The multiple-input minimal time regulator problem (General theory)

This work considers a two-input linear time-invariant discrete system whose state transition equation is given byXₖ₊₁ = AXₖ + Duₖ₊₁whereA = n × nconstant nonsingular matrix; xkis ann-rowed state vector of the system att=kT;Dis ann × 2constant control matrix with columns d1and d2; anduₖ₊₁is a 2-rowed control vector with componentsu¹ₖ₊₁andu²ₖ₊₁. The control vectoruₖ₊₁is restricted to be an admissible control, i.e.,|uⁱₖ₊₁| ≤ 1fori=1, 2andk=0, 1, .... The two-input minimal time regulator problem may be stated as follows 1) Given any arbitrary initial state of the system, find admissible control vectorsu₁, u₂, ...which will bring the system to equilibrium (i.e., the statex=0) in the minimum number of sampling periods. 2) Determine an optimal strategy, i.e., determine a vector valued functionu⁰(x)of the statexsuch that if the system is in statexat a sampling instant,u⁰(x)is an admissible optimal control for the next sampling period. First, the general necessary and sufficient conditions for the system to be controllable with admissible controls are established. For a controllable system it is shown that the optimal strategy at each sampling instant requires the following: For each componentuⁱₖ₊₁,i=1, 2,there exists a unique(n-1)-dimensional hypersurfaceεⁱ,i=1, 2. The optimal strategyu⁰(xₖ₊₁)is then a simple nonlinear function of each of the λi's where λiis the distance of x0fromζⁱalong a direction parallel toA⁻¹dᵢ, fori=1, 2. This optimal strategy therefore satisfies the operations of the feedback computer in order that the system returns to equilibrium in minimum time after any arbitrary disturbances. The results of this work are applicable to all discrete systems of the above form which are controllable by admissible controls irrespective of whether the eigenvalues ofAare distinct or multiple, real or occur in complex conjugate pairs. Furthermore, the theory is directly extendable to the case whereD=n × mconstant matrix anduₖ₊₁is anm-rowed control vector;m > 2, subject to the admissibility constraint|uⁱₖ₊₁| ≤ 1, i=1, 2, . . ., m.