A numerical method to optimize the stability of a linear dynamic system which was described in the last paper is generalized to the case of equality constraints in this report. The process is an application of the “steepest-descent method” as well, and it is inclusive of the procedure given in the last paper as a simple case. Hence, with application of this method, problems which are of practical interest but are difficult to treat with the previous method are expected to be solvable. A numerical example is presented for maximizing the damping of the Dutch-Roll mode of motion of an airplane

MAEDA, Hiroshi

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RIGHT:URL:CITATION:AUTHOR(S):ISSUE DATE:TITLE:A Method to Optimize the Stabilityof a Linear Dynamic System : II.With Equality ConstraintsMAEDA, HiroshiMAEDA, Hiroshi. A Method to Optimize the Stability of a Linear Dynamic System : II. WithEquality Constraints. Memoirs of the Faculty of Engineering, Kyoto University 1965, 27(2):166-1771965-05-20http://hdl.handle.net/2433/280623166 A Method to Optimize the Stability of a Linear Dynamic System. II. With Equality Constraints By Hiroshi MAEDA* (Received December 28, 1964) A numerical method to optimize the stability of a linear dynamic system which was described in the last paper is generalized to the case of equality constraints in this report. The process is an application of the "steepest-descent method" as well, and it is inclusive of the procedure given in the last paper as a simple case. Hence, with application of this method, problems which are of practical interest but are difficult to treat with the previous method are expected to be solvable. A numerical example is presented for maximizing the damping of the Dutch-Roll mode of motion of an airplane. 1. Introduction The method of gradients or "the steepest-descent method" was described for optimizing the stability of a linear dynamic system in the last report'\ However, in many problems of practical interest, the coefficients of the charac-teristic equation of a linear system are not only functions of control variables but also those of state variables, i.e. the subsidiary conditions are in general given by the following equations, where ,Bk (k=I,2,, .. ,M) and Tl (1=1,2,···,N) are M control variables and N state variables respectively. If the functions g1 are given analytically and solved for N state variables in terms of M control variables, the state variables can be eliminated and the problem is reduced to one without constraints, which was previously described in the last report. However, such an approach will not be practical in many applications, and, therefore, a general procecure applicable to the problems is stated in this report. * Department of Aeronautical Engineering A Method to Optimize the Stability of a Linear Dynamic System, JI. 167 2. Optimization of the Stability Problems with Equality Constraints The characteristic equation of a linear system is in general expressed by where the coefficients a; are given by i = 0, 1, ···, n-1 l = 1,2, ··•,N ( 1) ( 2) and T1 are the "state variables". The state variables are also related to other variables by the following known equations, ( 3) where fik (k=l,2, ···,M) are the "control variables" which are free to choose. Roots of the characteristic equation are in general expressed by ( 4) if the roots are real, w i = 0. Since the stability criterion is the value of the real part of the roots*, i.e. ni in Eq. (4), then in order to optimize the stability, I ni I should be maximized or nj should be minimized, because ni is negative for the stable state. Since J.i or ni are the functions of the coefficients a;, or Tl from Eq. (2), then it is necessary 'to determine fik so as to minimize n/T1) subject to the constraints of Eq. (3). This method starts with the starting values of the control variables /if. In order to determine the change in n(r,) for a small perturbation of the control variables d/ik, consider first the quantity where µ 1 are Lagrange multipliers. Hence, d</J = dn = [an, .. !!!!_]r~TI l +[µ1···µN] OT1 OTN : dTN * See Appendix ( 5) ag1 ag1 OT1 ······ OTN dT1 agN agN OT1 ...... OTN d/31 168 Hiroshi MAEDA ( 6) where the partial derivatives are evaluated at the starting point. Now let us chooseµ, through µN so that the coefficients of dr1 (l =l, 2, ···, N) vanish, i.e. ( 7) on + µ, og, + ... + µN OgN = Q orN orN orN or the Lagrange multipliers are determined by µ,•LI= on og2 ogN - orN orN ...... 8rN ( 8) og, OgN-1 on or, a:,:;- -ar, µN•LI = og, ...... ogN-1 on orN orN orN where og, OgN or, or, LI = : ( 9) og, OgN orN orN This reduces Eq. (6) to the expression (10) For the steepest descent, the following value of the positive definite quadratic form is defined, (11) where wk (k=l,2, •··, M) are the positive weighting numbers. To maximize A Method to Optimize the Stability of a Linear Dynamic System, II. 169 I dn I for a small perturbation d/3,. under a constraint condition given by Eq. (11), the similar process to the previous calculation is applied again, i.e. consider the quantity dlJI = (fi µt ~g/31 ) d/3, + ... + ( fi µt !13gt) d/3M l-l V l l-1 V M M +v[(dP)2 - I:: w,.(d/3,.)2] (12) k~l where µ, through µN are given by Eq. (8) and v is another Lagrange multiplier. The maximum I dn I is then obtained when (k = 1, 2, ... , M) (13) Accordingly, d/3,. = _1_(µ,8g, + .. ·+µN8gN) 2vw,. 8/3,. 8/3,. (14) Substituting Eq. (14) into Eq. (11), and solving in v, (15) Therefore, substituting Eq. (15) into Eq. (14), (16) Since dn should be negative, or n is to be decreased, - sign is used in Eq. (16). If the functions gi(/3,., r 1) are given analytically and Eq. (3) are solved for the state variables in terms of the control variables, i.e. Eq. (3) is reduced to or Hence, by Eq. (8) and (18), 8gt = 1 8rm 0 (17) (18) (19) (20) Consequently, the state variables are eliminated and Eq. (16) is identical with 170 Hiroshi MAEDA the result which was given in the previous report. For the next step, fit +d/3k are the starting values and the same procedure is repeated. The process should be repeated several times until the gradient dn/dP or ~ = - - ~-/ +···+- ~-/ dn [ 1 ( N ag )2 1 ( n ag )2]112 dP W1 !~1 8/31 WM H 8/3M (21) is nealy zero or the absolute values of all real parts of the roots become roughly the same. The maximum stability of the system will then be obtained. 3. Example-Optimization of the Dutch-Roll Stability of an Airplane As a numerical example, the optimization problem of the Duch-Roll stability of a light airplane is considered. Since the lateral equations of motion of an airplane are in general expressed by simultaneous linear differential equations in 3 variables, and the characteristic equation is the quartic, then it is ex-prssed by where the coefficients a; are2) a3 = -(0~+ ~lp+G_nr) 2µ tA tc - 1 (C C -c C ) +Cnfl+Cyfl(Cnr C1p) ll2 - -. -. nr Ip np Ir -.- -2 -.- +---,--tAtc tc µ tC tA ll1 = 21 -~Y~(C1,-Cnp-CnrC1p)+A(C11iCnp-Cn/lC1p) µ tAtc tAtc _l.-J..c Loe 1/l 2µ tA (22) (23) For the simplification of calculation, small quantities, e.g. small stability derivatives, product of inertia, etc., are left out in this expression. According to the expressions des-:::ribed in the previous section, the state variables r, are, therefore, as follows: r1 = Cyfl r1 = C1r r2 = Cnfl rs = iA r3 = C1/l rg = ic (24) r4 = Cnr r10 = µ rs= C1p ru = CLo rs= Cnp A Method to Optimize the Stability of a Linear Dynamic System, II. 171 i.e. M=ll. The control variables j3k which are freely chosen in the design of an airplane are considered as follows : /3, = Sv/S0' : vertical tail area 82 = lv/lv'-1< vertical tail length /33 = ZTT/z0' /3, = I' /I'* vertical tail height effective dihedral angle (25) i.e. N =4. In the above expressions the variable quantities are nondimen-sionalized for the convenience of the following calculation. By the definition of the state variables, the constraining equations are* g, = r,-K,/3,-K,' = 0 g2 = r2-K2/3,82-Ki' = 0 g3 = r3- K3/3,/33- Ks' /3, - K311 = 0 g, = r,-K,8,/3~-K,' = 0 gs =rs-Ks=0 g_s =rs-Ks= 0 g1 =r1-K1=0 gs = rs·-Ks = 0 gg = r9-K9 = 0 glO = r,o-Kw = 0 g11 = r11-K11 = 0 (26) where K, through Ks'' are assumed to be constants. Strictly speaking, g5 through g11 are the functions of the control variables too, but the influences are considered so small that they are neglected and rs,•··, ru are assumed to be constant in this calculation. The steepest-descent method described in the last section is, therefore, applied as follows : ** 1) control variables (starting conditions) 8,* = Sv/S0' = 1 /32* = lv/l0' = 1 2) equations of constraints /33* = zv/zv* = l /3,* = I' /I'*= 1 r,* = -0.157 -0.245 /3,* = -0.402 r2* = -0.055 +0.0955 /3,*82* = 0.0405 * This example is not so suitable for the case presented in this report, for this example can be computed by the procedure described in the last report. However, since it is easy to understand the procedure given here by this example, it was chosen as an illustrative problem. ** The numerical data of a typical light airplane are employed in this example. 172 Hiroshi MAEDA f3* = 0.0365-0.085 ,B.*-0.034 ,81*,83* = -0.0825 r.* = -0.0085-0.074 81*,82*2 = -0.0825 rs'l< = -0.490 r6* = -0.0505 r1* = 0.101 rs* = 0.032 fg* = 0.0615 r10* = 16.26 r11* = 0.405 The characteristic equation, Eq. (22), was solved by the use of NEAC-2101 at the Department of Applied Mathematics and Physics, Kyoto University, and 4 roots of the characteristic equation at the starting condition are obtained as follows: ).1 = -0.001277 : spiral mode ).2 = - 0.4828 : rolling mode A3, ).4 = -0.02230±0.1620i : Dutch-Roll mode Hence, the real value to be optimized or minimized is n = -0.02230 First of all, substituting Jr1 = -0.040, Llr2 = ··· = Llru =0 into the characteristic equation, it was solved again. By the small change of n or Lln, Lln = (an )* = 0.015 ilr1 8r1 The same procedure was repeated for the case of Llr2 , Llr3 , ••• respectively, and (~;J*, • • • were obtained as follows: By Eq. (26), and, therefore, by Eq. (8), ( an )* = - 0.0395 ar2 (~;J* = -0.091 ( an )* = 0.2315 ar. A Method to Optimize the Stability of a Linear Dynamic System, JI. 173 Furthermore, in Eq. (10) ~t are calculated by Eq. (26), i.e. (8g,)* = _8r, = 0.245 8/3, 8/3, (8g2)* = _8r2 = -00955/3* 8/3, 8/3, . 2 Employing those results, the following expression was computed, where w,= .. ·=w4 =1 are the weighting numbers. When ldPI =0•1, For the second step, the starting values of the control variables are therefore /3, = /3,*+d/3, = 1.0483 /32 = /3,*+d/32 = 1.0855 /33 = /33*+d/33 = 0.9930 /34 ~= fi4*+d/34 = 0.9826 The characteristic equation was solved for those fik, and the roots are, at the second point, ;i, = - 0.001052 J.2 = -0.4826 J.3, A4 = -0.02702±0.1798i i.e. the real part of the complex roots is n = -0.02702 that is, the stability of the Dutch-Roll mode was augmented about 2196 by one step of the calculation. 174 Hiroshi MAEDA The same procedure was repeated several times, and the results are shown in Table 1 and Fig. 1. The figure shows clearly that the stability of the Dutch-Roll mode is augmented remarkably by the application of this method, i.e. the damping or the time to half amplitude at the starting point is 3.12 sec. in this example, but it is 1.55 sec. after 5 times of iteration of the computation. The combination of control variables corresponding to this last damping mode is as follows : Sv/S0' = 1.196 l v/l 0' = 1.342 zv/z0' = 0.975 r I I'* = 0.942 That is, it is found that the vertical tail area and the vertical tail length should be larger and the effective dihedral angle should be smaller than the original configuration. Table 1. Dutch-Roll mode Control variables Characteristic roots Time to I ½ amplitude Period /3 1=Sv/Sv*=1 ,l1 = -0.001277 (sec.) (sec.) (1) P2=lvllv*=l ,l2= -0.4828 3.12 3.92 /33=Zv!Zv*=l ,l3 }= -0.02230±0.1620i /34 = r I I'*= 1 ,l• /31 =1.0483 ,l1 =0.001052 (2) /32=1.0855 A2= -0.4826 2.58 3.53 /i3 =0.9930 ,l3 }= -0.02702±0.1798i /34 =0.9826 ,1, /31 =1.0970 ,1, = -0.000950 (3) /32=1.1712 ,l2 = -0.4827 2.16 3.19 /33=0.9870 ,l3} = -0.03229±0.1986i /i4 =0.9680 ,l• -- ------! I /31 =1.1463 ,l1 = -0.000909 (4) /32=1.2569 ,l2= -0.4826 1.82 2.94 /33=0.9810 ,l3 }= -0.03823±0.2159i /34 ~0.9542 ,l• /31 =1.1963 ,l1 = -0.000903 (5) /32= 1.3424 A2=0.4826 1.55 2.73 /33~0.9753 ,l3 }= -0.04485±0.2327i /34 =0.9419 ,l• A Method to Optimize the Stability of a Linear Dynamic System, II. 175 sec. ?:: 4 ~ \ -----"( )----, ..___ ~ ~ Starting point 2 s;,:;* S, Period -- -~ ---------- ----o .____ --. r------, 3 4 5 6 No. of iteration 1.21------+---+---+---c='-.>-12~ QB ¾* 1.0 'rl--~=~·;cc;~:.::Z!=' ::t==~-~ QB QB --==x\---l-*---+---t---------i ~,.r-r +------11.0 r;/,* 112 r 08 Fig 1. An example of steepest-descent computation. 4. Summary A numerical method for optimizing the stability of a linear system .,<.1bJect to subsidiary conditions is presented. The practical procedure is described for the problems in which the subsidiary conditions are given as constraining equations in control and state variables. The process is an application of the "steepest-descent method" too, and is similar to that of the last report. Consequently, the procedure described in the last report is involved as a simple case in which the functions are given analytically and the set of con· straining equations can be solved for the state variables in terms of the control variables. A numerical example is presented for maximizing the damping of the Dutch-Roll mode of an airplane, and it is found that, to obtain the better stability of this mode, the vertical tail area and the vertical tail length should 176 Hiroshi MAEDA be larger and the effective dihedral angle should be smaller than the original configuration. Acknowledgement The author wishes to express his sincere thanks to Prof. H. Tokumaru, the Department of Applied Mathematics and Physics, for his kind help in the numerical computations in this work. Nomenclature a; coefficients of characteristic equations CLo lift coefficient (steady state) Cy13, Cn/l, C1/l, etc.: aerodynamic stability derivatives iA, ic: moments of inertia about X- and Z-axis lv vertical tail length N112 cycle to ½ amplitude Sv vertical tail area T112 time to ½ amplitude w,, weighting numbers zv vertical tail height /31, control variables r dihedral angle Tt state variables Aj roots of characteristic equations ().j=nj±iwj) µ relative density factor µ,,)) Lagrange multipliers References 1) H. Maeda: This Memoirs, Vol. XXIV, Part 4, 1964 2) B. Etkin : Dynamics of Flight, John Wiley and Sons, 1959 3) C. D. Perkins and R. E. Hage : Airplane Performance, Stability and Control, John Wiley and Sons, 1949 4) H. Maeda : Attitude Stability o{ Articulated Gravity Oriented Satellites, Part 2, UTIA Report No. 93, 1963 5) A. E. Bryson and W. F. Denham : A Steepest-Ascent Method for Solving Optimum Problems, Journal of Applied Mechanics, Vol. 29, No. 2, 1962 6) G. Leitman (ed.) : Optimization Techniques, Academic Press A Method to Optimize the Stability of a Linear Dynamic System, II. 171 Appendix- On the Stability Criterion of a Linear Dynamic System, Especially of the Oscillatory Mode In general, the values of the real part of the characteristic roots are taken as the stability criterion of a linear system, i.e. when the roots of the charac-terisitc equation are expressed by J.i=nj±iwi (if the roots are real, wi=0) and I n1 I> I n2 I or n1 < n2, the motion corresponding to n1 is considered more stable than that of n2 • The analysis described in this report is dependent upon this criterion, too. This criterion is quite right in the case of aperiodic mode of motion, but in the case of oscillatory mode another criterion is sometimes considered more reasonable. More specifically, since the real part of the characteristic root is related to the time to ½ amplitude, i.e. T 0.69 1 2=7nT sec. then large In I corresponds to small T11 2 , and the disturbed motion will dis-appear more quickly. This situation is the same in the case of the oscillatory mode as well, but for the oscillatory motion the frequency w should be taken into consideration together with n. For instance, by a flight test it is reported that, even if the time to ½ amplitude is smaller, when the frequency w is larger simultaneously the air-plane motion is not recognized as more stable. This means that pilots, crews or passengers are so sensitive to the acceleration of motion that the cycle to ½ amplitude N112 is important as well as the time to ½ amplitude T1; 2 • Consequently, in such a special case as the airplane motion, we suggest that the cycle to ½ amplitude is the more reasonable criterion of the system's stability. More specifically, since N112 = 0.110 I~ I then I~ I should be the smallest in order to minimize N112 or optimize the stability. When the frequency w is constant, this criterion is identical with the ordinary one, because In I should still be maximized. 

A Method to Optimize the Stability of a Linear Dynamic System : II. With Equality Constraints

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A Method to Optimize the Stability of a Linear Dynamic System : II. With Equality Constraints

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