Number Of Sampling Periods Research Articles

An On Line Solution to Sampled-data Time Optimal Control†

ABSTRACT The problem considered is that of designing a special purpose computer that is rapid enough to provide an on line solution to the sampled-data time optimal control problem in which the inputs are subjected to amplitude constraints. The computer is used to simulate a differential equation whoso steady state solution yields the input sequence that will take the sampled system from a given initial state to the reachable (in a given number of sampling periods) state closest to a specified terminal state. The time optimal control problem is solved by increasing the number of sampling periods until the specified terminal state is reachable. Preliminary results with a fourth order system indicate that computer settling times of a. few milliseconds can be obtained.

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 by X_{k+1} = AX_{k} + Du_{k+1} where A = n \times n constant nonsingular matrix; x k is an n -rowed state vector of the system at t=kT ; D is an n \times 2 constant control matrix with columns d 1 and d 2 ; and u_{k+1} is a 2-rowed control vector with components u^{1}_{k+1} and u^{2}_{k+1} . The control vector u_{k+1} is restricted to be an admissible control, i.e., |u^{i}_{k+1}| \leq 1 for i=1, 2 and k=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 vectors u_{1}, u_{2}, ... which will bring the system to equilibrium (i.e., the state x=0 ) in the minimum number of sampling periods. 2) Determine an optimal strategy, i.e., determine a vector valued function u^{0}(x) of the state x such that if the system is in state x at a sampling instant, u^{0}(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 component u^{i}_{k+1} , i=1, 2, there exists a unique (n-1) -dimensional hypersurface \varepsilon^{i} , i=1, 2 . The optimal strategy u^{0}(x_{k+1}) is then a simple nonlinear function of each of the λ i 's where λ i is the distance of x 0 from \zeta^{i} along a direction parallel to A^{-1}d_{i} , for i=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 of A are distinct or multiple, real or occur in complex conjugate pairs. Furthermore, the theory is directly extendable to the case where D=n \times m constant matrix and u_{k+1} is an m -rowed control vector; m > 2 , subject to the admissibility constraint |u^{i}_{k+1}| \leq 1, i=1, 2, . . ., m .

An optimal strategy for a saturating sampled-data system

Consider the usual sampled-data control system in which the sampler is followed by a zero-order hold and the transfer function is G(s) = 1/s(s+a). Saturation is represented by the fact that the forcing function applied to G(s) may not be larger than 1 in absolute value. The problem is to determine a saturating zero-order hold forcing function which forces the system from an arbitrary initial state to equilibrium in the least number of sampling periods. Such a forcing function is defined as an optimal strategy. The state plane is divided into boundary states and interior states. To each boundary state corresponds a unique optimal strategy. To each interior state correspond infinitely many optimal strategies. From the system parameters a polygonal curve, called the critical curve, is defined in the state plane. An optimal strategy is then proposed in which the required forcing function is simply obtained by computing the distance of the representative point in state plane to the critical curve. A simple computer is proposed to implement this optimal strategy. Finally, the proposed optimal strategy is shown to reduce in the limit as T→0 to that of the corresponding continuous system.