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ALGORITHMS FOR OUTPUT FEEDBACK, MULTIPLE-MODEL, AND DECENTRALIZED CONTROL PROBLEMS Nesim Halyo and John R. Broussard Information and Control Systems, Incorporated Hampton, Virginia First Annual NASA Aircraft Controls Workshop NASA Langley Research Center Hampton, Virginia October 25-27, 1983 ABSTRACT The optimal stochastic output feedback, multiple-model, and decentralized control problems with dynamic compensation are formulated and discussed. Algo- rithms for each problem are presented, and their relationship to a basic output feedback algorithm is discussed. An aircraft control design problem is posed as a combined decentralized, multiple-model, output feedback problem. A con- trol design is obtained using the combined algorithm. An analysis of the design is presented.
ADVANTAGESOF STOCHASTIC OUTPUT FEEDBACK The stochastic optimal output feedback problem 11-81 is a significant extension of the "full-state feedback" LQG problem [9]. Its formulation ad- dresses some important limitations encountered in practical systems and provides a flexibility useful in configuring the control law for ease of implementation.
Some of the advantages of the stochastic output feedback problem are shown below.
Output feedback introduces a rich class of control law structures which can be used in modern control designs.
l DESIGNER CAN SELECT THE STATES FOR FEEDBACK PROVIDES A METHODTO DESIGN OUTER LOOP CONTROL LAWS a ACCOUNTS FOR ACTUATOR DYNAMICS WITHOUT NECESSITY FOR ACTUATOR STATE FEEDBACK ACCOUNTSFOR PHASE SHIFTS INTRODUCED BY PREFILTERS AND OTHER ESTIMATORS WITHOUT NECESSITY OF FEEDBACK a PROVIDES A SYSTEMATIC METHODTO INCREASE OR DECREASE GAINS BY ADJUSTING PLANT AND MEASUREMENT NOISE COVARIANCES a PROVIDES CONSIDERABLE FLEXIBILITY IN THE CONTROLSTRUCTURE IN A MODERNCONTROLSETTING FORMULATION OF THE STOCHASTIC OUTPUT FEEDBACK PROBLEM The discrete stochastic optimal output feedback problem is formulated below. The control Uk feeds back the output Yk through a constant gain matrix K.
The term ZI is the set of gains K for which JN(K) converges to a finite value J(K).
The term 5' is the set of gains which stabilizes the closed-loop system. The opti- mization problem can be posed as: Find a stabilizing gain K* (K* e S) which minimizes the cost J(K), i.e., J(K*) I J(K), K c D.
= $ xk + r uk + wk 'k+l Yk = c Xk + Vk Uk = - K Yk
= WAki VT) = Mki x1s, = so
E(Wk w:, E(Vk E(Xo
E(Wk) = 0 E(Vk) = 0 V;) = E(Wk XL) = E(Vk XL) = 0 E(WI< + U; R Uk) JN(K) = &i-) ,io E(X;+, Q 'k+, J(K) = lim JN(K) < 03 KEV EXAMPLE Some important characteristics of the stochastic optimization problem posed are illustrated in a simple first-order example. In this example, the domain of optimization V is the semi-open interval (0, 2), while the set of stabilizing gains S. consists of the open interval (0, 2). The system is completely con- trollable and output stabilizable. However, a.s illustrated by the example, out- put stabilizability alone does not guarantee the existence of a solution to the optimization problem. The cost function J(K), for this example, has no minimum in V or in S. Furthermore, the example illustrates that the continuity of the cost function J(K). over its domain D is not guaranteed, as K.= 0 is a point of discontinuity. Therefore, it is desirable to determine conditions under which an optimal solution exists.
= Xk + Uk Yk 5 Xk + Vk 'k+l Q=l R=O V=l w=o so = 1
rTQI'+R=l>O c w CT+ v =l>O
O<K <2 K=O J(K) 3- 2- OUTPUT FEEDBACK EXISTENCE CONDITIONS As illustrated by the example, the domain of optimization is not necessar-: ily a closed set, and can be unbounded, although S is always open. Thus, it is necessary to determine conditions under which the minimum cost is attained Such conditions which guarantee the existence of a at an interior point of S.
solution to the optimization problem are shown below. Under these conditions, the domain of optimization V coincides with the stability set S, which in- On the other sures that the optimal gain stabilizes the closed-loop system.
hand, it can be shown that the cost function J(K) is always continuous on S Note that the example considered previously fails to satisfy the con- I103 - dition W 2 E r rT, but satisfies all the remaining conditions. While the conditions 1, 2, and 3 ensure the existence of a stable global minimum, they are not necessary for the existence of a solution to the optimization problem.
However, the class of optimization problems covered is quite broad, and because the existence conditions are expressed in terms of known system parameter matrices, verification is a simple task. Note that the measurement noise and control penalty terms are not necessary for existence, which is a major dif- ference between the discrete and continuous output feedback problems. Also note that Q and W need not be positive definite, but must satisfy 1. Condi- tion 1 is intriguing, as it corresponds to a method of improving robustness in control and filter designs [ll]. The uniqueness of the solution is not ensured, except for special cases such as full-state feedback.
SUFFICIENT CONDITIONS FOR EXISTENCE: 1. FOR SOME E > 0 Q ;? E CT C w 2 E r rT 2. rTQr+R>O CWCT+bO 3.
(C, 4, I-') IS OUTPUT STABILIZABLE LET 1 AND 2 HOLD. J(K) HAS A STABLE MINIMUM IF, AND ONLY IF, (C, 4, r) IS OUTPUT STABILIZABLE For gains which stabilize the closed-loop system, the cost function J(K) can be expressed more explicitly in terms of K, as shown below. An expression which provides more insight can be obtained by considering the incremental cost AJ(K, AK). As the incremental cost is the total change in the cost due to a change AK in the gain, the optimization problem can also be treated as that of finding a AK* which minimizes the incremental cost for a fixed K c S. Due to the almost quadratic form of the incremental cost, a "natural" direction is the one which would minimize the incremental cost if it were actually quadratic in AK. The following theorem exploits this direction, d(K).
Theorem: Let the existence conditions 1, 2, and 3 hold, and K. be in S. Then there exist f3 > 0, a sequence {Ki, i 2 O}, and a limit point, say K*, of the sequence such that and $(Ki) -f $(K*) = 0 J(Ki) ~ J(K*) K = Ki + o! d(Ki) i+l d(K) = p(K) -' rT P(K) @ S(K) CT S(K)-' - K whenever 0 < cx I B.
J(K) = i trip(K) WI + i triKT p(K) K VI KcS P(K) = $(K)T P(K) 4(K) + CT KT R K C + Q S(K) = 4(K) S(K) @(K)T + r K V KT rT + W i;(K) = rT P(K) r + R s(K) = C S(K) CT + V AJ(K,AK) = J(K+AK) - J(K) =--i
tr{2 AKT[?(K+AK) K s(K) - rT P(K+AK) $I S(K) cTI
+ bKT B(K+AK) AK S(K)1 K, K+AK E S NECESSARY CONDITIONS ii K* S^(K*) = rT P(K*) I$ S(K*) CT K* c S - OUTPUT FEEDBACK ALGORITHM Convergence Theorem: Let {Ki, i 2 01 be a sequence of gains obtained from the algorithm, starting with K. E S. Then, any limit point, say K*, satisfies the necessary conditions for optimality, stabilizes the closed-loop system, and J(Ki) ~ J(K*).
a0 = 1 1. CHOOSE K, 6 S z>l i =0 2.
SOLVE THE LYAPUNOV EQUATIONS P(Ki) = IT P(Ki) 9(Ki) + CT K: R Ki C + Q S(Ki) = I S(Ki) $(Ki)T + I Ki V Ki rT + W IF P(Ki) OR S(Ki) IS NOT NON-NEGATIVE DEFINITE GO TO 5 3. COMPUTEd(Ki), Ki+, d(Ki) = ij(Ki) -' rT P(Ki) 4 S(Ki) CT S^(Ki)-' - Ki K = Ki + ai d(Ki) i+l 4. COMPUTETHE COST J(Ki) J(Ki) = i triP(Ki) WI + k tr{Ki P(Ki) Ki Vf SET i = 1 IF i = 0, AND GO TO 2.
tr d(Ki_l)T~(Ki l)d(Ki ,)?(Ki ,) ] GO TO 6 IF J(Ki)-J(Ki-l)<-$ ai-1(2-ai-l) t 5. REDUCEa a.
= ai/z Ki = Ki 1 d(Ki) = d(Ki_1) K = Ki + ai d(Ki) cli+l = ai i = i+l GO TO 2 i+l 6. COMPUTEGRADIENT I = - I d(Ki) I IF 1)$(‘i) // ’ El OR IJ(Ki) - J(Ki-l)l > ~29 ai+l = ai, i = i+l, GO TO 2 7. STOP OPTIMAL DYNAMIC COMPENSATION Most control systems for complex plants use some form of dynamic compen- sation. The dynamic compensator may simply consist of an integral feedback or a rate command structure, or may be a Kalman filter or an observer. Classical control designs make considerable use of dynamic compensation in the form of various filters, washout loops, etc. The basic form of a digital control sys- tem making use of dynamic compensation is shown below. The design of dynamic compensation in an optimal control setting can be imbedded into the optimal output feedback formulation by augmenting the state with the compensator states In this form, the order of the dynamic compensator is a design [12,131 - parameter and can be selected so as to obtain a low order, easily implemented compensator. For systems which are not stabilizable with the available measure- ments, such as some cases of flutter suppression, dynamic compensation is a necessary rather than a simply desirable structure [141. The design of the dynamic compensator can be obtained using the output feedback algorithm pre- sented earlier.
PLANT 'k _ 'k , c (4, r) 'k COMPENSATOR = @ xk + r uk + wk Yk = c Xk + Vk 'k+l ADVANTAGESOF THE MULTIPLE-MODEL OUTPUT FEEDBACK APPROACH While output feedback introduces significant flexibility in the structure of a control law, it does not directly address some of the objectives and re- quirements encountered in designing control laws. The multiple-model output feedback approach provides a design method which can be used to obtain impor- tant design requirements while preserving all the advantages inherent in output feedback. Some of the advantages of the approach follow.
l PRESERVESALL THE ADVANTAGESOF OUTPUT FEEDBACK l PROVIDES A DESIGN METHOD FOR ROBUST CONTROL LAWS a PROVIDES A DESIGN METHOD FOR MULTIPLE CRITERIA l PROVIDES A DESIGN METHOD FOR ACTUATOR FAILURE ACCOMMODATION a PROVIDESA DESIGN METH.ODFOR SENSOR FAILURE ACCOMMODATION WI .- PLANT1 U1 ' (42 r-1) - - , I I W .
.
.
d za PLANT P P Xp Yp % ) cP - (q& r,)
1 /lr
/
1’1
+qY, ,
I P-4
.
L - DYNAMIC , J ( z COMPENSATOR - C- -K2 w#Jz, rz' FORMULATION OF THE MULTIPLE-MODEL OUTPUT FEEDBACK PROBLEM The multiple-model output feedback formulation considers the problem of igning a fixed control law to meet design object ives expressed in terms of des - various plant models, measurement models, and performance criteria as shown below. The control law structure can contain dynamic compensation and output feedback. For example, the design objective of insensitivity to variations in some plant parameters can be addressed by selecting plant models ($s, r.)
which include these variations. Some types of actuator, sensor, or ot er +l iJlant subsystem failures can be addressed in the design by appropriate selection of
the parameters rj, Cj, $j, V., and Wj. Various other design objectives can be
addressed in a similar manne 4.
Let Sj be the set of gains whick stabilizes the jth plant model, while Vj is the set of gains for which the jt cost remains finite. The intersection S of the S*'s determines the control gains which stabilize all the plant models, whi e the intersection V of the Vj’s is the set on which the total i cost J(K) is finite. The optimization problem can be posed as: Find a gain K* which stabilizes all the models (i.e., K* E S) and minimizes the cost J(K), i.e., J(K*) i J(K), K F V.
= +j xjk + rj ujk + wjk lsjsp
'jk+l Y. =c.x +v lsjip Jk J jk jk U -KY =-KC.X - K V.
jk = 0 J jk Jk Jj(K) = lim K E Vj N- & p E(xik+l Qj 'jk+l ' 'ik Rj 'jk) < m k-0 J(K) = f Y. J.(K) yj >o KcV j=l J J v= v. s= s.
R R
j=l J j=l J MULTIPLE-MODEL OUTPUT FEEDBACK EXISTENCE CONDITIONS As for the case of the (single-model) output feedback problem, the exis- tence of a solution to the optimal control problem posed is not always ensured.
However, as seen by the sufficient conditions given below, a solution does exist for a large class of optimization problems. It should be noted that the constraint that the optimal gain stabilizes the closed-loop models excludes problems where no stabilizing gain exists, so that this class of problems must be treated separately. The sufficient conditions for the problem posed can be It can be shown that obtained by extending the results for output feedback.
the cost function J(K) is always continuous on S, but not necessarily on V.
However, for the class of problems satisfying the sufficient conditions, V and S are equal. The conditions given here are not necessary for the existence of a stable global minimum, and the uniqueness of a solution is not ensured.
Nevertheless, the class of problems included is broad enough to cover most parameter sensitivity objectives, control, or sensor failure accommodation objectives, as well as other significant objectives.
SUFFICIENT CONDITIONS FOR EXISTENCE
1. FOR SOME c > 0, AND ALL Qj 2 E CT C. wj 2 E r. 2
jw J J J J 2. I'; Qj rj + Rj > 0 cj wj c; + vj > 0 lsjgp 3.
S IS NON-NULL LET 1 AND 2 HOLD.
J(K) HAS A STABLE MINIMUM IF, AND ONLY IF, P s = n sj CONTAINS AN ELEMENT j=l MULTIPLE-MODEL INCREMENTAL COST AND NECESSARYCONDITIONS For gains which stabilize all the plant models, the cost function J(K) can be expressed in terms of the gain K, as shown.
Similarly, the incremental cost AJ(K,AK) or the change in the cost due to a change in the gain, is seen to resemble a quadratic form in AK.
The necessary conditions can be easily obtained from the incremental cost by letting dK approach zero. The direction d(K), which would minimize the incremental cost if it were a true quadratic form, is selected to obtain an algorithm. It is seen that the direction d(K) is the solution of a'linear equation which requires a larger number of compu- tations than the output feedback case. Also note that setting p equal to one results in the output feedback equations.
KcS J(K) = i jf, Yj[tr{Pj(K) Wjl + trjKT pj(K) K vjl] Pj(K) = oj(K)T Pj(K) 9j(K) + C3 KT Rj K Cj + Qj Sj(K) = ~j(K) Sj(K) Q,(K)T + rj K Vj KT ri + W.
J Bj(K) = r; Pj(K) rj + Rj = Cj Sj(K) C; + V.
'j (K) J AJ(K,AK) = k trj2 AKT jf, vj[i;j(K+AK) K sj(K) - ri Pj(K+AK) @j Sj(K) c:] + E Yj AKT Gj(K+AK) AK Sj(K)1 K K+AK E S j=l
g(K) = jfl Yj‘j(K) K‘j(K) - I': Pj(K) ~j Sj(K) CJ
yj pj(K) d(K) jj(K) = - g(K) E j=l NECESSARY CONDITIONS P y. P.(K*) K* jj(K*) =
1 y. r-7 P.(K*) @j Sj(K*) CJ K* e S
jil J ^J j=l J J J MULTIPLE-MODEL OUTPUT FEEDBACK ALGORITHM z>l, i=O 1. CHOOSE K,eS, ao=l, . . . , 2.
SOLVE THE LYAPUNOV EQUATIONS FOR j = 1, P Pj(Ki) = ~j('i)T Pj(Ki) ~j(Ki) + C3 K: Rj Ki Cj + Qj
Sj(Ki) = $j(Ki) Sj(Ki) ~j(‘i)T + rj Ki Vj K; ri + Wj
IF Pj(Ki) OR Sj(Ki) IS NOT NON-NEGATIVE DEFINITE GO TO 5 3. SOLVE FOR d(Ki), Ki+, Y- h.) d(Ki) Sj(Ki) = - $Ki) E j=l J J 1 K = Ki + ai d(Ki) i+l 4.
COMPUTETHE COST J(Ki) J(Ki) = i E Yi [tr{Pj(Ki) Wj + tr(KJ Pj(Ki) Ki Vj/] j=l IF i = 0, SET i = 1 AND GO TO 2
IF J(Ki)-J(Ki-1 )<-i ai-1(2-ai-1) ,f Yj trfd(Ki-l)T?j(Ki-l)d(Ki-l)?j(Ki-l)~
j=l GO TO 6 5. REDUCE a Ki = Ki-1 a. = CXi/Z d(Ki) = d(Ki-1) i = i+l K = Ki + ai d(Ki) ai+, = ai GO TO 2 i+l 6. CHECK CONVERGENCE
IF II$(Ki) // ’ ~1 OR IJ(Ki) - J(Ki_l)i > E?, ai+l = ai, i = i+l, GO TO 2
7. STOP DECENTRALIZED CONTROLFORMULATION It is well known that the optimal decentralized control problem for linear plants with Gaussian statistics and quadratic cost criteria does not necessar- ily result in a linear system, and.when constrained to linear systems may result in infinite order systems [15-171. Given these negative results, it is natural to constrain the class of decentralized controllers to linear systems of fixed finite order, i.e., the class of decentralized controllers with .fixed- order dynamic compensators as local controllers [.lS$. However, since the dy- namic compensator problem can be imbedded into the output feedback problem, it suffices to consider the class of decentralized output feedback controllers.
Furthermore, the decentralized output feedback problem can be posed as a con- strained output feedback problem, where the gain K is restricted to block diag- onal form [18]. Let S be the collection of block diagram gains (of appropriate dimensions) which stabilize the decentralized system, while V is the set of block diagonal gains for which the cost J(K) is finite. Then the problem can Find a stabilizing gain K* c S which minimizes the cost function be posed as: over V, i.e., J(K*) 5 J(K), K E V. It can be shown that if a block diagonal stabilizing gain exists with rT Qr + R and CWCT+ V positive definite, and if there exists some E > 0 such that Q > E CTC and W > E r rT, then the optimal decentralized control problem posed has a solution.
The necessary conditions are easily obtained from the incremental cost [18].
‘k+,
= ’ ‘k + b rR Ullk + wk ’
a=1 Y = '2 'k + vRk ak U 1sRsL ak = - Ki?, %k J(K) = lim m h ,i b E(X;+, Qg,Xk+' + ';k Rg ',k) -0 R=l = 4 Xk + r uk + Wk Yk = c Xk + Vk 'k+l Uk = - K Yk cl c= :
r = (r,, . . . . rL)
.
( >
cL K = BLOCK DIAG iKa. lsIIsL{ J(K) = lim K E v
Fkoo & ,I, E(X;+, Q 'k+, + u; R 'k) < O3
MULTIPLE-MODEL DECENTRALIZED CONTROLALGORITHM One approach to obtain an algorithm for the decentralized control problem posed is to close the control loop over all the local controllers, except one, solve the resulting unoonstrained output feedback problem; then iterate .on the next local controller.
While this method does not make use of some of the analytical tools developed, which would result in a more efficient algorithm, a solution to the combined decentralized multiple-model output feedback can be obtained using the previously developed algorithm for multiple-model problems.
1.
CHOOSEA BLOCK DIAGONAL K" in S i=l a=1 2. SOLVE THE MULTIPLE-MODEL OUTPUT FEEDBACK PROBLEM FOR THE SYSTEMS C ic r) j = 7, 2, . . . .
ajs 4j - v R, R r!L1j KI1' t'j' tj PI f; WITH WEIGHTING MATRICES )Qj + 1 CT,. Ki: Ra,j K;, Ce, j = 1, . . . .
PI &'#a. R J R R j = 1, . . . , pI AND STATISTICS )Vej j = 1, 2, . . . . pI Rj ' iT .
i+l w. - a,$R ri,j Kg, Va,j KiI r,jj j = 1, . . . . p1 TO OBTAIN K J 3.
IF R = L, GO TO 4 SET K; = K;+' R=R+l GO TO 2 = K; 4. SET K;+' l<R<L IF IJ(Ki+') - Jo < El AND I~(Ki+')lj I ~~ STOP 5. i=i+l g=l GO TO 2 AN APPLICATION TO RESTRUCTURABLE CONTROLS The optimal stochastic multiple-model, decentralized control, output feedback and dynamic compensation problems formulated provide powerful techniques to inves- tigate a 1arge class of control system design problems. The stochastic output feed- back, dynamic compensation, and decentralized control problems. provide a wealth of control structures; however, the "best" structure(s) for a given design problem On the other hand, are not determined and depend on the practical constraints.
the multiple-model formulation provides a powerful technique to describe design objectives in an optimal control setting, with computable algorithms and imple- mentable structures.
As an illustration, the combined multiple-model decentralized control algor- ithm is used to design a simple aircraft control law which accommodates some While the performance of a control law under types of control actuator failures.
normal conditions is a primary design goal, a practical control design must also The consider the implications of various scenarios, such as actuator failures.
restructurable control problem addresses methods of modifying the structure of the law to accommodate failures. However, the detection of the exact nature of the failure requires a period of time. Depending on the actual failure, during this period of time the aircraft may be forced into a condition from which re- covery is difficult, and sometimes not possible. Therefore, it is reasonable to restructure the control system in stages. As soon as the existence of a failure is known, or even highly likely, the system may be restructured into a control law which can accommodate a large number of failed components. This first stage restructuring can provide the valuable time necessary to identify the exact failure, decide the best second-stage structure, and implement it before the aircraft is forced into a possibly irrecoverable condition. The multiple-model formulation provides a control design method where the law isatleast stable in the failed condition as well as under normal circumstances, when possible. Since the control law is stable for the case of no failure, a false alarm does not produce harmful effects. In the following example, a wing leveler with dynamic compensation and a decentralized structure is modeled with normal and failed aileron for failure accommodation, and at two airspeeds to provide insensitivity.
DISTURBANCE SENSOR MEASUREMENTS 3 1AIRCRAFT t------ t. --__ __ -------d
r
FAILURE L FACTOR 6, r H r------- 1 I VARYING FLIGHT CONDITION V=135 kt V=165 kt HGT HEIGHT FAILURE ACCOMMODATION RUNWAY FLIGHT CONTROL DESIGN EXAMPLE The advantages offered by the'multiple-model decentralized approach are demonstrated using an aircraft digital flight control system design problem.
The control problem is the simplified design of the lateral dynamics, inne,r- loop control system for the NASA ATOPS research aircraft (a Boeing 737). The aircraft model includes the body-axis states, v, p, r, and 4. Also included in the model are aileron and rudder actuators dynamics, a one-state dynamic compensator, and aileron and rudder control states caused by weighting the control difference in the quadratic cost function. Noisy sensor measurements The dynamic comperisator state and control states are noise- are p, r, and 4.
free measurements. The dynamic compensator state is quadratically weighted to "follow" the aircraft v state. The closed-loop eigenvalues using optimal out- put feedback at one trinnned flight condition (V, = 135 kn, Wt = 85,000 lbs, ho = 1,000 ft.) are shown below.
The other table shows the closed-loop eigen- values for the multiple-model, decentralized design at the same flight condi- tion with the same quadratic weights and noise covariances. The Dutch roll mode has the lowest damping in both designs.
OUTPUT FEEDBACK DESIGN MULTIPLE-MODEL DECENTRALIZED DESIGN MAPPED EIGENVALUES (135 KN) MAPPED EIGENVALUES (135 KN) REAL IMAG. REAL IMAG.
- .521 0.00 - .436 0.00 - .903 1.47 - -925 1.93 - .903 -1.47 - .925 -1.93 - 1.53 1.72 - 1.08 1.22 - 1.53 -1.72 - 1.08 -1.22 - 1.25 0.92 - 1.23 0.00 - 1.25 -0.92 - 2.62 0.00 -14.4 0.00 -13.4 0.00 -36.5 0.00 -42.1 0.00 FEEDBACK GAINS FOR SINGLE- AND MULTIPLE-MODEL DESIGNS The lateral dynamics flight control design for the multiple-model case uses four models: two models at 135 kn and 165 kn with the aileron opera- tional, and two models at 135 kn and 165 !cn with the aileron failed. The design is decentralized as four gains in the output feedback gain matrix are forced to be zero. The controls are the dynamic compensator control u, In the multiple-model decentralized aileron rate 6'A, and rudder rate 6'R.
design, no control states are fed back to the dynamic compensator, and aileron and rudder control state crossfeed gains are forced to be zero.. The primary differences in the two gain matrices are the 4 and p gains to 6R which change sign. The fixed-gain multiple-model design stabilizes all four models. The single-model design causes the closed-loop system with the aileron failed to be unstable.
OUTPUT FEEDBACK DESIGN CONTROLGAIN MATRIX COMP. r 6A 6R P @ 0.0032 -3.47 0.829 0.661 0.721 -3.94 -7.09 -3.96 -2.94 0.698 -0.499 3.77 -0.447 0.0055 -2.77 MULTIPLE-MODEL DECENTRALIZED DESIGN I CONTROLGAIN MATRIX L COMP. r 6A BR P 4J QJ -0.80 -0.527 -4.58 0.731 0. 0.
&'A 0.104 -2.49 -4.51 -1.96 -1.94 0.
6-R -0.516 0.721 7.74 0.860 0. -4.05 SINGLE-MODEL DESIGN SIMULATION - NO. FAILURES A linear simulation of the single-model optimal output feedback design is shown below.
Roll attitude is initially 5 dtig at the beginning of the simulation and is smooth1.y returned to zero by the control system.
Lateral velocity v is kept small, and the dynamic compensator state is similar to V.
- DYN COMP ---Z_I__ .----_- I - 1.
i----..- 1 1 1 L-- ____ ________.. - _-- .-..- TIME SEC TIME SEC
r
DA CMD DEG DEG '0. 10 .TIME SEC TIME SEC SINGLE-MODEL DESIGN SIMULATION - FAILED AILERON The linear simulation shows the aileron failing 1.5 set into the simu- lation and remaining fixed at approximately 2 deg. The closed-loop aircraft system is unstable and roll attitude is seen to diverge.
DYN -- \ ROLL COMP DEG [ --.'\ -4t ..---Lee- ._.J.--.____l--.-_--L---~.~'l -61--__.-.eJ __.---.-J-...-.-.m-.mi. ...__. .-_..!........-b!
0 10 0 10 TIME SEC TIME SEC I DR ;_- -__.--.__--_ . _----- _-----.
CMD DEG --- '-‘.\ .__.. / ._-- -1L .___. LL_ ~.~L.-~~~I~~
-I--.. _-..-I ___ -I-.--.__- _I
0 0 TIME SEC TIME SEC FIRST-STAGE RESTRUCTURING: MULTIPLE-MODEL DECENTRALIZED DESIGN - FAILED AILERON The linear simulation shows the aileron failing at 1.5 set into the simu- lation and remaining fixed at that level in the following period. The single- model output feedback design controls the aircraft until 10 sec. As the closed- loop system is unstable in this condition, the aircraft continues to roll past the level wings condition. It is assumed that by 10 sec.,or 8.5 sec. after the aileron failure, the decision that a failure has occurredismade, and the first stage restructuring is engaged. The control law simulated after 10 sec. is the multiple-model decentralized design. As shown by the simulation, the restruct- ured control arrests the roll of the aircraft, and brings it to a non-zero but easily manageable and stable bank angle, providing the time necessary for the second-stage restructuring.
r
DYN ROLL COMP DEG .
-7 -6 I TIME SEC r 3i- DR DA CMD CMD + DEG
I
I
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