Piecewise Continuous Controls

Let us use to denote a piecewise continuous control with jump discontinuities at times ti and t2- Thus, is made up of three continuous 

At the minimum of /, additional conditions called, Weierstrass-Erdmann corner conditions, must be satisfied on the corners. These conditions require the continuity of the Hamiltonian and costate at the corners. The details will 

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Figure 3.4 A piecewise continuous control Uti,t2 and the accompanying state y(t)...

The controls belong to the class of piecewise continuous controls defined in Section 5.2.1. 

The optimal control analysis developed above is also applicable when a control function is piecewise continuous with a finite number of jump discontinuities, as shown in Figure 3.4. Observe that a jump discontinuity suddenly changes y or the slope of the state variable. The result is a corner in the graph of y t) at the time of discontinuity where y is not defined. Geometrically, there is no unique slope of the curve at a corner. One can draw an infinite number of tangents there. 

The controls ttjS are piecewise continuous with respect to t. Figure 3.4 (p. 77) showed one such control, which is made of three continuous curves. When two such curves meet, there is a jump discontinuity, e. g., at time ti, as shown in the figure. On either side of ti, the control is provided by the curve on that side. At t we take the control value from the right-hand side curve. 

Also note that if Ns is chosen sufficiently large, the piecewise constant optimal control policy will be sufficiently close to the continuous optimal control policy. 

The terms Pmax and Pmax denote the limiting capacities of the inspiratory muscles, and n is an efficiency index. The optimal Pmus(t) output is found by minimization of / subjects to the constraints set by the chemical and mechanical plants. Equation 11.1 and Equation 11.9. Because Pmusif) is generally a continuous time function with sharp phase transitions, this amounts to solving a difficult dynamic nonlinear optimal control problem with piecewise smooth trajectories. An alternative approach adopted by Poon and coworkers [1992] is to model Pmus t) as a biphasic function... 

Even though the tacit assumption is that the variables of a process do not ordinarily change in jumps, that they normally behave as smooth continuous functions of time (and position), there are situations in which output variables are deliberately sampled and control action implemented only at discrete points at time. The process variables then appear to change in a piecewise constant fashion with respect to the now discretized time. Such processes are modeled by difference equations and are referred to as discrete-time systems. 

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