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1 Hierarchical Optimization in the Presence of an Intelligent Aversary  an H Approach Muralihar Ravuri, hungyao Kao an Alexanre Megretsi Laboratory for Information an Decision Systems Massachusetts Institute of Technology Abstract Level S In this paper, e consier the H optimization problem for systems ith a hierarchical structure. We propose an iterative algorithm to treat the problem. In each iteration of our algorithm, e nee to solve a problem of H optimization feature, but ith a much smaller imension. We also sho that the solution of the optimization problem in each iteration can be obtaine by solving a set of linear matrix inequalities. Level Leve r3 S3 r S r4 S4 S2 Keyors: hierarchical structure, H control synthesis Figure : Hierarchical structure in the military operations moel Motivation The chain of comman in military operations gives a natural hierarchy an ecomposition strategies. The notion of hierarchical optimization is not a ne one an has been stuie for over thirty years. The large scale nature of the military operations' ecision maing problem in the presence of uncertain ata an an intelligent aversary maes the problem more ifficult. This class of problems have not been previously ealt ith. The recent avances in robust control theory an the associate H optimization problems havegiven us ne tools an techniques unlie those consiere in the past. Some of the relevant issues that e consier here inclue the presence of a large number ofvariables an constraints, ynamic nature of the optimization problem, presence of an intelligent aversary. We begin by formulating the mathematical moel. 2 Mathematical Moel In this section, e ill formulate a mathematical moel of an hierarchical H optimization problem". We assume that there is a natural ecomposition of the system hich has been alreay chosen, as shon in Figure. Each noe in the figure represents a sector" that is a representative of a certain level of granularityofthe moel. Typically, this coul represent a specific geographical region here a person is in comman. As e move from bottom to top in the figure, e assume that e are moving higher in the chain of comman. The topmost noe represents the top level of the comman hierarchy. At each sector, there are several types of resources present. Examples inclue number of tans, aircrafts, bombers, raars, men, foo supplies etc. Our objective is to istribute these resources among the various sectors in orer to suppress the enemy activities. This problem is combinatorial in nature an the complexity increases ramatically even ith a fe choices of resources an sectors. The ynamics of the problem introuces even larger number ofvariables that maes the problem intractable. We oul therefore lie toapprox imate the problem an also use the hierarchical structure to esign a suboptimal control strategy. There are to ifferent ays of approximating the problem: one can formulate a etaile an fairly accurate nonlinear moel of the system an use a generic optimization metho to obtain a suboptimal solution for this highly nonconvex problem, or on the other han, one can approximate the mathematical moel instea an obtain the optimal solution to the optimization problem using the interior point methos. We ill use the secon approach an pose the problem as an H optimization problem. This oul allo us to inclue the enemy as an intelligent aversary as oppose to a isturbance that eoul lie to suppress. The mathematical moel use is a linearize moel of the flo of resources. We use a symmetrical moel for both frienly an enemy resources in any given sector
2 an they are represente in any given sector as x s (n +)x s (n)+r s (n) u s (n) ff s y s (n) () y s (n +)y s (n)+ s (n) s (n) fi s x s (n) (2) here x s represents frienly resources in sector s, y s the enemy resources in the same sector, r s are frienly resources coming from upper level, u s are frienly resources leaving sector s, s are enemy resources from his upper level an s the enemy resources that leave the sector. The term ff s y s (n) represent a linearize conflict loss moel for frienly resources an similarly there is a corresponing term in enemy state evolution equations. The parameter ff s represents a percentage of loss the enemy coul inflict in this sector. The conflict loss term is nonlinear in general, but by choosing variables that are variations about a nominal value, e coul linearize this term an represent them as above. Our objective then is to regulate about these nominal values. One ay to see the importance of this approach is by assuming that an initial strategy has been esigne that brings the state of the system to a esire value. We then use the metho that ill be escribe here to maintain this esire state even in the presence of an active intelligent aversary. 2. entralize H Optimization Approach The combine statespace equations (ritten in continuous time) of all the sectors can be ritten in the folloing stanar form _x Ax u (3) here x represents both the frienly states an the enemy states. The control variables that e nee to esign are given by u an the enemy isturbances are represente as. Here, not all states are measurable (clearly not all enemy states are non) an the measurements o have some errors. Let us represent all the measure variables by y. In aition, e have some performance variables z. These variables are chosen such that they represent the cost in moving frienly resources, superiority ineach sector, etc. This moel is shon in Figure 2. u G(s) Figure 2: entralize moel The objective of the centralize H optimization problem is to esign the control u as a function of y such that the orst case 2norm of the performance variables z over all isturbances ith jjjj 2» is minimize. This centralize H optimization problem has been extensively stuie in the last to ecaes an the solu z y Level Level Leve uc S2 r2 r S S Our Moel r3 S3 3 c3 S3 S 3 S S2 Enemy Moel Figure 3: Moel ith hierarchical structure tion can be obtaine by either solving the Riccati equations associate ith the system [], or solving a system of Linear Matrix Inequalities (LMIs) [2]. Recently, very efficient interior point methos that have been evelope to solve these LMI's in particular. Hoever the large imensionality of our problems mae the size of the LMI's sufficiently large that the interior point methos become intractable. So our approach is to ecompose the problem by using the hierarchical structure an rerite them in such aay that e still preserve the H structure at each level. Further, the aim is to have a small imensions at each level enabling us to use the efficient interior point methos on these loer imensional LMI's. We escribe this in greater etail in the folloing section. 2.2 Hierarchical H Optimization Approach We formulate a hierarchical moel by iviing the problem into several levels as shon in Figure 3. In this formulation, the performance objectives are chosen from the top level to the bottom level hereas the control esign problem is solve from bottom to the top level. The noes in the figure are represente as sectors (shon as S ;S etc). As e have mentione earlier, one of the main rabac in the centralize approach is that the combine statespace has a very large imension. Therefore, our primary objective informulating a hierarchical approach is to reuce the size of the statespace at each sector. The secon important feature is to have the same structure of equations at each level in the hierarchy. The thir feature e nee in orer to claim that it is a hierarchical H problem is that the objective that is passe from one level to the higher level is in such aay that e still have a suitable generalize H problem at the higher level. In other ors, e oul lie to formulate the hierarchical problem ith the above three features by satisfying the same global performance objective as that of the centralize problem. We further nee to ensure internal stability of the combine system.
3 One of the main ifficulties that e encounter in trying to formulate a hierarchical problem is that the ifficulty of the control esign problem as e go higher in the hierarchy, shoul not be increase ue to an unreasonable choice of objective at the loer levels. This is reflecte in the choice of objectives as ell as the choice of the reuce orer moel that is passe to the upper level. In the rest of this section, e ill propose an iterative algorithm hich has the three feasures mentione above to solve the hierarchical H optimization problem. Let us consier sector S shon in Figure 3. A brief escription of the signals involve are (see Figure 3): : current sector enemy resource allocations. : upper level enemy resource allocations. r: upper level frienly resource allocations that are passe onto the current sector. : inex of all sectors of the loer level connecte to the current sector. : enemy isturbances of all sectors in the loer level (represente by inex ) connecte to the current sector. This is similar to the variable escribe above, but that corresponing to the sector of the loer level. : similar to the variable corresponing to the sector in loer level. u c : current sector frienly resource allocations that nee to be esigne. r : current sector frienly resource allocations that are passe onto the loer level sector. This signal correspons to the signal r escribe above of the sector in the loer level. y: all the measurement signals. z: current sector performance variables. The approach taen to solve the hierarchical optimization problem can be outline conceptually as (taing as a specific example from the Figure 3). the reuceorer states an the objectives of sectors S 2 an S 3 are passe to S. 2. current objective an objectives from S 2 an S 3 are combine to form a generalize H problem. 3. solve theh problem for the controls an simultaneously the reuceorer states an objective of S to pass to S. 4. the same process is repeate at all the other sectors. We no escribe the above steps in greater etail. Figure 4 shos the plant ynamics expresse in the stanar form for a current sector level. Some of the sectors may nothave all the signals shon in the figure (for example, the topmost level ill not have ; an r, hile the loest level ill not have ; an r ). We have split the isturbance an control signals so that the hierarchy becomes apparent. We begin by first riting the statespace equations of the current sector level as r uc r G(s) Figure 4: Plant ynamics of a current sector expresse in stanar form given in equations () an (2) in a more general form (an in continuous time) as _x Ax r + 4 u c r z x + D + D 2 + D 3 r + D 4 u c + D 3 r y 2 x + D 2 + D 22 + D 23 r; (4) here y is the measurements in the current sector, an z is the current sector performance variables. We have use the notation that repeate inex P as in, for example, 2 is actually a summation 2K 2 here K is the set of all loer level sectors connecte to the current sector. The states of a suitable reuceorer" system passe from the loer level are represente by x Π an the corresponing statespace equations are given by _x Π A Π x Π r r (5) an the objectives from the loer level sectors are given by ~ Π () ~ ; ~ z y r : (6) We no represent the current sector generalize H optimization problem as minimizing the maximum eigenvalue over all frequencies of Π() of the form Π() in other ors, e ant to solve fi (j) fi 2 (j) fi (j) fl () f (j) A (7) fi 2 (j) f (j) f2 () max max(π()) min; subj to jjzjj ~ Π () ~» (jjjj ) ~ Π() ~ 8 ; ; r; an ; (8) for a given value of. Here, ~ r. Note that in (8), n l enotes the number of sectors that is uner control of the current one. For example, n l 2 for the sector S of the system in Figure 3. The esign
4 variables are u c ;r ;fi (j);fi 2 (j);fl ();f (j) an f2 (). Note that this optimization problem is a convex optimization problem jointly in all the esign variables if e rerite equation (8) in terms of the Youla parameter. Let us represent the states of the system Π() as x Π ith statespace representation given by _x Π A Π x Π r r: (9) These are the reuceorer states of the current sector hich, together ith the objective ~ Π() ~ ill be passe to the upper level. We use the same proceure for all the sectors for the same value of starting from the loest level. The global objective in the hierarchical optimization problem is to obtain the smallest value of by using a binary search. We illustrate the algorithm by the folloing simple example. u c u c u c2 our moel S r S r 2 S enemy's moel E E 2 E 2 for some. Let ~ 2 2 r 2 r, an ~ 2. y the propose algorithm, e first select a,say.then e try to solve the optimization problem max max(π 2 ()) min; subject to; z 2 2» ~ 2Π 2 () ~ 2 ; 8 ~ 2 () to obtain the control strategy u c2 an a reuceorer moel Π 2 () for sector S 2. Suppose () is solvable. We then pass Π 2 () to the upper sector, move up to the next level an solve max max(π ()) min; subject to; z ~ 2Π 2 () ~ 2» ( ) ~ Π () ~ ; 8 2 ; 2 ; ~ ; (2) to obtain u c, r 2 an Π () for the sector S. Finally, e move up to toppest level an solve Fin u c an r ; such that; (3) z 2 + ~ Π () ~» ( ); for all,, an. It is obvious that if e o solve () (2) an (3), then e inee obtain a control strategy (u c, r ) such that() hols for the e select. We then tae 2 :5 an repeat the proceure. If any one of () (2) (3) is unsolvable, e increase the value of an restart the proceure from the buttom level, i.e., (). We stop the preceure hen the e obtain matches a certain preset precision. Figure 5: Example for hierarchical H esign algorithm Example. onsier the hierarchical system shon in Figure 5. The frienly resources are ivie into three sectors an the sector locate in the upper position has control of the resources in the loer sectors. We assume that our enemy has the same comman hierarchy. Each represents the influence on the frienly resources at the sector from the enemy at the same level. The enemy's resources movement shoul also have some influence on our resources. These influences are capture by, another isturbance to the corresponing sector, as shon in the figure. Let z enote the performance measurement ofeach sector, an our goal is to esign u c an r such that 2X z 2» ( 2X 2 + 2X 2 ) () 3 Solution for Hierarchical H ontrol Synthesis In the previous section, e propose an iterative algorithm for H control synthesis of hierarchical systems. In each iteration of the algorithm, e have to solve an optimization problem: For a given, min u totky; Π() n l jjzjj 2 + X 2 + here ~ u tot max max(π()); subj to ~ Π () ~» jjjj 2 + (4) ~ Π() ~ 8 ~ ; an ; r, ~ r, u c ;r ; ;r nl,ank enotes the ynamic
5 controller that e oul lie to esign. In the folloings, e ill sho that solving optimization problem (4) is equivalent to solving a set of Linear Matrix Inequalities (LMIs). The avantage of having LMI formulation is that the optimization can be one by efficient numerical algorithms [5, 4]. We first observe thateachπ () in(4)hasthesame structure as Π() an satisfies Π I () :Π ()+ I A > ; 8 : (5) The reason for (5) is that each Π () as obtaine by the same type optimization problem as (4) in the loer level sectors, in hich Π () plays the role of Π(). In problem (4), since the inequality has to be satisfie for all ~,,e conclue that, by setting, z r fl () 22 ()r» 2 ~ Π() ~ ; 8 ~ : (6) Inequality (5)follos the fact that the lefthan size of (6) is positiveefinite. Since Π I () is positive efinite, it can be factorize as Ψ (j) Λ Ψ (j). Let Π an the statespace representation of Ψ (j) be ( Ψ _xπ A x Π + Π + 2 r : : z x Π + D Π + D 2 r Incorporate the openloop system in (4) ith all Ψ, ; 2; ;nl,e obtain a combine system G tot : 8 >< >: _x tot A tot x tot + tot tot + tot2 u tot z tot tot x tot + D tot tot + D tot2 u tot ; y tot2 x tot + D tot2 tot here x tot x ;x Π ; ;x is the vector Πnl of all state variables, z tot z ;z ; ;z nl is the total performance measurement, an tot ; ;r ; Π ; ; is the collection of all istur Πnl bances. Integral inequality in(4) can be equvalently ritten as z tot 2 tot 2» r 2 + ~ Π() ~ : As oppose to Π (), Π() in (4) plays the role of a reuceorer moel of the current sector an is one of the ecision variables that ill be etermine by the optimization process. In orer to mae a finite imensional optimization problem, e select a set of bases f s + a : a > ; ; 2; ;n p g an let Π() ~Π(j)+~Π(j) Λ, here ~Π(s) ~Π + ~Π n X p ~Π (;2) ~Π (2;) ~Π (3;) ~Π ; (7) s + a ~Π (2;2) ~Π (3;2) ~Π (;3) ~Π (2;3) ~Π (3;3) A ; ; ; ;n p ; an every one of ~Π (i;j) is a matrix variable that ill be etermine by the optimizer. y (7), e can express jjrjj 2 + in the timeomain as ff (x Π ; tot ): Z ψ x Π ~ Π() ~ (8) ψ ψ F tot F R x Π t; tot _x Π A Π x Π + Π tot ; (9) here A Π, Π are constant matrices, an F, R R are matrix variables hich consist of ~Π (i;j). Since their structures are irrelevant to the LMI formulation, e ill neglect the etaile construction of A Π, Π, F, an R. As e mentione before, Π() serves as a reuceorer moel of the current sector an ill be passe to the next level of the hierarchy asapartofthecontrol esign objective. Therefore, it is important that the bases (s + a )echoose are able to reflect the frequencyomain features of the current sector. It is also important that the orer of Π() is significantly smaller than the orer of the current sector, otherise e miss the main point ofintroucing the hierarchical structure. ; ;r. y the Let fl max max (Π()) an ~ efinition of fl,ehave the integral quaratic constraint ~ (Π() fli)~» ; 8 ~: (2) Similarly, e can express the integral form in(2)as ff 2 (x Π2 ; ~) : Z ψ x Π2 ~ ψ ψ F 2 F 2 R 2 x Π2 ~ t; _x Π2 A Π2 x Π + Π2 ~; (2) for some constant matrices A Π2 Π2 an variable matrices F 2 R 2. Notice that F 2 an R 2 are also functions of Π (i;j) an not inepenent fromf an R.
6 y (9) an (2), e can express problem (4) as min fl subj to; (22) K; fl; Π ( ~ jjz tot jj 2 jj tot jj 2» ff (x Π ; tot ) ff 2 (x Π2 ; ~)» ; 8 tot ; here z tot, tot, ~, x Π,anx Π2 satisfy the folloing ynamical systems G tot : G Π : G Π2 : 8 >< >: _x tot A tot x tot + tot tot + tot2 u tot z tot tot x tot + D tot tot + D tot2 u tot y tot2 x tot + D tot2 tot _x Π A Π x Π + Π tot _x Π2 A Π2 x Π + Π2 ~: It is shon in [3] that problem (22) can be caste as an optimization problem over a set of LMIs. Theorem : Assume that D tot2 is of full column ran an has the structure D tot2 ^D 2 μd 2 ith an invertable ^D2. Let 2 ~ ^D tot2 2 an D ~ 2 ( ^D2 ) Dtot2. Let W r W r 2 be the null space of tot2 D tot2, an W l D2 μ ^D 2 I, W 2 ~ be such that ψ ~ 2 D ~ 2 I : W l W l2 Then there exist a controller K, ~Π for ; 2; ;np, an fl hich solve the optimization problem (22) if an only if there exist P P >, E E >, E 2 E 2 >, E 3, fl, ~Π for ; 2; ;np,an ψ S S S S 3 S > ; 3 S 2 hich solve the folloing problem ( L < ; L 2 < inf fl; subj to L 3 > ; L 4 < here L L 2 L 3 (23) W l W l W l tot E 3 W l D tot E 3 tot W l 3 5 E 2 Π2 F D tot W l 4 S 2 A Π + A Π S I S S 3 I S 3 S 2 E 2 E 3 I E 3 E Π 2 E 2 F R A A A L 4 ψ A Π2 P + PA Π 2 P Π2 + F 2 Π 2 P + F 2 R 2 W l ( tot E W l2 ) 2 A tot E + E A tot + E tot W + W tot E W W l2 3 (A tot + W tot )E 3 E 3 A Π 4 tot + W l2 D tot E 3 Π2 5 E 2 A Π + A Π E 2 6 (S 3 A tot + A Π S 3)W r +(S 3 tot + S 2 Π2 )W r2 F W r2 7 W r (S A tot + A tots )W r + W r S tot W r2 + W r S 3 Π2 W r2 + W r 2 ( tots + Π 2 S 3)W r W r 2 R W r2 8 W r tot + W r 2 D tot Suppose that (23) is solvable. A stabilizing controller K hich solves (22) can then be recovere by again solving a set of LMIs. See [3] for the etails. 4 oncluing Remars We propose an iterative algorithm for H hierarchical optimization problem. The algorithm has three important feastures: First of all, the size of the optimization problem in each iteration is significantly smaller than the one of the centralize optimization approach. Seconly, the optimization problem in each iteration has the same structure. Finally, in each iteration, the optimization problem is a generalize H problem hich e no ho to solve efficiently by numerical algorithm. Ho ell can the algorithm reuce computation complexity hile still obtain a satisfactory suboptimal solution is subject to the future research. References [] J. Doyle, K. Glover, P. Khargonear, an Francis. Statespace solution of stanar H 2 an H control problems. IEEE Transactions on Automatic ontrol, 33:83 847, 989. [2] P. Gahinet an P. Aparian. A linear matrix inequalityapproachtoh control. International Journal of Robust an Nonlinear ontrol, 4:42 448, 994. [3]. Kao, M. Ravuri, an A. Megretsi. ontrol synthesis ith ynamic integral quaratic constraints  LMI approach. To appear in 39th IEEE D, 2. [4] Y. Nesterov an A. Nemirovsi. Interior point polynomial methos in convex programming, volume 3 of Stuies in Applie Mathematics. SIAM, 994. [5] L. Vanenberghe an S. oy. Semiefinite programming. SIAM Revie, 38():49 95, March 996.
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