Finite Horizon H∞ and Related Control Problems by M.Bala Subrahmanyam

By M.Bala Subrahmanyam

HIS e-book offers a generalized state-space conception for the research T and synthesis of finite horizon suboptimal Hoo controllers. We de­ rive expressions for a suboptimal controller in a common surroundings and suggest an approximate approach to the Hoo functionality robustness challenge. the cloth within the e-book is taken from a set of analysis papers written by means of the writer. The booklet is geared up as follows. bankruptcy 1 treats nonlinear optimum keep watch over difficulties during which the fee practical is of the shape of a quotient or a manufactured from powers of yes integrals. the issues thought of in Chap­ ter 1 are very basic, and the consequences are valuable for the computation of the particular functionality of an Hoo suboptimal controller. Such an software is given in Chapters four and five. bankruptcy 2 provides a criterion for the overview of the infimal Hoc norm within the finite horizon case. additionally, a differential equation is derived for the possible functionality because the ultimate time is diverse. A basic suboptimal keep an eye on challenge is then posed, and an expression for a subopti­ mal Hoo nation suggestions controller is derived. bankruptcy three develops expressions for a suboptimal Hoo output suggestions controller in a truly common case through the answer of 2 dynamic Riccati equations. Assuming the adequacy of linear expressions, bankruptcy four provides an iterative process for the synthesis of a suboptimal Hoo controller that yields the mandatory functionality even less than parameter variations.

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In the Appendix we show that the performance of this feedback controller is greater than >.. We can also get an idea of the actual performance of the state feedback controller given above. For this, let K(t) and = U1(t)P(t) + U2(t), (66) W1 = W1 + W2K + K*W2' + K*W3K, (67) W2 = W4 + K*W6, (68) Sec. 6: A Suboptimal Feedback Controller 29 Then the system given by (1) can be written as (70) and (3) becomes It: ~v*(t)R(t)v(t) dt (71) The exogenous input v needs to be selected such that (71) is minimized.

Denotes an inner product. We have In the above expression iT(_~B;'1/J,V)dt = iT(_~1/J,B2V)dt to to = _~iT(1/J,X - Ax - BIU)dt. (33) to An integration by parts and equations (16), (17), and (31) yield -~ iT (B;'1/J, v) dt = ~ iT {x*Wlx+2x*W2U+U*W3U+X*W4V+U*W6V} dt. to to (34) Substituting (34) in (32), we get i T v* Rv dt to = ~ iT {X*WIX + 2X*W2U + U*W3U to Thus, the value of (4) associated with (u, v) is ~ provided that the right side of (35) is nonzero. Hence, if (x,1/J) is a nontrivial solution of the boundary value problem given by (27) and (31) for the smallest parameter ~ > 0 with rT z*W z dt > 0, then ~ is the value of (3), and (x, 1/J) is an optimal pair.

39 0 Simultaneous solution of (7), (8), (11), and (12) yields a saddle point solution (uO,vO) for the functional given by (6). We now express the above minimax solution in a simpler form. 1/;(T) + 1](T) = o. 1/;(t) + 1](t) = 0, t E (14) [0, TJ. Thus the saddle point solution is characterized by ° ° d1/;= - A * 1/;+W1X+W2U +W4V, dt uO(t) = Wi 1{Bi1/; - W2'x - W6VO}, 1/;(T) =0, (15) (16) (17) Let 0 < ). opt. We define a full state feedback controller in the fol- lowing way. Note that we do not require W3 to be positive definite in the following formulae.

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