Optimal output feedback for nonzero set point regulation

D. Bernstein, W. Haddad
1987 IEEE Transactions on Automatic Control  
6 4 1 function K are given by (1 1) and (19), respectively, the optimal control is u= [ 2b2x:/q2Ix2(x;+2x;)I 1 ' -261x1/ql [x? + 2x;1 Proof: Evaluating the mathematical expectation in (8) we find that exp [ -F(x, t)/2] =x,/[x:+ 2x:] (21) and formula (20) then follows from (IO). The termination set D considered in this section depended only on xz(T), so that we had a one-barrier problem. In the next section we will obtain the optimal control in the first quadrant; that is, we will solve a
more » ... will solve a two-barrier problem. m. OPTIMAL CONTROL IN THE FIRST QUADRANT In this section we assume that the process x ( t ) starts in the region x 2 ( t ) ) : x,(t)>O, x2(t)>OI (23) and we want to find the control that will allow us to leave C at minimal cost. We consider the terminal loss function That is, we want the process x ( t ) to leave the continuation region C through the origin. When the continuation region is the first quadrant, the first passage from ((x" xd, t ) to ((yl, yz), T ) for the uncontrolled process dx,/dt=c, has the probability density r where s = Tt and ~( x , y , z)=exp { -[ ( x -y ) ' + z 2 ] / 2 s } -e x p { -[ ( x +~)~+ z~] /~s ) . We may check that so that the optimal control can be obtained from (8) and (IO). find that If the proportionality constant c in formula (7) is equal to 2, we easily exp t -F ( x , 0121 = U ( x l , x d + W X Z , X I ) (28) where U(x,y)=[2x/n(yZ+2x2)"2] arctan [y/(y2+2x2)'/2]. Proposition 3: When the continuation region C is the first quadrant and the terminal loss function is given by (24) then, if c = 2, the optimal control is r where G = exp [ -F(x, t)/2] is given by (28) and GXl = 2~; / n M~'~ arctan ( X~/ M~/~) -X~X~/ ?~N~/~ arctan ( X~/ N I /~) + ~x Z ( X I -x : ) / r N M with M = x : + 2 x : and N = x : + 2 x : (and GX2 is the same as GXI but with x1 and x2 interchanged). Proof: Equation (29) follows from (10) since F,= -2G,/G. REFERENCES P. Whittle, Optimization over Time, vol. I. Chichester: Wiley, 1982. M. Lefebvre, "M&k of hazard survival," Ph.D. dissertation, Cambridge Univ., Cambridge, England, 1983. -, "Optimal stochastic control of a class of processes with an exponential cost function," Ann. Sci. Math. QuPbec, to be published. P. Whittle and P. A. Gait, "Reduction of a class of stochastic control problems,"
doi:10.1109/tac.1987.1104678 fatcat:6ypmyqmbjvadpkawd2fd52jbyu