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# Optimal Control Theory ECE 620

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This 15 page Class Notes was uploaded by Antonina Wuckert on Monday September 28, 2015. The Class Notes belongs to ECE 620 at George Mason University taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/215041/ece-620-george-mason-university in ELECTRICAL AND COMPUTER ENGINEERING at George Mason University.

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Date Created: 09/28/15

Minimum Fuel Optimal Control Example For A Scalar System A Problem Statement This example illustrates the minimum fuel optimal control problem for a particular rstorder scalar system The derivation of the general solution to this problem is found in the course textbookl The expressions for the switching curve the switching time and the minimum time will be developed and several examples with different initial states and different target states will be presented The general form for the system model control constraint and minimum fuel performance index are T W f WM 3 lWLtluW WW S 17 W 6 17m J CT luildi 1 0 where at 6 ER and u E ERm The speci c expressions for the rstorder system for this example with n m 17 are given by T m own lan s 1 J lutldt lt2 0 For this system the terms in the general state space model and performance index have the following values f act7 t 71 B actt l and CT l The Hamiltonian for the system is HO MW Mimi WWO 3 where Mt is the scalar costate variable Lagrange multiplier for this system The solution to the minimum fuel problem is the control signal ut that satis es the state and costate equations and that minimizes the Hamiltonian such that H aciut g H aciut V admissible ut 4 which for this problem becomes Minimize lutl Atut lutl g 1 5 The solution to this minimization problem follows the form for the general case and is bTA t ut idez idezt 6 Ci where dez is the deadzone function de ned by OUT 17 when N gt 1 dez OUT 71 when N lt 71 7 OUT0 when lINl 31 B Costate Trajectories andAdmissible Controls The costate equation and its solution are 72H W m Mt i0e 8 f There are ve possible solution trajectories for Mt that affect the control signal These different trajectories depend on the initial condition M0 and its relation to 0 and i1 The trajectories are shown in Fig l where Case 1 corresponds to MO gt 1 Case 2 corresponds to 0 lt M0 lt 1 Case 3 corresponds to 71 lt M0 lt 0 Case 4 corresponds to M0 lt 71 and Case 5 corresponds to M0 0 The time indicated in the gure as t1 is when the trajectories for Case 2 and Case 3 cross i1 Since the solution to the costate differential equation is a single growing exponential Mt never decreases and is always increasing if M0 7E 0 If 0 lt lMO g 17 the initial control will be u0 0 As t increases the costate variable will be 1FL Lewis and VL Syrmos Optimal Control 2nd Ed WileyInterscience 1995 Chap 5 Costate Mt 2 Case 4 3 4 I I I 0 01 02 03 04 0 5 06 07 08 09 1 Time s Fig 1 The 5 possible costate trajectories that affect the optimal control A t 1 at t t1 At that time the control signal will switch to u t1 1 It will remain at that value for t1 lt t g T If 0 gt 17 then 1 for 0 g t g T Therefore if 17 it will never switch to ut 0 Control sequences of the form 10 10 107 1 107 1 etc are excluded from being admissible controls The only admissible control sequences for this system are shown in Table 1 The Minimum Principle effectively lters out all other control sequences and provides us with the only admissible ones regardless of the initial condition or the target state Other control sequences could obviously be applied to the system but they would not represent minimle fuel solutions Table l Admissible control sequences for minimum fuel control Case ut Comments 1 1 No switching 2 07 1 Switch at t t1 3 07 1 Switch at t t1 4 5 1 No switching 0 No switching At the nal time t T7 the minimum fuel problem ends The control that is applied for t gt T depends on the performance requirements that are imposed for that time period If if is desired to maintain the state at the target state so that act 1T for t 2 T7 then for the system model used in this example the equilibrium control is ut 1T for t gt T This control will make act 0 The control used to maintain this equilibrium condition is not included in the minimum fuel performance index since J only includes time in the interval 0 g t g T C Stale Trajectories The state trajectories can be found by substituting each of the allowed values for the control signal ut into the state equation and then solving the state equation for This will determine what target states can be reached and whether or not the reachable target states depend on the initial condition For the general linear timeinvariant system described by 32 Am But 9 the solution to the differential equation is2 t 35t2 eA11 tl3ct1 eA11 TBurdr 10 ii For the system model used in this example A 71B 1 and the allowed control values the state trajectories are given by the following expressions 0 For ut 0 32 7m j 302 30t1e t17t1 11 which indicates that 35t decays toward the origin 30 0 in accordance with its natural dynamics given by the openloop eigenvalue at s 7 o For ut 1 1 30t 735t 1 3 30t2 3ct1eitrtl eimi dr 12 ii 35 2 50m 35 1 7 1 1 showing that the state asymptotically approaches 35t 1 at Q 7 t1 7 oo regardless of the initial condition 350 o For ut 71 M 32 73ct71 35t23ct1e 11quot17 07 13 t1 mg e 1quot13ct1171 showing that with this input signal the state asymptotically approaches 35t 71 at Q 7 t1 7 00 Figure 2 shows typical state trajectories for the three values of From the trajectories that are shown the following observations can be made 0 lfthe speci ed nal time T is nite then ut 0 cannot be the only control value used ifthe target state is 35T 3cT 0 or if the target state has the opposite sign of the initial state 350 that is if 300307quot lt 0 o The target state has to be in the open interval 71 lt 3cT lt 1 in order for the target state to be reached in a nite time from any initial condition If the target state were outside that interval 3cT 2 1 there would be some initial conditions that could not reach the target state at all For example no state trajectory with initial condition 350 gt 71 can ever reach any target state 3cT lt 71 and it can only reach 3cT 71 in in nite time On the other hand an initial condition 350 gt 1 can reach a target state 350 gt 3cT gt 1 in a nite time with the optimal control being either u 0 or u 0 71 depending on the value of T The control u 1 would not be used since it has the same fuel cost as u 71 and yet takes longer to drive the state toward its target value However 3cT 2 1 does limit valid initial conditions For ut 1 or ut 71 the state trajectory tends toward 35t ut as t 7 oo asymptotically approaching that value Therefore the state trajectory cannot cross 30 t 1 from either direction when ut 1 A similar statement holds for 35t 71 with ut 71 Any target state 3cT gt 71 can be reached in a nite time from any initial state 350 gt 3cT with the control ut 71 Therefore the possible minimum fuel control sequences for this scenario would be ut 71 or ut 0 71 depending on the value of T Likewise any target state 3cT lt 1 can be reached in a nite time from any initial state 350 lt 3cT with the control ut 1 Therefore the possible minimum fuel control sequences in this case would be ut 1 or ut 0 1 again depending on the value of T Example I Let the system model be given by 2 and let the target state and initial condition be 3cT 3cT 0 350 350 5 If only ut 71 is used then the target state can be reached and it will be reached in minimum time 21G Reid Linear System Fundamentals 1983 McGraW Hill NY Chap 7 Control Signal is ut 0 2 I I I I I 1 r r r r 1 D E I 1 I I I I I 2 I 0 05 1 15 2 25 3 Control Signal is ut 1 2 I I I State xt State xt 0 05 1 15 2 25 3 Time s Fig 2 State trajectories for ut 0 ut 1 and ut 71 The performance index is J1 fOT 71 dt T and the value of T necessary to reach 1T from 0 with u 71 is T Ln 1 5 1792 seconds If we specify that ut 07 71 and specify the switching time to be t1 2 seconds then 2 T J2 0dt 71dtT72 l4 0 2 and T 2 Ln 1 562 2517 seconds so J2 0517 This is less than the minimum time solution using only ut 71 If we let the switching time be t1 3 seconds then 3 T 13 0di 71diT73 15 0 3 and T 3 Ln 1 563 3222 seconds so J3 0222 This is less than either of the two previous solutions This implies that the longer we allow the system to take in order to reach the target state the lower the cost in terms of fuel to reach the target state In this example J3 lt J2 lt J1 This is generally the case However there is a scenario in which letting the system take longer to reach the target state actually increases the cost as well as the time This scenario will be presented later in Examples 8 and 9 In each of the cases in the example above the value of the performance index J for this system is the length of time that a nonzero control is being used This is a general result for singleinput systems Therefore T t1 T J uidt 0dt i1dt 16 0 0 ti J T 7 t1 17 If T TM 7 the minimum time solution then the switching time is t1 07 and J T TM For T gtgt TMm7 the performance index approaches a limiting value If the control law needed to drive 350 to TT includes ut 1 or ut 717 the limiting value of J depends only on the absolute value of the target state xT The limiting value of J depends neither on the sign of 1T nor on the value of the control signal used to drive act toward xT In this case the limiting value of the performance index is JLimmaX7Ln1acT7 7Ln17acT l8 The validity of 18 is seen in the next section in Eqn 24 and the discussion following If 1T 07 or if ut 0 will allow the state to decay from To to TT in a nite time then JLim 0 otherwise JLim gt 0 In order for J to actually represent a quantity of fuel that is being minimized the control signal ut must have the units of fuel used per unit time The integration over time then yields a value for J having units of total fuel consumed In these notes values for J will be given in each example without any associated units D Determining the Switching Time t1 and the Minimum Time TMm Assume that the target state is 71 lt 1T 3 0 and the initial condition is To gt 0 so that the control ut 0 will not force the system to reach the target state in a nite time Therefore either ut 71 or ut 07 71 must be used depending on the value of T Assuming that T gt TMm so that the control sequence will be ut 07 71 let i t1 be the time at which the control switches from ut 0 to ut 71 The initial time will be taken as t 0 From ll and 13 the state trajectory will be 1i1 acoeitl l9 T 1T JET ac i1 eiu ih 7 67T7Td739 roe T 7 e T ET 7 ell 20 t1 acT xoe T 7 1 711 xoe T 7 1 Tell 21 6 1 eT JET 1 7 0 or T 1T 1 7 Toe T 22 The rst expression in 22 can be solved for the relationship between t1 x0 and xT 21 Ln eT 1T 1 7 x0 23 An alternate equation for t1 can be developed by solving the second expression in 22 for T 7 t1 T7i1J7LnacT17acoe T 24 If T gtgt 1 then the exponential on the right side of 24 will be negligible and T 7 t1 m 7Ln acT 1 If the target state is the origin then T 7 t1 m 0 for large T This means that ut 0 is used for most of the time so the system coasts for as long as possible Since J T 7 t1 for the control sequence ut 0 71 the cost can be made arbitrarily small for 1T 0 by letting the nal time be arbitrarily large However there is no solution to the optimal control problem if the target state is the origin and T is free If 1T 7E 0 then the limiting value of the performance index will be positive with its value depending only on acT If the nal time is xed then 24 can be solved for the switching time t1 t1 T Ln JET 1 7 roe T 25 The nal time has to be xed at a value greater than or equal to the minimum time TMm in order for there to be any solution to the minimum fuel control problem and T gt TMm is necessary for t1 gt 0 The time optimal solution can be obtained from 19 and 20 by setting t1 0 and it is given by 1T roe T 7 e T ET 7 1 e T 0 1 7 1 26 J 1T 1 ace 1 27 7 V 7 0 1 T 7 TMm 7 Ln LET 1 28 The expressions developed above for the performance index and minimum time are also valid for 0 lt 1T lt 350 and when 71 lt 1T lt 350 g 0 The governing rule for this condition is that ace 1 acT 1 gt 0 and 1350 1 gt WET 1 and 1T gt 71 When the third condition based on requiring the target state to be in the open interval 71 lt 1T lt 1 is included this simpli es to x0 gt acT Likewise the expressions for the switching time are valid as long as the relationship between 0 acT and T are such that switching occurs Otherwise t1 0 The expressions are summarized in Table 2 Table 2 Summary of equations for u 71 or u 0 71 Characteristic Equation Eqn No Switching Time t1 Ln ET JET 1 7 x0 23 Performance Index T 7 t1 J 7Ln JET 1 7 xoe T 24 Switching Time alternate t1 T Ln xT 1 7 xoe T 25 Minimum Time T Tm Ln 1 1 28 1T 1 Similar expressions can be developed for the case where 350 lt 0 and 0 3 1T lt 1 which requires u 1 The general rule for this case is ace 7 1 acT 7 1 gt 0 and 1350 7 1 gt acT 7 1 and 1T lt 1 which simpli es to 350 lt 1T when the restriction on the target state is imposed The expressions are developed in the same way and are given in Table 3 Table 3 Summary of equations for u 1 or u 0 1 Characteristic Equation Switching Time t1 Ln l7eT acT 7 1 350 Performance Index T 7 t1 J 7Ln 790T 1 xoe T Switching Time alternate t1 T Ln 795T 1 xoe T 7 1 Minimum Time T TMm Ln x0 1T 7 1 Example 2 Using the same initial condition are 5 and target state 1T 0 as in Example 1 Table 4 shows the switching time and total cost for several different values of T including the minimum time T 1792 Table 4 Switching time and total cost as a function of nal time T T t1 J 1792 0 1792 2 0871 1129 25 1972 0528 3 2714 286 X 10 1 5 497 343 X 10 2 10 99998 22703 X 10 4 Relationship Between Final Time and Interval of NonZero Control x0 5 XT 0 Tmin 1792 sec T t1 sec i i i Limition J 0000 sec 4 5 6 Final Time T see Fig 3 Plot of performance index J vs nal time T The entries in the table clearly show the reduction in J as T increases For large T7 t1 m T and J m 0 Figure 3 graphically shows the relationship between the speci ed nal time T and the performance index J T 7 t1 The vertical dashed line in the gure is at T TMm 1792 seconds The graph clearly shows that when 1T 07 the limiting value of the performance index is J Lim 07 and that J decreases fairly rapidly with increasing T For a speci c value of T7 the corresponding value of J can be obtained from the graph so the cost of operating the system can be determined in advance The value of J 0528 for T 25 seconds is shown for illustration 6 E The Switching Curve and ClosedLoop Control Once the target state acT the initial condition are and the nal time T are known the value of the switching time t1 can be computed from the appropriate entry in Table 2 or Table 3 The system can then be operated with ut 0 for 0 g t 3 t1 and ut 71 or 1 for t1 lt t g T The only requirement is to keep track of time This is openloop control since the current value of the state act is not used to determine the control signal It would be preferable to use closedloop control for the usual reasons of reducing the effects of uncertainty and disturbances This can be done by comparing the current state with the value of a switching curve Assume that the initial condition and target state satisfy the condition are gt acT requiring the nonzero portion of the minimum fuel control to be ut 71 Then the switching curve is the state trajectory that reaches the target state acT 1T at the nal time T using only the control ut 71 Call this switching curve The expression for 2t has the same form as 13 and is given by 2t 5 270 l 7 1 29 where the initial condition is 20 5T 1T l 7 1 30 Substituting 30 into 29 and letting t T yields the result that 2T xT Note that the switching curve depends only on the target state 1T and the nal time T not on the initial condition 350 The corresponding equations for the condition 350 lt acT with ut 1 are 2t 571270 7 1 1 31 20 5T 1T 7 1 1 32 The closedloop control law is shown in Table 5 which can be simpli ed as shown in 33 and 34 Table 5 Closedloop minimum fuel control law x0 gt W x0 lt W ut 0 if xt lt 2t ut 0 if xt gt 2t ut 71 if xt 2 2t ut 1 if xt S 2t ut 0 if x0 7 JET 7 lt 0 33 ut 7SGN10 7 1T if 10 7 JET 7 2 0 34 The minimum fuel control law allows this system7which is openloop stable7to respond naturally for as long as possible with zero control A nonzero control is applied only at the time necessary to force the state so that it arrives at the target state at the speci ed nal time Figure 4 shows the switching curve for T 225 seconds and 1T 0 The initial condition for the state is are 5 The heavy line in the gure is the actual trajectory that the state variable would follow reaching the origin at the speci ed value of T The switching takes place at t1 1501 seconds State xt J 3 5 xt xtlilultlzo 5E5wiihut1 39 1 i i i i 15 Time s Fig 4 State trajectory and switching curve for 0 5 mT 0 and T 225 sec 1 1501 sec and J 0749 sec E Illustrative Examples 1 Overview of the Examples Several examples will be presented in this section to illustrate the use of the various expressions developed in the previous sections and to point out one scenario that has a peculiar twist to it These examples are certainly not exhaustive but they do show the major aspects of minimum fuel control for this rstorder stable system Changing the signs of the initial condition are and the target state 1T only changes the sign of the control signal ut there is no change in the fundamental properties of the control or state trajectories The examples to be presented are listed below and summarized in Table 6 Ex 3 x0 gt 0 W 6 710 T gt TMm EX 4 x0 gt 0 JET 6 710 T TMm Ex 5 x0 lt 0 W e 01 T gt TMm Ex 6 x0 gt 0 W e 0 TMm lt T lt The Ex 7 0 gt 0 1T 6 010 TMm lt Tu0 S T Ex 8 x0 gt 0 1T 6 x01 T gt TMm T xed in value Ex 9 x0 gt 0 1T 6 x01 T gt TMm T set equal to TMm The parameter The is the time required for the state act to decay from are to 1T only using the control ut 0 This can only occur for the situation de ned by 1T 7E 0 and new gt 0 and and gt ac If The 3 T then ut 0 is the minimum fuel control and T will be reduced to The if the speci ed T is greater than The Examples 8 and 9 illustrate the scenario that has the peculiar twist that was previously mentioned ln Example 2 shown on page 6 it was seen that increasing the value of the nal time T reduces the amount of fuel used to drive the state from are to acT This characteristic will also be seen in Examples 377 in this section For the conditions in Examples 8 and 9 however increasing the nal time actually increases the amount of fuel used The true minimum fuel solution for these conditions is also the minimum time solution Those two examples will illustrate the results when T is required to be xed at its speci ed value and when it can be decreased to equal the optimal minimum time value Table 6 Summary of examples Example 350 1T T sec t1 sec J TMm sec JLim ut 3 3 705 25 1129 1371 2079 0693 0 71 4 3 705 2079 0 2079 2079 0693 71 5 73 05 25 1129 1371 2079 0693 0 1 6 075 025 05 0271 0229 0336 0 0 71 7 075 025 1099 NA 0 0336 0 0 8 035 075 175 0581 1169 0956 1386 0 1 9 035 075 0956 0 0956 0956 1386 1 2 Description of the Examples The examples will be described on the following pages The important points of each example will be highlighted to illustrate how that example is similar to or different from the other examples For each of the examples three plots are provided The upper left graph shows the state trajectory act solid line and switching curve 2t dashed line The upper right graph shows the control signal In each case it is assumed that the goal is to maintain the state at the target value for t 2 T Therefore the control signal takes on its equilibrium value ut 1T for t gt T The bottom graph shows the relationship between the value of the performance index J T 7 t1 and the nal time T In each of the examples the relationships between the initial state the nal state and the nal time should be noted as well as the relationship between the nal time and the minimum time TMm and the relationship between the state act and the switching curve Example 3 x0 37 1T 7057 T 25 sec Fig 5 The graphs for this example are shown in Fig 5 Since 350 gt 0 and 1T lt 07 the control signal must have the value ut 71 for at least part of the time since ut 0 cannot force act to change sign From 28 TMm Ln 350 1 acT 1 2079 sec so it is clear that T gt TMm This allows the control signal to be ut 0 for part of the time From 23 the time at which the control switches from ut 0 to ut 71 is t1 Ln ET acT 1 7 x0 1129 sec The state response is seen to decay naturally from t 0 to t t1 with the applied control being ut 0 At time t t1 the state trajectory intersects the switching curve trajectory and the control switches from ut 0 to ut 71 The switching curve and state reach the desired value 1T 705 at t T 25 sec It is assumed that the goal is for the state to be held at that nal value for t gt T7 so the control signal is switched to ut 1T 705 at t T This makes act 735T 1T 0 for t 2 T Since lel lt 17 the control signal can take on this equilibrium value under the speci ed control constraints The control signal is seen to be piecewise constant switching values once at t t1 and again at t T in order to establish equilibrium at at 1T The value of the performance index for this example is J T 7 t1 1371 The bottom plot in Fig 5 shows that relationship between the nal time T and the value of J The maximum cost would be incurred if T TMm 2079 sec In that case the switching time would be t1 0 the control starts at ut 7 7 and the value of the performance index would be J T 7 t1 2079 TMm The plot shows that increasing T decreases the value of J7 but the performance index does reach a limiting value Since lel gt 0 and ut 7E 0 is required the limiting value of J is nonzero In accordance with 18 Jun 7Ln 1 1T 0693 State T 2500 sec t1 1129 sec J 1371 Control Signal ut O 2 2 Time s Time 5 4 5 6 Final Time T see Results from Example 3 mo 3 mT 705 T 25 sec Example 4 x0 37 1T 705 T 2079 sec Fig 6 The graphs for this example are shown in Fig 6 As in Example 37 the control signal must have the value ut 71 for at least part of the time since 350 gt 0 and 1T lt 0 so that ut 0 cannot drive the system state to the target state The minimle time depends only on the initial and target states so it has the same value as before TMm 2079 sec The nal time in this example is chosen to be equal to the minimum time so the control signal cannot be ut 0 any time during the trajectory Thus the time at which the control switches from ut 0 to ut 71 is t1 0 sec the control starts at ut 71 There is no natural decaying of the state in this situation The switching curve has the initial condition 20 are so it and the state reach the desired value 1T 705 at t T 2079 sec As before it is assumed that the goal is for the state to be held at that nal value for t gt T7 so the control signal is switched to ut 1T 705 at t T This makes act 0 for t 2 T The value of the performance index for this example is J T 7 t1 20797 which is obviously higher than in Example 3 Since T is the independent variable in the bottom plot of the gureicost vs nal timeiand the cost depends on T7 350 and acT the curve showing that relationship for this example is the same as in the previous example since 350 and 1T are unchanged As seen in the bottom plot the maximum cost has been incurred since T TMm 2079 sec Since the limiting value of J depends only on land except when ut 0 is a valid control by itself that limiting value is the same as in Example 3 namely JLim 7Ln 1 1T 0693 6 State T 2079 sect1 0000 sec J 2079 3 Control Signal 4 5 6 Final Time T see Results from Example 4 mo 3 mT 705 T TMm 2079 sec Example 5 x0 73 1T 057 T 25 sec Fig 7 The graphs for this example are shown in Fig 7 This example is just the mirror image of Example 3 The signs of the initial condition and target state are both reversed from the previous example The values of switching time minimum time performance index and limiting value for the performance index are unchanged although the times are computed from different expressions because of the changes in signs of are and xT The correct expressions for t1 TM 7 and J are obtained from Table 3 and the expression for JLim is JLim 7Ln 1 7 acT The control sequence is ut 07 1 rather than ut 07 71 as it was in Example 3 Reversing the signs on both are and 1T has no effect except to change the sign on the nonzero portion of For any of these examples the bottom plot can be used to determine the value of the nal time T needed in order to achieve a particular value of the performance index J J0 in the interval TMm 2 J0 gt J Lim This can be done by substituting the desired value J0 into 24 or the corresponding expression in Table 3 and solving for the value of T The resulting value of T is 1 7 10 W 6 35 1 T7Ln 0 100 If x0 07 then J JLim max iLn 1 1T 7Ln 1 7 ac regardless of the value of T 6 State T 2500 sec t1 1129 sec J 1371 Control Signal 4 5 6 Final Time T see Results from Example 51mg 73 mT 05 T 25 sec Example 6 x0 0757 1T 0257 T 05 sec Fig 8 The graphs for this example are shown in Fig 8 This example and the following one illustrate the situation when the target state can be reached from the initial state using ut 0 only if T is suf ciently long Since the openloop system is stable ago will decay naturally to 1T in a nite time if lapel gt lel gt 0 The time needed for this decay is The Ln aceaw If the speci ed nal time T is greater than or equal to Th07 then the minimum fuel control would be ut 07 and the cost would be J 0 If T lt The7 then ut 1 or ut 71 is required depending on the sign of ago For the values given in this example The Ln3 1099 sec so with T 05 sec the control signal cannot be ut 0 by itself The switching time is computed in the normal fashion and the control signal switches from ut 0 to ut 71 at t t1 The top two graphs in Fig 8 show the state trajectory switching curve and control signal As with the previous examples the cost decreases with increasing T7 as seen in the bottom graph with a limiting value of J 0 Unlike the examples shown in Fig 3 and Figs 577 however the limiting value is not reached asymptotically in this example The graph of J vs T in Fig 8 appears to be approaching a negative value as T increases The limiting value JLim 0 is reached at T Th0 instead of as T 7 00 This can be seen from J 7Ln 1T l 7 xoeiT 7Ln JET l 7 roeTLMxO q 36 J 7Ln acT 1 7 Jaw 50 7Ln JET 1 7 ate 2 37 0 J 7Ln1 0 when T Tu0 38 For T gt The7 the switching time t1 is greater than the nal time This would make the performance index negative which is not a valid situation Therefore for T 2 Th07 T 7 Th07 no switching occurs ut 07 and J 0 T is is illustrated in the next example 6 State T 0500 sect1 0271 sec J 0229 Control Signal xt ut 02 04 06 08 1 Time s 04 06 Time s x 075x 025T 03365ec 0 T min JTt1 15 Final Time T see Results from Example 6 m0 075 mT 025 T 05 sec Example 7 x0 075 1T 025 T 1099 sec Fig 9 The graphs for this example are shown in Fig 9 The initial state and target state are the same as in Example 6 but now the nal time has been increased to T Th0 1099 sec The system state can decay naturally from the initial state to the nal state so no control is necessary The optimal minimle fuel control is ut 0 and the corresponding cost is J 0 The state and control trajectories are shown in the top two graphs There is no switching curve shown in the gure since 2t would correspond to the control signal ut 71 and the only control used in this scenario is ut 0 The state xt will not coincide with 2t anywhere except at t T where they both have the value acT 2T xT If the nal time is speci ed at a larger value T gt The then the state will still decay from are to 1T in Tuio seconds There is no way to increase that length of time and still implement a minimum fuel control law Since ut 1 0 and ut 1 0 71 are not valid control sequences for this system with the minimum fuel performance index there is no way to move the state away from the target state in order for the trajectory to take a longer period of time When the initial and target states are such that lapel gt lel gt 0 then Tuio is the maximum time that is acceptable for minimle fuel control Therefore TMm g T g The is required in this situation u0 1099 sec J 0000 Control Signal State T T um 15 2 0 05 1 Time s 1 Time s x 075x 025 T 0336 sec 0 T min JT g Limit 25 3 15 Final Time T see Results from Example 7 mo 075 mT 025 T Tuio 1099 sec Example 8 x0 035 1T 075 T 175 sec Fig 10 The graphs for this example are shown in Fig 10 Examples 8 and 9 illustrate the scenario mentioned previously that has a peculiar twist to it relative to the other scenarios studied This scenario is characterized by the fact that the control signal ut 0 allows the system state xt to move in the direction away from the target state The target state can still be reached using a control sequence of ut 0 1 or ut 0 71 depending on the sign of ago However the true minimum fuel control for this scenario is also the minimum time control ut 1 or ut 71 It actually takes less fuel to force the system to move from are to 1T as quickly as possible rather than allowing the system to coast The reason for this is that when the system is coasting with ut 0 it is coasting in the wrong direction When the control signal 1 is nally applied the state is farther away from 1T than it initially was and the nonzero control has to be applied for a longer period of time Thus the cost goes up with increasing T The relationship between 0 and 1T to have this scenario is one of the following 39 40 1gt 71lt acTgtacogt0 x0gt0 xTltxolt0 x0lt0 In this example the nal time is set at T 175 sec which is longer than the minimum time TMm 0956 sec However this example forces the speci ed nal time to be used for the simulation Therefore the initial control is ut 0 The switching time t1 is computed in the normal fashion and the control switches to ut 1 at t t1 The state will reach the target state at the speci ed nal time as seen in the upper left graph of Fig 10 The bottom graph in the gure shows the increase in cost with increasing T This is opposite of what was seen in all the other examples and this characteristic of the J vs T curve is speci c to this scenario For the speci ed nal time ofT 175 sec the cost is J 1169 State T 1750 sect1 0581 sec J 1169 Control Signal ut Time s Time s x 035 x 075 T 0956 sec 0 T min 15 I I I i 39 39 5 Limit on J 1386 J1169forT17505ec 4 5 6 Final Time T see Results from Example 8 m0 035 mT 075 T 175 sec

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Recurring Subscriptions: If you have canceled your recurring subscription on the day of renewal and have not downloaded any documents, you may request a refund by submitting an email to support@studysoup.com

Satisfaction Guarantee: If you’re not satisfied with your subscription, you can contact us for further help. Contact must be made within 3 business days of your subscription purchase and your refund request will be subject for review.

Please Note: Refunds can never be provided more than 30 days after the initial purchase date regardless of your activity on the site.