of t; V=V(q.sup..zeta.,d)=W(q.sup..zeta.)[J(q,d)A.sub..zeta..sup.-1 (q.sup..zeta.)B(q)W(q.sup..zeta.)].sup.-1 ; q is a minimal-order vector of dynamic variables that completely describe the dynamics of the system, insofar as those dynamics affect the output vector y; q.sup..zeta. is a matrix q.sup..zeta. =(q q.sup.(1). .. q.sup.(i). .. q.sup..zeta.-1)), wherein q.sup.(i) is the ith derivative of q with respect to time, and .zeta..gtoreq.0 is the highest order derivative of q needed to describe the system; y=g(q,d) is expressible as an algebraic function g of q and d; d is a vector or vector function of the external disturbances affecting the system; J is the Jacobian matrix (.differential.g/.differential.q); A.sub..zeta. =(.differential..function..sub.1 /.differential.q.sup.(.zeta.)), a matrix or matrix function determined by the dynamics of the system; .function..sub. =.function..sub. 1(q, q.sup.(1),. .. , q.sup.(.zeta.)), a .zeta.-th order differential vector function of q which represents the dynamic properties of the system; B=(.differential..function..sub.2 /.differential.b), a matrix or matrix function determined by the dynamics of the actuators: .function..sub.2 =.function..sub.2 (u, u.sup.(1),. .. , u.sup.(i),. .. , u.sup.(.eta.)), an .eta.-th order differential vector function of u, wherein u.sup.(i) is the ith derivative of u with respect to time, and .eta..gtoreq.0 is the highest order derivative of u needed to described the control input; b=b(u, u.sup.(1),. .. , u.sup.(.eta.)) is a vector function of u and its derivatives describing the effect of u on the actuators; is the dimension of vector u; n is the dimension of vector y; W(q.sup..zeta.)=[J(q,d)A.sub..zeta..sup.-1 (q.sup..zeta.)B(q)].sup.T if r&gt;n; W(q.sup..zeta.)=I, the identity matrix, if r=n; (d) means for deriving the external control vector u from the internal control vector w from a relationship which is expressible in a form substantially equivalent to EQU b(u, u.sup.(1),. .. , u.sup.(.eta.))=w(t); (e) transmitting the external physical control vector u to the actuators, whereby a physical response in the actuators is induced, physically causing the physical output vector y to converge toward the selected physical output vector y.sub.d ; and (f) repeating steps (a) through (e) sufficiently to physically force the physical output vector y to converge to the selected physical output vector y.sub.d in accordance with the selected vector tracking property T.