Global Solution of Hierarchical Optimal Control Problems Based on Simulated Annealing and Neural Networks

Copyright © IFAC Youth Automation, Beijing, PRC, 1995

GLOBAL SOLUTION OF HIERARCmCAL OPTIMAL CONTROL PROBLEMS BASED ON SIMULATED ANNEALING AND NEURAL NETWORK It

Ruo-Li Yang and Cang-Pu Wu Department ofAutomatic Control, Beijing Institute of Technology Beijing 100081, P.R.China

Abstract: A new method for solving hierarchical optimal control problems is proposed in this paper which combines the accelerated simulated annealing algorithm and the constrained optimal control neural network in an appropriate manner. Two different forms of explicitly defined high-level constraints of the hierarchical problems are treated by the different methods. To cope with the highlevel constraints depending only on the high-level control variables, an auxiliary problem is introduced at the upper level, avoiding using the penalty function method to deal with such constraints in an inefficient way. The computational results for numerical examples demonstrate the validity and better performance of the proposed method in terms of solution quality and computational efficiency. Keywords: Hierarchical decision-making; hierarchical systems; global optimization; optimal control; neural networks; algorithms.

different levels does not hold generally for the HOCP arising from the second situation. As a result, the HOCP arising from the second situation is nonconvex in nature even if each problem at different levels is a linear-quadratic optimal control problem. It is obvious that the existence of the nonconvexity may cause the HOCP of this kind to possess many local optima within the domain of interest.

1. INTRODUCTION

A hierarchical optirnal control problem (HOCP) can, in general, arise from two different situations. One is the decomposition of an overall one-level optimal control problem with some special structure into a coordinator problem at the upper level and a number of independent local problems at the lower level, and the other is the treatment of the dynamic hierarchical decision-making problem which is composed of a high-level decision-making unit (IIDMU) and a number of independent low-level decisionmaking units (LDMU) operating in a noncooperative manner. The first situation occurs when the hierarchical computational structure is to be used to solve the one-level optimal control problem with a separable structure, while the second situation occurs when the dynamic decision-making problem with a hierarchical structure is to be treated.

The HOCP arising from the second situation consists of a high-level problem and a set of low-level problems with possibly the different system dynamics. The performance index of the high-level problem depends not only on the control variables of its own but also on the optimal control and the corresponding state trajectory of each low-level problem which is determined under the given value of the high-level control variables. Therefore, it may be partly conflicting with the performance indices of the low-level problems. In fact, the dependence of the high-level problem on the low-level optimal solutions imposes implicitly additional constraints upon the high-level problem. The major computational difficulty in solving this kind of HOCP lies in the existence of the inherent nonconvexity resulting from inharmonious performance indices between upper and lower levels. The nonconvex nature makes it difficult to develop an effective computational method for finding the global optimal solution to this kind of HOCP . In the dynamic noncooperative game theory CBasar and Olsder, I 982a; Basar and Cruz, 1982b), the necessary optimality conditions for the open-loop Stackelberg dynamic game problem which is similar to the HOCP being considered in this paper are derived under some convexity assumptions from an equivalent one-level problem produced by substituting the low-level optimal control problems

Different kinds of HOCP's arising from these two different situations have quite different characteristics. An important aspect of the HOCP arising from the first situation is that the performance index of the coordinator problem at the upper level is a strictly order preserving function of the performance indices of the local problems at the lower level. This makes it possible to use the various decomposition-coordination techniques (Mesarovic, et aI., 1970; Singh and Titli, 1978; Findeisen, et aI., 1980; Jamshidi, 1983) to solve this kind of HOCP to reduce the computer memory requirement and computation time. Nevertheless, such a coincidence between the performance indices at the • Project supported by the Science Funds of the National Natural Science Foundation of China

265

which are parameterized by the given value of the highlevel control variables with their corresponding necessary optimality conditions. In spite of the conceptual usefulness of the necessary optimality conditions for the two-level hierarchical optimal control problems, the numerical solution method based on them can only yield a local optimum. Zhong et al. (1992) proposed a solution method for solving the general form of open-loop Stackelberg dynamic game problem based on the decomposition-coordination techniques. The overall two-level hierarchical problem is transformed under some convexity assumptions imposed upon the low-level problems into an equivalent onelevel nonlinear programming problem with a separable structure by the same way as mentioned above. This transformed one-level problem can be solved by using a decomposition-coordination technique based on the Lagrange multiplier method with appropriately chosen penalty terms related to only the nonconvex constraints. Although this solution scheme can reduce the computation time for solving the general open-loop Stackelberg dynamic game problem due to the use of the decomposition-coordination techniques, it can only fmd a local optimal solution to the overall two-level hierarchical optimal control problem. In order to obtain the global optimal solution to the HOCP arising from the second situation, we will propose a new solution method based on an appropriate combination of the accelerated simulated annealing (ASA) algorithm (Yang and Wu, 1993) and the constrained optimal control neural network (COCNN) (Yang and Wu, 1994). The main idea of this solution scheme is that the high-level problem is solved by the ASA algorithm, and at each ASA iteration the low-level problems are solved in parallel by the COCNN under the given value of the high-level control variables. The use of the COCNN allows for a fast and exact location of the low-level optimal control and the corresponding state trajectories for each given value of the high-level control variables under some convexity assumptions, which is indispensable for the efficient and exact evaluation of the high-level performance index at each ASA iteration. The major advantage of this new solution scheme is that the global or near-global optimum of the HOCP with the inherent nonconvexity can be located with high probability and high efficiency. In addition tQ some convexity assumptions imposed upon the low-level problems, the high-level problem is allowed to be of any form, even nondifferentiable. Furthermore, two types of explicitly defmed high-level constraints for the HOCP will be considered in this paper. One is the general constraints depending on both the high-level control variables and the low-level optimal solutions, and the other is a special class of constraints depending only on the high-level control variables. In the former case, the penalty function method has to be employed at the upper level to deal with such general high-level constraints due to its simplicity and general applicability, though it may have an adverse effect on the solution quality and computational efficiency in some circumstances. In the latter case, the specialty of the highlevel constraints can be utilized to improve the solution quality and computational efficiency. To achieve this goal, an auxiliary optimization problem is introduced for each value of the high-level control variables generated directly by the ASA algorithm at the upper level which violates the high-level constraints. This auxiliary problem is solved by using the nonlinear progranuning neural network (NLPNN) (Yang and Wu, 1995), and the optimal solution obtained

266

is guaranteed to satisfy exactly the high-level constraints and is taken as the new candidate value of the high-level control variables being assigned to the low-level problems. This avoids using the penalty function method to treat the explicitly defined high-level constraints depending only on the high-level control variables, and hence helps improve the computational efficiency. The computational results for two numerical examples demonstrate the validity and better performance of the proposed method for finding the global or near-global optimal solution to the HOCP in terms of the solution quality and computational efficiency.

2. PROBLEM FORMULAnON Two formulations of the HOCP with different kinds of explicitly defined high-level constraints will be considered in this paper. The HOCP with the general high-level constraints can be formulated as minJ(u,x,v)

(la)

u

S.t . H(u,x,v) =0 G(u,x,v) S; 0 .

rrunJf

(lb) (lc)

_ N-\ k

=I

k

k

k

{vJ}

k=O

S.t. xfk .. \

k k) , = jk1 (ukJ ,xf,v f

XJ

X

(le)

k =O,I, .. ·,N-I

(If)

g; (uj ,xj ,vj) s; 0,

k

= O,I, .. ·,N-l

(Ig)

= 1,2, .. ·,M E

Rm) , xj ERn) , vj

N-\ 0 N-\) Rm ( 0 u= u\''',u\ ,···,uM,··,uM E , X-

vt, .. ·,vZ-\) ERs, Rn

=0 ,1, .. ·, N -1

hj(uj ,xj ,vj) = 0,

where for each 0 s; k s; N - 1, uj s R· "

k

(Id)

is given

j

E

N

Lf(uf,xf,vf)+Kf(xf)

J:R m X Rn

X

R S ~ R,

R S ~ RP, P < rn, G:R m X Rn

X

H :R m X

RS ~ Rq , L~:

R m, x Rn, xRSj ~R, K/R n, ~R, jf:R m, x Rn, k

m

n

S·

pi

h·R'xR'xR'~R' J' -',

gj :Rmj xRnj

X

k S R ' ~Rq"

j

= 1,2,··,M.

k

pJ·
that in this formulation of the HOCP, the general highlevel constraints as given in (lb) and (lc) depend not only on the high-level control variables but also on the optimal control and the corresponding state trajectories of the lowlevel optimal control problems which are obtained under the given value of the high-level control variables. The HOCP with a special class of high-level constraints depending only on the high-level control variables can be described as minJ(u,x, v)

(2a)

u

s.t . H(u) = 0

G(u) s; 0

(2b) (2c)

.

m!nJj {v}

_ N-l

=

I

L

k=O

k

k

k k

N

Lj(uj ,Xj ,vj ) + Kj(xj )

(2d)

S.t. xj+l = ff (uj ,xj ,vjL k = 0 , 1,·· ·, N -I (2e)

o.

Xj

IS

.

given

k=O , I,· ··, N -I

(2f)

k=O,l, " ',N-I

(2g)

where the symbols have the same meaning as those given in (I) except that H :R m ~ RP and G:R m ~ Rq . It is noted that the only difference between (I) and (2) lies in the form of the explicitly defined high-level constraints. From the viewpoint of hierarchical decision-making, the HOCP is solved by a lIDMU and a nwnber of independent LDMU's operating in a sequential and noncooperative manner. The lIDMU chooses at first a decision u and assigns it to each LDMU. For each I ~ j $; M, the j-th LDMU

Since the proposed solution scheme requires that the lowlevel problems be solved at each iteration of the ASA algorithm used at the upper level, the computational efficiency of the method employed to solve the low-level problems may have a great effect on the whole computational efficiency of the proposed solution scheme. Furthermore, the exact solution of the low-level problems for each given U En is also necessary to guarantee the exact evaluation of the high-level perfom1ance index at each iteration of the ASA algorithm. As a result, the appropriately constructed dynamic neural networks with massively parallel computation capacities are desirable means to perform the fast and exact location of the low-level optimal solutions for each given U En. The COCNN as proposed by Yang and Wu (1994) can satisfy the basic requirements on the computational efficiency and solution quality, and hence can be used to solve the low-level problems for each given U En. For the j-th low-level optimal control problem as given in ( Id)-{Ig) with fixed U En, the model of the corresponding COCNN can be described by a system of differential equations with some lower bound constraints as follows

determines its optimal decision vj =(vJ,. .. ,vl- 1) and the corresponding state trajectory x j

=(xJ ,. .. , xl)

(3a) un-

der the given lIDMU's decision, and returns them to the fIDMU as its rational response to the given lIDMU's decision. Since the high-level performance index is related to the high-level control variables and the rational response of each LDMU, the lIDMU should adjust its decision such that the high-level performance index is minimized.

(3b)

(3c)

To facilitate the presentation in the sequel, let .0 be a set of high-level control variables u which satisfy the explicitly defmed high-level constraints and ensure that there exist feasible solutions for each low-level problem. Apart from the difficult nature of the inherent nonconvexity of the HOCP as given in (1) or (2), the multiplicity of the low-level local optimal solutions for each U En may allow the HOCP to become much more difficult to solve. This is because the high-level performance index may have a different value for each low-level local optimal solution under a fixed U En, and the global optimal solution to the HOCP may correspond to a local optimal solution to the low-level problems rather than a global one. Therefore, the global optimal solution to the general HOCP with multiple low-level optimal solutions for each given u En is impossible to obtain unless all tlie low-level local optimal solutions for each given U En can be found . Since fmding all the local optimal solutions to a nonconvex optimal control problem is much more difficult than locating only a global one, the general HOCP with multiple low-level 10cal optimal solutions for each given U En seems intractable. For this reason, we will confine ourselves to a tractable class of HOCP by making the following asswnption. Assumption 1. For each given U En, each low-level problem has a unique optimal solution. '

.

where }.~ ~

k

E

RP} is Lagrange multiplier vector of stage k,

k

E

Rq J is Kuhn-Tucker multiplier vector of stage k,

k+ (k+

gj

=

k+)

k+

g j,I ,· ··,gj,qJ ' gj,i

k . = max{ 0 , g),i}' I = 1,.· ·,

qj , gj,i is the i-th component of the vector valued function gj , r) > 0 and r2 > 0 are given penalty parameters, V denotes the gradient operator, and

Ft =VK/xl)

(4a )

~k =Vx~L~+Vx;h; .[}.~ +rlhn+Vx;gj. (4b) Ilk [,..)

+"2gk + ] + V fk . F kt I ) x~ )) k

= 1,2,.· ·,N -I

xj +l=ff(uj,xj, vj ), k=O , I,. ··,N -I

(4c )

xJ is given. It is noted that the Kuhn-Tucker mUltiplier neurons as described in (3c) operate in a one-sided saturated mode. being different from the dynamics of other neurons. The saturation characteristics of such neurons can be implemented by means of the diode connected to the feedback loop of the corresponding analog integrator.

It is noted that no convexity asswnption is made for the functions related to the high-level problem.

3. NEURAL NETWORK MODELS 267

For the general high-level constraints as in (lb) and (Ic), the penalty function method can be employed to transfonn the original high-level problem into one without explicitly defined high-level constraints, though this treatment may have an adverse influence on the solution quality and computational efficiency of the resulting solution scheme in some circumstances. However, for a special class of highlevel constraints as in (2b) and (2c), the use of the penalty function method can be avoided by introducing an auxiliary optimization problem for each given value of the highlevel control variables generated directly by the ASA algorithm at the upper level. This auxiliary problem takes the following fonn

min !llu _DI12

2 s.t . u EU == {uIH(u)

the candidate solutions can ·be produced at random from the current solution. The latter is used to reduce the value of temperature successively as the annealing goes on. The temperature Updating rule which is inversely proportional to a polynomial function of annealing time with an order greater than or equal to 2 is so derived that it can match the generation function according to a heuristic criterion. Due to the limited space, we will only present the ASA algorithm for solving the high-level problem of the HOCP as described in (2) in which the auxiliary problem (5) has to be solved to obtain the new candidate value of the highlevel control variables satisfying the high-level constraints (2b) and (2c). The ASA algorithm for solving the highlevel problem of the HOCP as described in (I) can be obtained in a similar way except that the general high-level constraints (I b) and (I c) are dealt with by penalty function method. The procedure of the ASA algorithm is as follows.

(Sa)

u

where

=0,

G(u):$; o}

(5b)

u is the given value of the high-level control vari-

ables, and

11 . 11

Step 1. Let

denotes the Euc1idean nonn. It can be seen

trol and

that whether DE U holds or not, the optimal solution to (5) satisfies the explicitly defined high-level constraints as given in (2b) and (2c).

du.

be an arbitrarily given initial high-level con-

be a given initial temperature with a suffi-

uo .

Otherciently large value. If DO E U, then set uO = wise, solve the auxiliary problem (5) with u = uO using the NLPNN as described in (6). The optimal solution obtained is denoted by uO which is feasible with respect to the explicitly defmed high-level constraints (2b) and (2c). Taking uO as the given value of the high-level control variables, the corresponding low-level optimal control problems can be solved by using the COCNN as described in (3). If there exist the low-level optimal control vO and the corresponding state trajectory xO associated with uO, the

To perfonn a fast and exact location of the optimal solution to this auxiliary optimization problem, the NLPNN as proposed by Yang and Wu (1995) can be used. The model of the corresponding NLPNN for solving the auxiliary problem (5) can be expressed as P

- = u-u- L(J-, +rH;(u»)VHi(U) dt i= 1

DO

To

(6a)

high-level perfonnance index value J(uo,xO,vo) can be calculated. Otherwise, assign a sufficiently large number to the high-level perfonnance index value. Then set k = 0,

- f(pj +rGj (U»)VGj(u) ) =1

dJ.. dt

= H(u)

(6b)

umm .

dp dt

= G(u) , pe. 0

(6c)

Step 2. If a prescribed termination condition is satisfied,

=uO

and J mm .

=J(uo "xO

vO).

then stop and take umin as the fmal high-level optimal control. The low-level optimal control and the correspond-

where all the symbols have an analogous meaning as in the previous text. It is noted that both the COCNN and the NLPNN as described in (3) and (6) respectively take advantage of the same way to treat the inequality constraints so that the introduction of the slack variables for converting the inequality constraints into the equality ones becomes unnecessary, preventing the number of neurons for implementing the neural network from increasing. Using the same way as that given by Yang and Wu (1994 , 1995), it can be shown that the equilibrium point of the COCNN and the NLPNN is asymptotically stable and corresponds to the optimal solution to the original problem under some mild conditions.

ing state trajectory associated with umin can be obtained accordingly. Otherwise, continue with the following steps. Step 3. Generate a m-dimensional random vector zk with

the i-th element

z; being specified by

i = 1,2,···,m

where

(Ji'S

are independent random variables with the uni-

fonn distribution in the interval (0,1),

4. ASA ALGORITHM

(7)

OJ/s

are inde-

pendent random variables with the unifonn distribution in

To overcome the inherent nonconvexity of the HOCP and find its global optimal solution, we will apply the ASA algorithm as proposed by Yang and Wu (1993) to solving the high-level problem of the HOCP as given in (I) or (2). The high reliability and efficiency of the ASA algorithm are mainly attributed to the use of a new kind of generation function and an appropriate temperature updating rule. The fonner is used to generate random trial steps so that

the interval [-1,1], Br and

OJ j

are mutually independent

for all i and j , Tk is the value of temperature at the k-th iteration, and (Ye. 2 is an integer number. Step 4. From the current value u k of the high-level control variables and the random vector z k as generated above, a

268

new candidate value il+ 1 of the high-level control variables can be produced at random in the following way

(8) Step 5. If zik+ 1 E U, then set u k + 1

= zik+ 1

Otherwise,

The computational results for two numerical examples are presented to demonstrate the validity and better performance of the proposed method for finding the global or near-global optimal solution to the HOCP as in (1) or (2). Example 1.

m~n~ ~~Jluk +a21 +Ix k +b21 +Iv k +c21]+~lxN +b~1

solve the auxiliary problem (5) with zi = zik+ 1 using the NLPNN as described in (6). The optimal solution obtained is denoted by uk + 1 which satisfies uk + 1 E U .

k

k

S.t. Cu )2+Cxk)2+Cv )2:S;1.6,

-lO:S;uk:S;IO, k=O,I" " ,N-l

Step 6. Taking uk+I obtained above as the given value of the high-level control variables, the corresponding lowlevel problems can be solved by using the COCNN as described in (3). If there exist the low-level optimal control v k + 1 and the corresponding state trajectory xk+ 1 associated with uk + I , the high-level performance index value J(uk+I ,xk+I, v k + I ) can be calculated. Otherwise, assign

-I[

s.t.xk+l=xk+uk_vk,

k=O,I,"',N-l

xO = I where N = 4, u k , xk, v k ER, and the parameters are given in Table 1. Table 1 Parameters for Example 1

Step 7. Let /jJk+I ;: J(u k + I ,xk+l, vk+I) _ J(u k ,xk, v k ).

a2 0

b2

bk

c2

Ck

0

0

-I

0

I

I

-I

I 0

0

I

2 3 4

0

0

I

-I

0

-I

0

0

k

(9)

If /jJk+I :::;; 0, then replace u k with u k +1 and go to Step B. Otherwise, generate a random number X distributed uniformly in the interval (0,1) and examine the condition

p

]

1 NI (x k +bk)2+(vk +ck)2 +_(x I N +b1)2 mink {v } 2 k=O 2

a sufficiently large number to the high-level performance index value.

where

k=O,I," ',N-l

0

This example includes the general form of explicitly defmed high-level constraints which can be dealt with by the penalty function method due to its simplicity and general applicability. The augmented high-level performance index 1 with an appropriately chosen penalty term is defined as

is some positive real number. If it is satisfied,

the new candidate value uk + I of the high-level control variables is accepted and is replaced for u k as a new current value of the high-level control variables. Otherwise, uk+ 1 is rejected and the current value u k of the high-level control variables is retained. Then, go to Step 9.

N-l

l=J+s

I

max{O, (uk)2+(~)2+(vk)2_1.6}

k=O StepB. If J(Uk+I,Xk+I,V k +I ) < J rnin , set urnin =u k + I and J rnin = J Cu k +1, xk+I , v k +I ) .

°

Step 9. Update the value of temperature by the following updating rule 1',

To

-

k+I-

l+yCk+l)O'

k=0,1,2,' "

°

where To > is the initial value of temperature, y > some real number, and a~ 2 is an integer number.

where J is the original performance index for Example 1 and s > is the given penalty parameter. By applying the proposed method, in which the general high-level constraints are dealt with by the penalty function method, to solving this example problem with the augmented highlevel perforffiance index 1 as defmed above, the computational results as shown in Table 2 can be obtained, where the following parameters are employed: the initial value of the high-level control variables is set to zero, S = 3,

(11 )

°

=105, Y = 1,

To

is

a= 3, and the prescribed number of it-

erations as the termination condition of the ASA algorithm is 2000 .

Step 10. Increase the iteration index k by one and go back to Step 2.

Table 2 Computational results for Example 1 with the general high-level constraints by the proposed method

It is noted that if there are lower and upper bound constraints imposed explicitly upon the high-level control variables, a simple limiter as proposed by Yang and Wu (1993) can be applied to allow the new candidate value of the high-level control variables as produced in Step 4 to keep within the given lower and upper bounds.

5. NUMERICAL EXAMPLES

r

0.773903

0.193365

3.30707

-0.419462

-0.387166

u k·

0

-0032733

1

-0.999992

2

0.000004

-0.032297

-0.354865

3

-0.999999

-0.677432

-0.677432

Example 2.

269

xk+l.

k

v k·

Table 4 ComEutational results for Example 2

~n~:~Jluk +a21+lx k +b21+lv k +c21]+~lxN +b~1 N -l

L

5 .t .

using the penal~ function method

(u k )2=1

k~O

-!O~uk~!O, k=O, I, ···,N-I

{v

k

-l[

]

1 NL (x k +bl)2 +(vk +ck)2 +_(x 1 N +b1)2

min}

2

2

k=O

ko

xk+l.

r

0.11925

1.000008

0.119242

3.20194

k

u k·

0

v

I

-0.452865

-0.119234

-0.214389

2

0.000837

0.095155

-0.308707

3

-0.88357

-0.596\38

-0.596138

s .t . xk+l=xk+uk_v k , k=O,I ,-··,N-I

xO = 1

6. CONCLUSIONS

where N = 4, u k , xk , v k ER, and the parameters are the same as in Example 1. The explicitly defined high-level constraints of this example depends only on the high-level control variables. Byapplying the proposed method, in which an auxiliary problem is used to treat such special high-level constraints, to solving this example problem, the computational results as shown in Table 3 can be obtained., where the following parameters are employed: the initial value of the high-level control variables is set to zero, 1'0 = 104 , r = I, (j = 3, and the prescribed number of iterations as the termination condition of the ASA algorithm is 500. Table 3 Computational results for Example 2 with the special high-level constraints by the proposed method

k

u

ko

'v ko

xk+lo

JO 3.18918

0

0.014121

0.999974

0.014147

I

-0.16389

-0.014173

-0.13557

2

0.000024

0.121397

-0.256943

3

-0.986378

-0.62166

-0.62166

REFERENCES

To make a comparison, the penalty function method is employed to cope with the explicitly defmed. high-level constraints of Example 2. The augmented high-level performance index 1 with a proper penalty term is defmed as

where J and S have the same meaning as in the previous text The computational results obtained by using the penalty function method are shown in Table 4, where the following parameters are used.: the initial value of the highlevel control variables is set to zero,

5

= 1.5,

To

A novel solution scheme for fmding the global or nearglobal optimal solution to the HOCP is proposed in this paper which integrates appropriately the ASA algorithm and the COCNN. The former is used to find the global or near-global optimal solution to the high-level problem, and the latter is used to perform a fast and exact location of the low-level optimal control and the corresponding state trajectories for each given value of the high-level control variables. For a class of HOCP with the explicitly defmed high-level constraints depending only on the highlevel control variables, an auxiliary optimization problem can be introduced at the upper level to avoid using the penalty function method to deal with such high-level constraints in an inefficient way. The computational results for two numerical examples indicate the validity and better performance of the proposed solution scheme for fmding the global or near-global optimum of the HOCP in terms of solution quality and computational efficiency.

= 106 ,

r = 1, (j = 3, and the prescribed number of iterations as the termination condition of the ASA algorithm is 3000. It can be seen from the computational results for numerical examples that the proposed solution method combining the ASA algorithm and the COCNN is applicable to solving the HOCP with the general form of explicitly defmed high-level constraints. Furthermore, the proposed method involving the solution of an auxiliary problem can be used. to fmd more reliably and efficiently than the use of the penalty function method the global or near-global optimum of a class of HOCP with explicitly defmed high-level constraints depending only on the high-level control variables. 270

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