Integrated Control Using the SOFFT Control Structure

section apply to the feedback control law" complexity of implementation, subsystem validation difficulties, understanding behaviour in nonlinear regions, etc.

Because of all these reasons, we think that the best approach to the feedback control design is to introduce coupling only among variables which are already coupled. H" the variables and the subsystems are already coupled, the extra coupling produced by the control law does not really add new complexity since the designer will have to understand the coupling between the subsystems anyway. On the other hand, since the variables and subsystems are coupled, the control law is likely to achieve better performance when it is not restricted.

Since the open-loop system model for the integrated system ((29), (30)) has been augmented into the same form as the single or centralized system equations ((1),(2)), the integrated feedback control formulation follows precisely the same procedure until we are ready to introduce the coupling constraints.

For the PIF (Proportional-Integral-Filter) control structure that we are considering the feedback control law has the form (27) However, now each vector and matrix is an augmented version of the original case. To constrain coupling between any two subsystems through the control law, we simply must constrain the corresponding elements of the augmented gain matrix, K, to vanish. Thus, no communication link will be present between the corresponding variables of the subsystems involved.

While setting the desired elements of the gain matrix to zero formulates the problem to be solved, note that the Stochastic Output Feedback algorithm [3] does not provide for this type of constraint in the optimization algorithm. Therefore, we must develop an output feedback algorithm which can optimize the feedback gain matrix subject to the constraints set above. This problem will be addressed in greater detail in Section rtl both for time invariant or single model problems, as well as for the multi-model case with a variable-gain matrix for potential application to nonlinear problems [5], [1].

Now, consider the feedforward control integration problem. In the SOFFT methodology, the main feedforward control objective is to produce the desired system response to the input commands by tracking the command model outputs with no noise or disturbance present. It is important to note that the feedforward control law does not have any effect on the dosed-loop system stability. This can be easily verified by observing, say in Figure 2 or 3, that the feedforward control law is in series with the closed-loop plant.

Thus, as long as the feedforward control law itself is stable, it does not change the stability characteristics of the closed-loop plant even as the plant parameters vary :or other nonlinearities occur.

Therefore, for purposes of understanding the stability of the system, the complexity of the feedback control law makes the analysis of stability and its various robustness criteria more complex; however, the feedforward control law does enter this difficult part of the analysis.

Similarly, by the definition of the feedforward control problem in the SOFFT context, the feedforward control law leaves the objective of noise attenuation to the feedback control law. The feedforward control law uses only pilot input commands, but has no access to the measurements obtained by the sensors. Accordingly, it cannot attenuate the random noises which enter the system. Thus, the feedforward control does not contain any noise in its variables except for computer round-off errors which are occurring in the flight computer on board. In most of the computers used today, these noise levels are negligible. Therefore, introducing unnecessary noises from one subsystem to another is not a consideration in the feedforward control law.

It should be noted that the validation of plant subsystems can be performed irrespective of the type of feedforward used. As long as the feedback control is designed in a manner that maintains the identity of a subsystem, the feedforward subsystem inputs coming from u_ and y_ to that subsystem only can be used in the validation.

The accommodation of random or known disturbances is left: to the feedback control law which has access to the measurements and can feed them back to ensure that the system is, in fact, where it ought to be. The feedforward control law does not attempt to accommodate disturbances. As discussed in a previous section, ignoring the coupling among subsystems corresponds to treating them as unknown disturbances. If the coupling among the subsystems is neglected in the design plant model, the feedforward control law will generate the wrong feedforward control and measurement vector sequences, u_, and y_, respectively, which is its main objective. In other words, the desired response to input commands will not be generated correctly. It will then be necessary for the feedback control to try to correct the tracking error due to the coupling of the subsystems in a reactive manner.

Since generating the desired system response to input commands is the main objective of.

the feedforward control law, it seems that a significant benefit will be lost if the coupling is omitted. Therefore, we recommend using the centralized perfect tracking feedforward control law. Because the computation of the perfect tracking feedforward matrices is so straight-forward, it is hard to see many cases where this would produce any of the disadvantages present for the feedback control law. However, in cases where ignoring the subsystem coupling produces a benefit in the feedforward control, this can be easily accommodated using the SOFFT feedforward for each subsystem, thus obtaining an uncoupled feedforward control law.

The formulation of the integrated feedforward control law is the same as the one described Section II.A. 1.

C. INTEGRATED FEEDFORWARD CONSIDERATIONS While we have developed a promising approach to integrated control using the SOFFT philosophy for feedforward and feedback, some questions about the form of the open-loop system still remain. The first question is related to the measurements. While many sensor outputs can be described as linear combinations of the state, others require the use of the control as well as the state. We would like to include such control-dependent measurements in the formulation of the integrated SOFFT control problem.

The other question that deserves some attention is the stability of the feedforward control law when the open=loop system is unstable; i.e., the question of static instability for the

feedforward control. The feedback control stability has been studied extensively. The

feedforward stability for unstable open-loop plants will be investigated for the integrated control problem.

1. Control-Dependent Measurements Consider a sampled-data system; i.e., a continuous time system which witl be controlled by a digital control system. Suppose that the system has control-dependent measurements of the form (36) y=(t)=C=x(t)+D, ux(t) where t denotes time and the measurement noise term has been neglected so as to concentrate on the problem at hand.

While most measurements can be expressed without the use the control vector, some sensors measure a linear combination of the state and control vectors. These are usually sensors which measure an acceleration, a velocity or a force or moment. For example, the output of an accelerometer is a control-dependent measurement. In the example used in Section IV, we will use an accelerometer along the forward stability axis which requires the control vector. Accordingly, we will need to find a way of accommodating control- dependent measurements. One approach is given in the following.

The standard sampled-data formulation [10] assumes that the control remains constant over the sampling period, At.

(37) u_(t)=u_,, tk=kAt<_t<(k+l)At=tk+ _, k=0,1,2,--- Let t_ denote the time at which the k th measurement is sampled. Thus, the measurement, y= (t_), is obtained at time t_. At this time, we can start to compute the feedback vector which is usually obtained by multiplying the measurement vector by the feedback gain matrix. Once the flight computer computes the complete control vector, the control values can be sent to the control actuator at time t k = t: + Ac = k At (38) where Ac denotes the amount of time required by the flight computer to compute the control vector, usually in the order of a few milliseconds. Until the new control is sent to the actuators, the sampled-data commands the previous control vector, u_,__ .

From (38), it is clear that t_ < k At . Thus, when the/c th measurement is obtained, the control vector is u___ ; i.e., the control corresponds to the previous sampling instant. In mathematical form, (39) y.. = y_(t[,)=Cxx(t_,)+Dxux(t[,)=Cxxt +Dxu_t-, Therefore, for sampled-data systems, control-dependent measurements of the form shown in (36), take the following form when they are discretized.

y.+ = cx x++ z)x u ,_l (40)

If the original measurement was corrupted by noise, the discrete measurement will also be corrupted by a corresponding discrete noise process.

Now, we will embed a sampled-data system with control-dependent measurements into one without a dependence on the control vector. We achieve this by augmenting the state vector by the previous value of the control vector as follows. Define the vector

(41)

/'/¢ m Zg__ 1 Now, augment the state equations in the plant (1) by (41) using the augmented state vector shown below.

(42)

"-,-,)=("o, ,..., o,,,,.,.,,,.,),,.-,-(o-)

Now, if we express the control-dependent measurements in (40) in terms of the augmented state in the system of (42), we obtain 3o (43) kr_ where we have included the measurement noise in the formulation.

In (42) and (43), we have expressed the system with control-dependent measurements in the form of one which the measurements depend only on the state. Thus, the control- dependent measurement problem has been embedded in one of the form of (1) and (2).

Therefore, all the results we have developed in Section II apply equally to the case of control-dependent measurements.

2. Unstable Open-loop Plant As engineers have explored advanced aircraR new aerodynamic profiles, experimented with new aerodynamic control surfaces, tried forward-swept wing concepts and eliminated the tail section of the aircraft, the static stability of the aircraR has become a variable which depends on the particular aerodynamic design of the aircraft. Several advanced aircraft are unstable, at least in certain flight conditions. In particular, the modified F-15 SMTD aircraR which is used as an example in Section IV is open-loop unstable at the flight condition corresponding to 30 ° angle-of-attack.

In this section, we briefly investigate conditions under which the SOFFT feedforward control law stabilities an open-loop unstable plant model shown in Figure 3. We will consider the "optimal tracking" and "perfect tracking" eases in that order.

Referring to the feedforward cost function given by (9) in Section II.A. 1, recall that when the matrices Q_',R_" vanish, the tracking error can also vanish resulting in the perfect tracking case. If these matrices do not vanish, then the tracking error can only be optimized to obtain the smallest level of error attainable for that cost function.

Lemma 1. For the optimal feedforward control problem defined in Section I1.A.1, if the open-loop system is stabilizable and detectable, then the SOFFT feedforward control gain K" stabilizes the open-loop system if the cost matrices Q? ,R? are both positive definite.

The proof of the lemma follows directly from Theorems 6.30. and 6.31. in [12], p. 497.

It is clear that for a large class of open-loop systems, the optimal tracking feedforward control will stabilize the system even if it has static instabilities. The case of the perfect tracking feedforward control is more complicated. However, the following result applies.

Define the closed=loop system matrix as shown below.

(44)

l.emma 2. For the optimal feedforward control problem defined in Section II.A.1, where the cost matrices Q_ ,R_ are both null, the optimal control will stabilize the open- loop system if the system ( H_ ,-_ ,Ix ) is output stabilizable and [H_ F_] is invertible.

Proof" The proof follows directly from Theorem 1 in [3], p. 9, by noticing that the impulse response goes to zero, since (45)

r = u, r,]-' r, = o, k>_x

Thus, the impulse response matrix for this system converges to 0 as k gets large since it is null. Therefore, the closed loop system matrix for the perfect tracking case q), is stable.

The class of open-loop systems covered by these conditions is less clear. From our experience, it is quite a large class. However, we are not sure that it covers all systems of practical importance. It should be noted that, whenever necessary, one may add very small cost matrices Q_" ,R_" to include the class defined by Lemmal.

III. INTEGRATED/CONSTRAINED OUTPUT FEEDBACK CONTROL In this section, we will formulate and develop an algorithm for the integrated or constrained stochastic output feedback control problem. The formulation will be within the SOFFT context; i.e., we formulate a problem with a feedback control law that will cooperate with a SOFFT feedforward control law. However, the algorithm developed for the feedback control gain optimization is applicable to any discrete output feedback problem.

We will first consider the single model or time-invariant problem. Then we will extend the formulation and the algorithm to the variable-gain output feedback control case to accommodate nonlinear problems with wide variations in the operating range.

A. TIME--INVARIANT PROBLEM FORMULATION The problem formulation of the integrated output feedback control problem was started in Section II.B. 1 in the process of developing a SOFFT approach to the integration of the control law. Here, we wiU formulate the complete problem to find the optimal solution to the problem and obtain an algorithm to compute the gain matrices which define the optimal control law.

Consider a system composed of L subsystems some of which may be coupled in their dynamics and their measurements. Note that while some measurements may contain state variables fi'om more than one subsystem, the control variables and state variables can belong only to one subsystem. This does not apply to the measurements which may be considered part of more than one subsystem. However, care must be used in such cases since the number of measurements increases and the measurement matrix loses its full rank property.

Now, let the ith subsystem state vector at the kth sampling instant be denoted by x,,. Let the ith state vector, x_, have the dimension n_. Similarly, let the ith subsystem measurement vector, y=,, have the dimension ny. The ith subsystem model can be expressed in the form L xi,÷,=_=xi,+F.._u.,,+__._{_xj,+F_u_,}+w=,, i = 1,2,---,L (29) j=l L

i= 1,2,...,L (30)

j=l j,ti where the plant and measurement noise sequences are assumed to be zero mean white noise processes uncorrelated to each other and to the initial condition vectors.

The coupled set of subsystems can be integrated into a large single system of the usual form we have been considering in the previous sections; i.e., xk+ _ = _ x k +I x u_, +w_, (1)

(2)

y.. =C=x k + oxt For completeness, note that the integrated state and measurement vectors can be obtained by augmenting the subsystem state and measurement vectors as shown below.

Xlk Yx2k _'2k (31) Y.dr --" "_k -- Yxlk 1 Y_,, _'LIt The integrated control vector, as well as the plant and measurement noise vectors, can be obtained in the same manner.

I Nxl k I Wxlk 1 Ux2 k (32) _xk L_a, --" -- I ux2ki Uxl k 1 O,.tt \u,,r k The integrated open-loop system matrices in (1) and (2) can be expressed in terms of the subsystem matrices in (29) and (30) as follows.

_ = _2_ _ (33) f _)xil (I)x12 "'" (1)xlL f L---- I"x2! I"x22 (34) : : i 1"_11 L12 "'" I"xlL ,F_ F,.=

C= C=

05)

... 1

,C.,._ C= Thus, the integrated system can be expressed as a standard discrete linear system as given in (1) and (2) as shown above. Accordingly, the SOFFT developments described in Section H.A. 1 for both feedforward and feedback control apply to the integrated system.

Without repeating all of the equations in this section, we will consider a feedback control law with the PIF structure as described in equations (15) - (21).

The state, measurement and control vectors of the integrated system shown above can now be augmented to include the control and integrator vectors of the PIF structure as follows.

)

: I • | I = u,* = u_,_k i (47) lit • • .]

The integrated and PIF augmented system can now be expressed as

(48)

+w, = C _'k + vk (49) where the new control vector, vk , is defined by (19) as the rate of change of the original control position.

For the integrated control problem we are formulating, it is necessary to constrain the control vector so that some variables are not fed back into certain subsystems. This will allow the designer to choose which subsystems and variables to couple and which not to couple within the feedback control law.

First, we constrain the control vector to use only the measurement or feedback vector including internally generated variables such as the control position vector and the integrator vector. This constrains the control to the measurement vector in (49) as the standard stochastic output feedback problem.

K.K,)

(50) Showing the details of the subsystem partitions results in ¢.

Yxl t (5_) l'i x 1 k Ilk where K_, K w. and Klu have the dimensions n_ x nyj,n_ x nuj and n_ x nl,, respectively.

Equation (51) describes the integrated control feeding back the specified feedback vector for the PIF structure. However, to avoid coupling the subsystems, or more generally, to avoid coupling a feedback variable to a control component for whatever subsystem, it is necessary to place additional constraints on the form of the control law. We can achieve this result by constraining certain elements of the feedback gain matrix to vanish; i.e., the control designer sets certain gain elements to zero.

For example, note that if we constrain the partitioned gain matrices K., K. and Kj to be block diagonal, that is we constrain the off-diagonal blocks to be zero, the resulting feedback control system will be completely uncoupled. Since, the only nonzero blocks in the gain matrix are on the diagonal, each subsystem control will have the form %,, = -K=, y.,j, - K,,_ u-=k - K_ _k, i = 1,2,..-,L (52) Each control feeds back only the measuremems related to its own subsystem and, therefore, the control law is uncoupled. On the other hand, if the designer wants to allow some coupling between two subsystems, he simply does not set the corresponding blocks to zero.

In fact, note that the coupling through the feedback control law can be one-sided. In other words, we can feed back measurements from subsystemj to subsystem i by allowing some gains in the blocks K,_ ,K w. or K_# to be nonzero; i.e., by not constraining them to be zero. However, we do not necessarily have to feedback measurements from subsystem i to subsystem j. Thus, subsystem i may be influenced by subsystem j; however, subsystemj need not be influenced by any other subsystem.

It is important to note that we can distinguish between the particular variables within the subsystems. Thus, we may feed one or two variables into a given control component in some subsystem. This is achieved by setting all the gains except the desired ones to zero.

l_mally, note that it is not sufficient to set the off-diagonal blocks in K, (the sensor feedback gain partition) to zero if you want to avoid coupling subsystems. You must also set the off=diagonal blocks in K, and in K l to zero to avoid coupling the desired subsystems through cross-terms in integral feedback or control feedback.

A simple way in which we can set specified elements of a matrix to zero is by defining the following element by element multiplication of two matrices which we shah denote by "×'.

We define the ij element of the matrix product as the product of the ij elements of each matrix. Thus, let the matrices K and Z have the same dimension. Then, the elements of their x-product is given by (53) [ g x Z], j -- g, j Z, j Now, define the matrix Z to have elements which are either zero or 1. If you want to constrain an element of the gain matrix K to zero, then set the corresponding element of the matrix Z to zero while leaving all its other elements set to 1. Then, the product KxZ will have zero's for the specified elements while the others will be unchanged.

Select the feedback cost function = X_+, Q,_k÷1 + v[ R_k (54) J(X) tim 1E{_=o-r -) u-_ 2(N + 1) Then the optimal control problem for integrated/constrained output feedback can be posed as follows. Find a stabilizing control gain matrix K which minimizes the cost function in (54) subject to the constraints in (48), (49), (50) and (55).

K = KxZ (55) The constraint in (55) requires that the allowable gain matrices must have zero's: wherever Z has zero's as specified by the control system designer. Thus, we have an additional constraint beyond those in the standard output feedback problem. We will consider the optimization of this problem in the following section.

B. ALGORITHM DEVELOPMENT With the exception of the constraint in equation (55), the problem posed in the previous section was solved in [3]. A similar problem posed for continuous time, constant gain control laws is treated in [13] using a different approach. Here, we will follow a similar approach to the one used in [3 } and then extend this development to the case of the variable-gain output feedback treated in [5].

1. Necessary Conditions Following the approach in [3], let us define the set, S, of stabilizing gain matrices as follows.

(56) where p denotes the spectral radius of the matrix and q)(K) is the system matrix with the loop closed by the unconstrained gain matrix, K, as shown below.

O(K) = • - F K C (57)

For a stabilizing gain, the cost function shown in (54) is known to be finite. So if the system is output stabilizable, S is not empty. However, we now have the further constraint of (55) which zeros out certain elements of the gain matrix. Accordingly, we need to define the constrained set of stabilizing gains which we denote by Sz • (58) Sz = IK _2_K x Z = K} If the set of constrained stabilizing gains, Sz, is not empty, then a finite cost exists and the optimization problem posed is well-defined.

From Lemma 10 in[3] (pp. 19 - 20), the cost function is continuously differentiable on S and the gradient of the cost function exists and is given by 8d'( ^ ^ K _ S (59 a) K)= P(K)KS(K)- F r P(K)C_S(K)C r , 8K Since S z is a subset of S, the expression for the gradient also holds on S z . However, this expression of the gradient shows nonzero gradient values in all locations; i.e., even if an element of K has been set to zero, the gradient would show as a nonzero value at the corresponding location. The reason is that this expression does not take into account that the gain element is fixed at zero so that the derivative of the cost with respect to it is also zero. We want to maintain the matrix form of the gradient. Thus, we are setting the corresponding elements to zero. Another formulation could leave those elements completely out of the gradient at the expense that the matrix form would be lost.

For the integrated/constrained stochastic output feedback problem, the gradient can be expressed as

P(K)KS(K)-F r P(K)@S(K)C r xZ, K _S z (59 b)

j(K) =

^^ J

4O where j(K) denotes the constrained gradient and the matrices used in the gradient are defined by the Lyapunov equations below.

P(K) = q_(K) r P(K)@(K)+ C r K r RK C + Q (60) S(K) = _(K)S(K)_(K) r + F K V K r F r + W (61) A P(K) = F r P(K) F + R (62) A S(K) = C S(K) C r + V (63) Thus, using Lemma 3 below, we can obtain the necessary conditions for the Integrated/Constrained Output Feedback Problem by setting the gradient to zero.

A A

(64)

P(K)KS(K) xZ= [Frp(K)_S(K)Cr]xz , X _Sz Therefore, the optimal gain, K, must satisfy the necessary conditions shown in (64) and any gain that satisfies (64) is a critical point of the optimization problem.

2. Constrained Algorithm Whereas we know that the optimal gain must satisfy the necessary conditions, solving (64) directly is not an easy matter. Furthermore that would only give us a critical point of the cost function. To solve the op "tmaization problem, we must find a gain that minimizes the cost function. Aecordingiy, we use that idea to develop the algorithm. In other words, given a starting gain, we will try to fred a new gain which reduces the cost function and keep doing that until some convergence criterion is met. At each iteration, we shall add an increment to the gain going in a direction which reduces the cost.

Recall the incremental cost which was instrumental in developing the algorithm in [3].

We will use that concept in the current problem as weft.

(65) A J(K, AK)=J(K + AK)- J(K) = _tr 2AK r P(K +AK)KS(K)-Fr P(K + AK)_S(K) Cr (66) +M_2r p(K+MOM(S(K , K _S,(K+_K)_S

^ ^ )}

Rearranging the terms in (66), it can be rewritten as ([3], p.22, Eq. (80)) (67) AJ(K, AK) = _tr 2AK r 8J AK r ---_(K)+ P(K)AKS(K)+02(AK) Before continuing with the development of the eo_ed algorithm, we need to establish some elementary properties of the matrix product x defined in(53).

Let all the elements of Z Lemma 3. Let A, B and Z be matrices of appropriate order.

be l or O. Then, (68) AxZ=ZxA (69) (A+B)xZ= AxZ+BxZ (70) ZxZ=Z (71) (A x Z) r = A r x Z r (72) (73) Proof- The assertions in (68) - (72) are immediate implications of the defimtion in (53). We will only show the validity of (73) here.

(74) _,k i,k which shows the desired result.

Suppose that we have a stabili_ng gain, K, that satisfies (55). We want to find a direction, d(K), in which the cost can be reduced. Now, consider the direction matrix d(K)=- K)-' j(K)S(K)-' xZ , K _S z (75)

A 1

Thus, we will look for a new gain along the direction d(K). If K_ is the gain for the i th iteration of the algorithm, we define the next gain as K_+, = K_ +ctd(K_), i = 0,1,2,--- (76) where a is a positive number greater than zero. From (67), it is clear that if the first term in the trace is negative then the cost along the given direction will start by going down.

This is due to the fact that all the other terms in the cost function are of second order in a.

If a is selected to be small enough, then the first order term will dominate the incrememal cost and the next gain will reduce the cost. Thus, consider only the first term in (67) to determine its sign.

(77) (78) = -air P(K) -_ j(K)S(K) -_ xZ --_(K) Using the properties of the x-product given in Lemma 3, and manipulating (79)

tt A 1 T

Substituting and From (59), note that the last term in (79) is the constrained gradient.

manipulating the transpose, we get Va > 0 (80) tr AK r 8J = -cttr K)-' j(K) r S(K)-' j(K < O, Therefore, the direction selected in (75) is, in fact, a descent direction which will reduce the cost function. Thus, selecting a small enough by trial and error will produce a new gain to be used as the next gain in the iterations of the algorithra.

We think that it is possible to prove the convergence of this algorithm following the same arguments as in [3]. However, this is beyond the scope of this investigation and will not be pursued here. Our experience with this algorithm indicates that it is numerically stable and convergent.

We will give the detailed steps of the algorithm for the more general variable-gain problem which is investigated in the next section.

C. EXTENSION TO VARIABLF.,-GAIN SYSTEMS The Variable-Gain Output Feedback Control methodology [5] was developed to accommodate nonlinear systems with large variations in the operating range. This methodology allows the feedback gain matrix to vary with selected system parameters so that the control law can adapt to the changing dynamics of each operating point.

The Variable-Gain methodology was extended in [1] to allow the control system to use the SOFFT approach with both feedforward and feedback control laws. Let p represent the parameter vector which specifies the operating point of the system. Thus, the system matrices now vary with the parameter vector.

x,+_ = O.(p) x, +r'x(p) u. +w_ (81)

Yn = Cx (p) x k + u._ (82) Using the PI_" feedback structure as shown in(17) - (27), and using the same partitioning for the coupled subsystems as described in Section M.A, we results in the design model

X,+, = a,(p) X, +r(p) _ + w, (83)

(84) The difference fi'om the time-invariant ease treated earlier is that now the control law can also vary as the system moves through different operating points. Thus, we allow the control gain matrices to vary according to the parameter vector as shown below.

(85)

I

For a variety of reasons too lengthy to discuss here, we place the constraint of a linear relationship between the control gain matrix and the parameter p.

q K(p)= K 0 + _Pi K_ =K G(p) (86) i=1 where _ is an augmented gain matrix defined as follows.

(87)

G(p) = (88)

'I

p_I) We embed the variable-gain output feedback problem into the time-invariant or constant gain problem by redefining the measurement vector.

(89) Y_: = G(P)Yk = G(p)C(p) *k +G(P)Uk = C'(P) ffk + v't where the new measurement equation has the same form as before although it has a higher dimension. However, with this measurement vector, the control law can now be expressed as a constant gain feedback control law. Substituting (86) and (89) into (85), note that (9O) Now we have a constant gain control law trying to control a system which can vary with the parameter vector, p. Selecting a representative number, say M, operating points IpJ,j = 1,2,---,M} to cover the system's operating range, we have a multi-configuration output feedback problem with the cost function M (91) J'(_) = '_" f j J(K(pJ ),p j) j=l .](K(pJ),pJ)= run 1 E X_+, Q(pi).Xk+_ + _r R(pJ)_k (92 a) N--_ 2(N + I) _ (92 b) 3(K(p),p)= _tr(P(K(p)) IV(p)) +:tr K(p) r P(K(p)) K(p)V(p)

I A 1

where fj is the wdght attached to the jth operating point. Now, recall that for the integrated/constrained variable-gain problem under consideration, we have the additional co_t of setting specified elements of the gain matrix to zero. It is important to note that for the variable-gain control in (86), to avoid coupling the desired subsystems, we must zero out each gain matrix, K i , i = O,1,2,..-,q. Alternately, we must augrnem the Z matrix used for the time-invariant case. Thus, the control gains must satisfy (93) K ixZ=K_, i=0,1,---,q or alternately,

(94)

KxZ=K (95) Z=(Z Z To obtain the gradient for this case, we simply use the general expression (53) in [5] and specialize it to our case.

8J' (96) j, fK)=7_-(K)xZ, K eS,.

,, F ^ ^ , ,l

x z,_c es= (97)

J

= _fjp/[P_.(g)K(p')S(g)-r, Pj(K)q_,Sj(K)C, j(K)=(jo(_) j,(K)"'" jq(K)) (98) Integrated/Constrained Variable-Gain Feedback Algorithm 1. Embed the variable-gain problem into the MCC form by augmenting the gain matrix as described above 2. Initialize parameters: Select initial stable gain to, ao, i = 0, Z, etc.

3. Initialize and save matrices. Forj = 1, 2, ---, ,A/I

(I):=(1)(p j) r;=r(pJ) c,=c(p j) C;=C'(pJ)=G(pJ)Cj (99)

(ioo)

Qj = Q(pJ) R; = R(p j) (101) Wj =W(p j) Vj' =V'(pJ)=G(pJ)V(pJ)G(pJ) r 4. Test closed-loop stability at iteration i using (102) for j = 1, 2, .-., M. If any instability is found, go to 10.

_jCi_) = _j -F,. _ C_ = #oj -rj _ G(pJ)Cj = _j -Fj K(pJ)Cj (102) 5. Solve the P and S Lyapunov equations below forj = 1, 2,---, M r

(lOS)

, TFT (104) A Pj(_.) = r r Pj(_.) rj + Rj (105) J A

sj(_) = c_ sj(r_)c'. _ +v; (lO6)

J 6. Compute the cost J'(_Z_) using (91) and (92 b). If the cost is not lower than last iteration, go to 10.

7. Compute the integrated/constrained gradient j(K, ) using (96) - (98). If the gradient norm is smaller than the convergence criterion, stop.

8. Solve for the variable-gain direction d(g, ) in M A A _fj Pj(_) d(T_.) Sj(_)=-j(_) (107) j=l If M= 1 (i.e., single model case), use Equation (75) to solve for the direction d.

IfM > 1 (i.e., Multi-Model or Variable-Gain (V-G) cases), use the Kronecker product formulation given by (107 a).

(107 a)

.{A A

where ® denotes the Kronecker product and col denotes the column vector form of the appropriate matrix. Solve for the direction by inverting the matrix in brackets or use the approximation algorithm described in Equations (74) - (81) in [5], pp.30-31.

9. Zero out the direction and go to 11.

d(T,_)=d(T,_.)×Z (lOS) 10. Reduce step size a_ ; e.g.,

a,+,= 2 (109)

If step size is smaller than criterion, stop.

11. Compute new gain matrix and go to 4.

= + a, d( L)

(110) i.6..-i+l (111) The algorithm given above is intended for the Integrated/Constrained Variable-Gain Output Feedback Control problem. However, it also applies to the time-invariant or constant gain problem when the number of models, M, equals 1 and the dimension of the parameter vector, p ,is also 1; i.e., q = 0. Furthermore, this algorithm is also applicable to the Multi-Configuration Control (MCC) problem described in [5].

Computing the direction in which to search for a lower cost is relatively straight-forward for single model problems as seen from (75). However, for multi-model problems (107) can be computationally demanding. An approximate solution can be obtained with significantly less computational burden by approximating the Hessian as shown in [5]. On the other hand, this may result in a more shallow direction for the search.

D. FLOW CILKRT OF VARIABLE-GAIN (V-G) ALGORIT]B_ Start I

!

[ lnltlallzatlons i

, I

Test Closed-loop Stability 1102) Unstable Stable I

Solve P Lyapunov Equations (103) I

I

Cost up I Compute COst (91-92) I Cost down i Solve S Lyapunov Equations (104)

I

Compute Gradient for each model (97) I

I

Compute V-G Gradient & Norm (98)

I

, IJ_nv Single or Multi Model? _:_' / l Obtain Kronecker Product Formulation (107 a) Compute Direction (j7_ I

[ Zero Out Direction (75) I I Zero Out V-G ]Directi°n (1081 I

Reduce cL l

I

a Too Small ] I Compute New Gain 11101

I

5O IV. AN INTEGRATED FLIGHT AND PROPULSION CONTROL EXAMPLE The Integrated SOFFT Control approach developed in the previous sections was applied to the design of an integrated flight and propulsion control system for the modified F-15 SMTD alrcraR to provide an example of the design methodology developed. Although the methodology is applicable to both single model and multi-model variable-gain problems, here we will only design a single model control at one flight condition. Clearly, the Integrated SOFFT Control methodology is very general and has numerous applications in a variety of fields.

A. AIRFRAME AND PROPULSION MODELS The modified F-15 SMTD alrcrait model has an advanced propulsion system with thrust vectoring capability and has aerodynamic canards to complement the usual assortment of aerodynamic control surfaces. The aircraft has static instability in the longitudinal dynamics at a flight condition of 30 degrees of angle-of-attack. It is open-loop stable at the other high angle-of-attack flight conditions we have considered. Accordingly, we select the flight condition for 30 degrees angle-of-attack at level flight in order to observe the impact of the static instability on the methodology. From the results obtained in the investigation, the methodology stabilities the open-loop unstable mode without any noticeable consequences.

The thrust vectoring can produce moments in the pitch, yaw and roll axes. Thus, it has three control components. These thrust vectoring components have very similar effects as the aerodynamic control surfaces, namely, the stabilator, rudder and ailerons. In normal operation, it would be counter-intuitive to command positive rolling moment with the ailerons while simultaneously commanding negative rolling moment with thrust vectoring, thus canceling each other's effects. Accordingly, we select to combine the thrust vectoring controls with the corresponding aerodynamic control surfaces. This reduces the number of control components and reduces the computational load and avoids situations as the one mentioned above.

The aerodynamic surface and thrust vectoring controls are shown in Table 1.

Table 1. Airframe Controls : 6s > degrees Stabilator + pitch down 6 o degrees Pitch Thrust Vectoring + pitch down 6 A degrees Aileron + rollright ._.

6 a degrees Rudder + yaw left 8¢ degrees Yaw Thrust Vectoring + yaw left 6 5 _ degrees Roll Thrust Vectoring + roll right Let u._ be the new or combined airfi'ame control vector (having three components) which we will use in the plant model. Then the airfi'ame controls in Table 1 are obtained as follows.

:1 0 0

o o

0 1 0 8_ u_, x u.i = u_ (112) 0 0 .6

/"l

U¢ 0 0 .9 The airfi'ame state vector is given in Table 2.

Table 2. Airframe State Variables fl / sec speed along x s t21 degrees angle of attack deg / sec pitch rate ql OI degrees pitch angle Xl "-

El

degrees sideslip deg / sec roll rate Pl r I deg / sec yaw rate degrees roll angle ,¢') In Table 2 and elsewhere, x, denotes the x stability axis. In a similar vein, the engine controls were combined from a total of five components to the following three.

Table 3. Propulsion Control Variables Fuel Flow 1,000 lbs / hr 10ore 2 Exhaust Nozzle Area Ux 2 -" degrees Rear Compressor Variable Vane The propulsion state vector and units are given in Table 4.

Table 4. Propulsion State Variables

Low Rotor Speed High Rotor Speed N 2 1,000 rpm X2 -- IN, TI 1,000 rpm HPC 1,000 R o Rear Compressor Metal Temperature The state equations for the airfi'ame and propulsion subsystems are of the form x I = A I x I +B I uxl +Bl' l u,1 (113) x 2 = A 2 x 2 + B 2 ux2 + B_, u, (1 14) The cross-coupling terms, the last terms in the equations above, are given by (115) Ull = Dp = where F s and Dp are the gross thrust and the drag due to the engine. Substituting for the cross-coupling terms and manipulating, the integrated continuous open-loop system can be found as X 1 = AxH X 1 +Axl 2 x 2 +Bxl I tlxl +Bxl 2 /4x2 (116) x 2 = Ax21 x I +A.z 2 x 2 +B.. ux: (117) The details of the derivation are given in Appendix B. It should be noted that several of the system matrices in (1 16) and (1 17) are different than the ones in (1 13) and (1 14) (See 03-8) and 03-9) in Appendix B). Also note that the propulsion subsystem dynamics in (117) do not have a term for the airframe controls although there is still some coupling through the state vector even if small.

The integrated system A_ and W_ matrices are shown in Tables 5 and 6, respectively Table 5. Integrated System A s Matrix 1 2 3 4 5 1 -0.2201 -0.6139 0.0000 -0.4532 -0.0274 2 -0.0801 -0.2040 1.0000 0.0630 0.0030 3 -0.0849 0.4817 -0.2448 0.0000 -0.0030 4 0.0000 0.0000 1.0000 0.0000 0.0000 5 0.0000 0.0000 0.0000 0.0000 -0.1251 6 0.0000 0.0000 0.0000 0.0000 -24.8180 7 0.0000 0.0000 0.0000 0.0000 -0._008 8 0.0000 0.0000 0.0000 0.0000 0.0000 9 0.0000 0.0133 0.0000 0.0000 0.0136 10 0.0000 0.0076 0.0000 0.0000 0.0075 II 0.0000 0.0000 0.0000 0.0000 0.0000 6 7 8 9 10 1 0.0000 0.0000 0.0000 0.6337 0.0804 2 0.0000 0.0000 0.0000 -0.0695 -0.0088 3 0.0000 0.0000 0.0000 0.0687 0.0087 4 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.5000 -0.8660 0.1060 0.0000 0.0000 6 -0.7052 1.5406 0.0000 0.0000 0.0000 7 0.0150 -0.2254 0.0000 0.0000 0.0000 8 1.0000 -0.1086 0.0000 0.0000 0.0000 9 0.0000 0.0000 0.0000 -2.5764 1.7038 I0 0.0000 0.0000 0.0000 0.0213 -1.5592 Ii 0.0000 0.0000 0.0000 0.0175 0.0149 1 0.2598 2 -0.0285 3 0.0281 4 0.0000 5 0.0000 6 0.0000 7 0.0000 8 0.0000 9 0.4365 I0 0.3440 II -0.3846 Table 6. Integrated System B x Matrix 2 3 4 5 0.0000 0.0000 0.5901 -2.1013 1 -0.4182 0.0000 0.0000 -0.0647 0.2304 2 -0.1326 0.0000 0.0000 0.0639 -0.2277 3 -3.4895 0.0000 0.0000 0.0000 0.0000 4 0.0000 0.0060 0.0813 0.0000 0.0000 5 0.0000 3.8394 0.0031 0.0000 0.0000 6 0.0000 -0.1895 -3.7094 0.0000 0.0000 7 0.0000 0.0000 0.0000 0.0000 0.0000 8 0.0000 0.0000 0.0000 0.9496 2.3631 9 0.0000 0.0000 0.0000 0.5476 0.7002 10 0.0000 0.0000 0.0000 0.0019 -0.0068 11 0.0000 1 O. 0015 2 -0.0002 3 0.0002 4 0.0000 5 0.0000 6 0.0000 7 0.0000 8 0.0000 9 0.2151 10 -0.1431 11 0.0009 It should be noted here that the propulsion control vector and, therefore, the B x matrix was modified for the final design example. This new control vector and the motivation for the change are described in the next section Finally, the integrated system measurement or feedback vector was selected as shown in Table 7. Note that the first six measurements may be considered to be airframe measurements with the remaining four measurements corresponding to the propulsion system.However, both sets contain variables fi'om the other subsystem.

Table 7. Integrated System Measurement/Feedback Variables fl / sec 2 acceleration along x s a g$ ¢x angle of attack degrees pitch rate deg / sec q sideslip angle degrees roll rate deg / sec P Yx = r deg / sec yaw rate 1,000 rpm low rotor speed Nl 1,000 rpm high rotor speed HPCT 1,000 R ° rear compressor metal temperature SMttC % rear compressor stall margin Finally, although we have treated the problem as one with two subsystems, namely the airframe and the propulsion subsystems, it is possible to consider it as a problem with three subsystems: the longitudinal airframe dynamics, the lateral airframe dynamics and the propulsion subsystems. In fact, the most common integrated control problem in flight controls certainly has to be the longitudinal and lateral flight control systems. Although usually treated separately, these two subsystems are coupled when the aircraft has a nonzero roll angle.

B. FEEDFORWARD CONTROL DEVELOPMENT In this section, we will discuss the formulation and selection of the command model for the integrated system. However, most of the comments made here apply to centralized optimization problems as well. In particular, we will discuss the selection of the number of variables which should be commanded for best results in the next subsection.

1. Input Command Considerations The first feedforward design question is the selection of the pilot input commands. This selection is often the one mentioned to describe the overall control system. For example, we will say that we have a "pitch rate command system" or an "alpha-command system" or an "attitude rate command system" meaning that the pilot commands those variables.

It is generally accepted that we can have as many commands as the number of independent control variables. While this may be true in a strict sense, it is not always the most reasonable or desirable number to select as we shall discuss in the following. Other considerations may lead the designer to select a different number of input commands.

From the integrated model described in the previous section, the overall system order is 11 and the number of controls is 6. Therefore, we started out by selecting 6 input commands.

Not all of the commands need to be input by the pilot as real-time inputs. Some commands may be computed according to given formulas or algorithms. In other cases, some input commands may simply be constant commands.

Some examples of input command vectors are shown below.

f • • m q+.2a q+.2a q+.2a N: a= P P P

(1 is)

H, =

r F r SMF N1 _4ttC _C N2 All of the choices above resulted in very high cross-gain values fi, om the longitudinal airframe state variables to the propulsion control variables. Several other selections were tried to eliminate various possible causes for such high gains. For example, decoupling the engine dynamics fi'om the airfi'ame by setting A,2 _ to zero produced no significant change.

Decoupling the input command vector itself by commanding 3 airfi'ame and 3 engine states helped to reduce the cross-gains somewhat. However, the gains fi_om the engine states to the engine controls were still high. The last result, coupled with the previous ones, pointed to a new approach to the problem.

Our hypothesis was that the reason the gains are very high is due to the possibility of commanding the three engine state variables (or an equivalent set of variables) to unreasonable, mutually opposing values. For example, it is possible to command N_ to high rpm's and simultaneously command N 2 to lower rpm's while commanding HPCT to remain constant. The input command selections above make this a possibility even if such a command profile never actually occurs. Therefore, the control law must be able to achieve unrealistic commands. And it doesf But at the expense of very high gains. Of course, the propulsion system is not designed to track such commands; it is designed to produce thrust, a single variable.

Thus, even though, strictly speaking, it is possible to command as many variables as the number of controls, it may be totally unreasonable to do so. One must take into account the characteristics of each subsystem before selecting the input command vector. Just because it is possible to command many variables does not mean we are required to do so. It is of interest to investigate further the conditions under which such situations occur. However, such an investigation is beyond the scope of the present study.

To prove the hypothesis above, we reformulate the propulsion problem as a one-control one-command problem. Consider the single component pseudo control u, defined by ux2 = G: u, (119) where Uxe is the propulsion vector in Table 3 and G 2 is a 3x 1 matrix to be determined.

Let us choose G 2 so as to minimize the E-norm of the feedforward gain matrix K_. After considerable work, it is possible to show that the minimum occurs for l'0 G 2 = ¢ 3.34 (120) _,0.21J where c is a arbitrary scalar. We do not show the derivation here as it is not relevant to the problem under consideration. Sul_ce it to know that using this vector with c equal to 1, together with the input command vector shown below, immediately solved the problem of very high gains reducing K_ by several orders of magnitude. We consider the hypothesis proved heuristically by finding a solution which dramatically reduces the gain.

I q+.2ct)"

(121)

Accordingly, we select eq. (121) as the pilot input command vector for this example.

However, minimizing the norm of the gain matrix is not necessarily the best way to select our pseudo control. In particular, the choice of (120) results in a negative value for the (1,4) dement of B x . Thus, a positive step in the pseudo control would produce an initial tendency to reduce the thrust and the forward acceleration. This tendency would reverse itself after the transient response dies down. To avoid this counter-intuitive initial tendency, we modified the value of G 2 as shown in (122) so that the pseudo control would produce the desired initial response without introducing large moments in the pitch rate (q) and angle-of-attack (cz) equations. The resulting B x matrix is shown in Table 8.

(122)

2. Command Model or Flying Quafities Given the pilot input command vector, we need to select a command model which produces the flying qualities desired by the pilot. In the SOFFT methodology, the command model determines the response of the aircraft dynamics to the pilot's input commands. In this example, we select the command model to produce specific types of responses for each component of the vector in (121).

For the longitudinal and lateral dynamics, we will use [14], pp.511 - 525 as a guide in selecting the corresponding command models. For the propulsion related forward Table 8. Integrated System Modified B x Matrix 1 2 3 4 1 -0.4182 0.0000 0.0000 0.2605 2 -0.1326 0.0000 0.0000 -0.0286 3 -3.4895 0.0000 0.0000 0.0282 4 0.0000 0.0000 0.0000 0.0000 5 0.0000 0.0060 0.0813 0.0000 6 0.0000 3.8394 0.0031 0.0000 7 0.0000 -0.1895 -3.7094 0.0000 8 0.0000 0.0000 0.0000 0.0000 9 0.0000 0.0000 0.0000 6.3768 I0 0.0000 0.0000 0.0000 2.7474 11 0.0000 0.0000 0.0000 0.0016 acceleration dynamics, it is not clear that a flying quality criterion is available.

Accordingly, we witl select a first order model for this variable.

Note that the command model dynamics can be varied in real time as the parameter vector follows the aircraft's flight condition for the variable=gain case. Thus, different flying qualifies can be obtained at different flight conditions. We should point out that the command model selected here is not intended as a recommendation for flying qualities, but only as an example for use in demonstrating the Integrated SOFFT control methodology developed. With the advent of advanced aircraft with significantly different airfi'ame components and new propulsion systems, the question of what flying qualities are most appropriate is itself a subject of research.

We will specify the command model for _ch component of the pilot input command vector as a continuous transfer function. These will the be put into state variable form and discretized to obtain the command model in the form shown in (6) and (7) for the sampled-data control problem at hand.

(s) = co q

(123) u_, s2+2fqc°qs+c°:q , coq=3,(q=.7 03p (124) Y'z (S)= s2 s+o_ , co,=3,(p=.7 u.2 +2 _'p rap Y,3 (s)= 1 , r, = L5 (125) u:3 r,. s + 1 y,4 (s) = 1 , ro = 1 (126) u:4 ro s + 1

3. Feedforward Control Gain Matrices

With the integrated model given in (116) and (117), the pilot input command vector given in (121) and the command model given in(123) - (126), we can design the feedforward control law as described in Section II.A.1. Of course, the integrated system and command model were first discretized using the standard sampled-data discretization methods and the control-dependent measurement augmentation shown in (42) and (43) was applied to the integrated system before actually computing the feedforward gain matrices.

Note that, the system augmentation due to the control-dependent measurements increases the order of the integrated system from 11 to 13. As only two of the control components are used in the measurements, we only need to augment the integrated state by those two components. The feedforward control has the form (also see Fig 3, p. 16) (10) u_, = -K_ x_ - K, z, - K_, ua Using the perfect tracking feedforward control case, the three gain matrices in (10) were computed according to equations (11) - (13). These gain matrices are shown in Tables 9, 10and 11.

Table 9. Feedforward Control Gain Matrix : K" x .; V_" a; q;

g

• " 0. 0248 -1.5723 -7. 1625 -0.0128 0.0001 • 0.0000 0.0000 O. 0000 0.0000 ,,.

-6.4473 • 0.0000 0.0000 0.0000 0.0000 0.5715 ," -0. 4874 -2. 9462 -7. 0169 -1.0784 -0.0632 .. p; < ¢; N;.

N2.

W t • 0.0000 0.0000 0.0000 -0.0033 -0.0001 • 6.3787 0.2085 -0.0138 0.0000 0.0000 m -0.3263 -6.7200 0.0012 0.0000 0.0000 x" -0.0006 -0.0003 -0.0003 1.3406 0.2716 R t -0.0011 0.0000 0.0000 u , 0.0000 0.0000 0.0000 • 0.0000 0.0000 0.0000 w• 0.6279 -0.0003 0.0002 Table 10. Feedforward Control Gain Matrix : K z Z 1 z2 z 3 z 4 z 5 7.1230 0.2634 0.0000 0.0000 0.0000 0.0000 0.0000 -6.5824 -0.2434 0.2978 tit a v 0.0000 0.0000 0.3351 0.0124 6.5727 u* 6.879J 0.2544 -0.0007 0.0000 0.0025 Z6 -0.0187 0.0000 0.0000 II* -2.2904 Table 11. Feedforward Control Gain Matrix : K uz //zl U:2 I/z3 _'4 # 0.0488 0.0000 0.0000 -0.0008 < 0.0000 -0.0451 0.0080 0.0000 0.0000 0.0023 0.1776 0.0000 0.0471 0.0000 0.0001 -0.0935 < Some Important Observations In the Integrated SOFFT Control methodology we have developed in this study, much of the "integration" is done by the feedforward control law since we have chosen to have the feedforward control law be centralized while the feedback is more limited in its cross- coupling. The feedback control law also performs important integration functions.

However, the feedforward, not being constrained, is free to produce coupling as it finds necessary. Therefore, we may learn about which subsystems and which variables would be coupled if we did not have constraints.

With these thoughts in mind, observe the feedforward gain matrices for particular coupling trends. In K_ , note that the elements in the first row, on columns 9, 10, 11, have very small values. These elements correspond to the coupling from the propulsion system to the longitudinal dynamics subsystem. Clearly, the SOFFT feedforward does not produce much coupling from the propulsion to longitudinal subsystems.

On the other hand, the 4 th row dements on columns 1 -4 do not have negligibly small values. These terms correspond to the coupling from the longitudinal dynamics to the propulsion subsystem. In other words, the SOFFT feedforward uses longitudinal information in propulsion, but does not use the propulsion state in the longitudinal control. Also note that, with the possible exception of the sidelip term/C_(4,5), there is little coupling from the lateral dynamics to the propulsion subsystem. As would be expected in level flight, the lateral control does not use information from any other subsystem.

K, produces a leading control movement when the pilot moves his commands. Note that when an acceleration command is given, the initial control will be to increase the thrust through the propulsive control. However, only a small amount of stabilator control is used to maintain the pitch rate from moving. However, when a pitch rate command (with no acceleration command) is given, both the stabilator (with pitch thrust vectoring) and the propulsion control move in a coupled manner.

While these observations are often what an experienced designer might expect, it appears that the SOFFT feedforward control gain matrices contain important information as to which subsystems need coupling and maybe how much coupling. It is always a good idea to verify one's intuition with more objective methods.

C. FEEDBACK CONTROL DEVELOPMENT With the feedforward control obtained, we can follow the development in Section II.A. 1 and Section Ill to design a feedback control law for the integrated/constrained flight and propulsion control system. A Proportional-Integral-Filter (PIF) feedback control law was designed. The structure of the feedback law is as described in Section II.A. 1. We show the relevant equations here for convenience.

(20) With the control vector having 4 components and the integral feedback vector having 4 components as well, the system state is augmented fi'om 13 to 21. Accordingly, the feedback vector consisting of the 10 measurements together with the 4 controls and 4 integrators is augmented to a dimension of 18.

The sampling rate used for sampling the sensor outputs and for updating the control commands is 25 Hz corresponding to a sampling period of 0.04 sec. The integral error feedback is obtained by using (121) to select the state components in the integral (more correctly, the accumulator). The control component of the integral is determined by H u .

0 .05 0 0 /./ (127)

"'=o o .05 o

rO 0 0 0 0 0 0 .05 k The reason for using any control at all in the integrators is due to problems encountered in obtaining an initial stabilizing gain when the integrator produces a double eigenvalue in the system. For example, if the roll rate error is integrated for feedback, the integrator state is closely related to the roll angle component of the state. These conditions place at least numerical difficulties on the algorithms used. We have found, by experience, that adding a small amount of the corresponding control into the integrator differentiates it sutticiently to make the numerical problems disappear. We are not aware of a theoretical reason for this condition. Presumably, an algorithm specifically designed to handle multiple eigenvalues would not run into the same difficulties. However, we have never tried to test this hypothesis.

The form of the control is given by (50) and (51) which we repeat here for convenience.

(50)

("/

( w Yxlk (51) Recall that Kfs ,K,_. and K_o are dimensioned -_ x nyj ,.,_ x ..j and ._ x n1_ , resp.. As mentioned before, with 4 control components and 18 feedback variables, the integrated feedback gain matrix has the dimensions of 4x 18. The first 10 feedback variables are the measurements obtained f_om the sensors as shown in Table 7. The next 4 components are the control commands and the remaining 4 are the integrators.

1. Feedback Constraints The considerations discussed at the end of Section 1V.B.3 "Some Important Observations", can now provide some helpful hints as to how to select the constraints on feedback coupling of the subsystems. Accordingly, we first constrain the lateral controls to use only lateral variables for feedback. Then, with the exception of the sidelip angle, we constrain the propulsion and the longitudinal subsystems from using lateral feedback variables, i.e., measurements,controls or integrators. We allow the sideslip coupling because it isan inputintothepropulsion open-loopsystem.Accordingly, therealready is some coupling present. We are not really increasing the complexity of the closed-loop system by allowing further coupling. Only the extent of coupling may change.

Finally, we constrain the longitudinal control from using the propulsion measurements of HPCT and SMHC. However, we allow the use of the low and high rotor rpm's, N_ and N 2 , in the measurements, largely as an experiment, to see if it will have an impact. We did not see any important impact and would not recommend this coupling in general.

With the exception of the lateral feedback variables, the propulsion subsystem control was not constrained. Table 12 shows the constraint matrix, Z, used in the optimization of the feedback control gain matrix.

Table 12. Constraint (Zero) Matrix : Z a= a q fl p ue 1.0000 1.0000 1.0000 1.0000 0.0000 u¢ 0.0000 0.0000 0.0000 1.0000 1.0000 u_ 0.0000 0.0000 0.0000 1.0000 1.0000 u, 1.0000 1.0000 1.0000 1.0000 0.0000 ue r N l N2 HPCT SMHC u, 0.0000 1.0000 1.0000 0.0000 0.0000 u¢ 1.0000 0.0000 0.0000 0.0000 0.0000 1.0000 0.0000 0.0000 0.0000 0.0000 U, 0.0000 1.0000 1.0000 1.0000 1.O000 Ua U_ U¢ U, Iq ua 1.0000 0.0000 0,0000 1.0000 1.0000 u_ 0.0000 1.0000 1.0000 0.0000 0.0000 u_ 0.0000 1.0000 1.0000 0.0000 0.0000 u 1.0000 0.0000 0.0000 1.0000 1.0000 Ip I, I°, u° O.OOOO 0.0000 1.0000 u, 1.0000 1.0000 0.0000 u. 1.0000 1.0000 0.0000 u, 0.0000 0.0000 1.0000 2. Feedback Gain Matrix Using the algorithm developed in this work for the Integrated/Constrained Output Feedback Control problem, we computed the optimal feedback gain matrix. After some

trial-and-error experimentation, the gain matrix shown in Table 13 was computed. Simple

observation indicates that the algorithm placed zero gain values at the appropriate locations specified by the constraint matrix.

Table 13. Constrained Feedback Gain Matrix : K

a q P P

-0 -3.4588 -21.9572 -13.8515 -0.0042 0.0000 "* 0.0000 0.0000 0.0000 -67.4612 11.3506 u, 0.0000 0.0000 0.0000 7.1143 0.7541 ,, 0.4146 -2.2134 -1.4035 -0.1674 0.0000 r N, N: HPCT SMHC u° 0.0000 -0.2679 -0.6348 0.0000 0.0000 "* 5.1052 0.0000 0.0000 0.0000 0.0000 ", -9.2444 0.0000 0.0000 0.0000 0.0000 ,, 0.0000 2.5412 1.4876 2.1020 0.1145 Z/a /4# U¢ U. ]q U# 2.4268 -6.7740 ,, 10.6735 0.0000 0.0000 0.0000 12.4872 2.9060 0.0000 0.0000 "" 0.0000 0.6132 12.5980 0.0000 0.0000 ", 0.5549 O. 0000 O. 0000 8.0908 -1.8906 .. Ip I, I..

,, O. 0000 O. 0000 O. 4299 14.1108 -1.4198 0.0000 "" 2.2957 -3.1476 0.0000 "" O. 0000 0.0000 8.5901 The constrained feedback gain matrix provides the desired simplicity and other characteristics. However, an interesting question is whether the optimal gain matrix would not result in pretty much the same matrix as the one computed if we had not placed all the constraints. Aider all the lateral dynamics are uncoupled from the longitudinal and propulsion subsystems. In other words, maybe we can achieve approximately the same result without as much work. Table 14 shows the feedback gain matrix obtained by optimizing the same cost function without any constraints.

As can be seen from Table 14, the unconstrained feedback gain matrix is highly coupled.

In fact, the coupling seems to be present in all the subsystems. It is surprising to find the extent to which the lateral dynamics is now coupled with the other subsystems. Clearly, the new approach and algorithm are necessary if we want to obtain an integrated/constrained feedback control law.

Table 14. Unconstrained Feedback Gain Matrix : K a= u q fl p g 0 -2.1948 -21.8214 -13.8979 1.3406 2.4836 u, 1.0281 1.2752 -1.0107 -56.0643 20.0745 "v 0.3207 0.3319 0.1278 6.0044 0.2203 ,, 0.1629 -2.9952 -!.4927 -2.2007 -4.5740 r N, N2 HPCT SMHC um -0.1072 -0.9778 -0.2175 0.2034 0.2427 ,, 5.8336 -4.3872 -6.7735 -4.8243 13.5707 -9.2235 0.0733 0.6210 0.3184 -1.4593 "" 0.0168 2.7710 1.5883 2.8411 0.8239 m t u a u, u_, u, lq ,, Ii.5028 O. 4778 i. 1337 I. 3328 -6. 4635 u, 4. 3741 12. 2794 3. 1801 -28. 6426 3.2624 -0. 0300 0.7203 12. 6883 !. 6995 -0. 0859 ","" -0. 3811 -0. 4167 -0. 6839 14. 6445 -2. 4864 Ip Ir fa.

uo 4.0241 -1.2069 -0.6368 u, 23.5896 -4.4386 -22.8120 ", 1.6205 -3.2986 1.2380 u. -4.5364 1.2960 11.9095 D. SIMULATION RESULTS A digital simulation of the integrated/constrained SOFFT control law obtained in the preceding sections together with the combined dynamics of the modified F-15 SMTD aircraft was developed on the ACET sottware package. The simulation of the modified F-15 SMTD aircraft was at the flight condition of 30 degrees of angle-of-attack in level flight. The integrated/constrained SOFFT control law was simulated in the incremental implementation form (see [1], [2] and Appendix A).

Two cases were simulated. At first, the simulated plant model was the one used in the feedforward control design. In the second case, the simulated plant B_ matrix was 10% smaller than the control design model. In both cases, the simulation introduced no random plant or measurement noise into the system.

Figure 4 shows the results of the first simulation. This case did not provide a challenge for the feedback control law. However, we can see the feedforward control law producing the necessary control activity to track the pilot input commands and produce the response specified by the command model. In Figure 4a, we see that the pilot inputs are commanding an acceleration in the speed of the aircraft at the rate of 6£dsec/sec for a period of 4 seconds and a simultaneous negative pitch rate plus angle-of-attack of-5 deg/sec for 3 seconds. These pilot input commands have been simulated as pulses or steps that last a finite period of time. Also note that the ACET simulation module automatically generates the y-axis labels which correspond to the variables plotted; e.g., UZ(4) is the fourth component of u,, or the pilot input command of a=, YX(3) is the third component of y, or the pitch rate, etc.

............ i ...... Plot h_pUtCommand axs .. _ ........

.... : ...... .......... r .......

U .......... " ......... E .......................... _ ......... _............ ! ..............

-2 6 2 4 TI_ (SEO) 2.5 ........ • .... t ' • qe.2(x ..... :.. ::. ::.. :. - ....... ; . -.- . Pilot rout C or_ .... ! ...................

...... _........... ._ _Z _.'_-_ ._..."....'__..L Z _ ............

.......... _ _. .... L_.. Z._ ..L _.-.. '! ........... :_ ..................................

-2.5 N ............ . .........................

-5.0 -7.5 2 4 TI,,,'IE (SEC) Modified F- 15 SMTD Integrated SOFFT Control Simulation, ct=30 °, Figure 4a.

Pilot Input Commands 7O A N ................. i...................................

-6 0 2 4 6 TME (SEC) Figure 4b. Modified F-15 SMTD Integrated SOFFT Control Simulation, or=30 °, Command Model Outputs Figure 4b shows the nonzero command model state variables (z,) which correspond to the flying quality commands for the first and fourth elements in eq (121). Note that these represent the desired response of the aircraft to the input commands.

In Figure 4c, we see the response o£ the airframe variables to these commands. The acceleration tracks the commands with high accuracy. The pitch rate has a similar but somewhat different response than the command model output. The reason is that the command applies to the linear combination of the pitch rate and angle-of-attack _r )- 2_ 2 4 (_EC; -2 -4 .............. _................. T ...............: ............ { ................................. " ............................. _ ...................... " ..............._ .............

A ¢M -6 >-

i!!!!ii!!i!!i

-8 -10 0 2 4 6 TME tSEC) .......__. ............. : _ _ "_. i_ I_E:_I':_._.2_ I/. .......... _..., _ i_ _._'_'_i_._._, -: _ I__ ............ ........... _ ............. ÷................_ ................. I_1_ __'.__ __ .................................................... : -Z5 ::::::_:__ ":_:: _i_: ============================================================================================== _:i_:::i ;:_:' i:" _::::'_:.::: :: -5,0 -7.5 _P -10.0 i L -12.5 0 2 4 t) T_E _CI.

............................ _.............. _._.:_,;_'_:--'_2:2_I_::T_':T_I _._ ....':_ii___i_._i;_i_i_,121_._._ _: " ' . "". _Y ,] .

..................... ; ........... _': " V" ............... _.............. ¢ ...............................................

2O '_ ii2 _ . _- . :; il .: ! ,,i.i_....: ............ , ..................... 1 ...............................

2 4 (SEC_ Modified F-15 SMTD Integrated SOFFT Control Simulation, or=30°, Figure 4c.

Airframe Variables ..... _" .......... i ' ! ......... : ....... : .......... i ..... J ...... : ................

/ ........................... ! i ! : -2

i ! '

x i i . i ! i >- !............................ T ................. " ................................ r..............................................................

-4 ............... i.................. i.................. _....

-.6 0 2 4 6 TI_ tSEC) E -1 x -2 -3 0 2 4 6 TIME (SEC) .ZZZ Ii'_ZZZZZi_.IZZZZI_ZZIIZZIIII_Z_ZZZZZIIIZZZZIIZIIZII o _ ............ _'__ ............. _ .................. _ ................. !................. _ ................ : ................ _ .................

................. !.................. i.................. i.................. !.................. i............................... _ ................. 'i- ................. :.................. _ .................................

................. ;................. T ................. _ .................. _ .................. _ .................. !............. +................. _ .................. f................. !................. _ ................

A '_ ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::i, ....._..........................................._:::::::i .................

>-- .................. _ ................. i.................. !................. t................. _ .................. _ .................. ;........... ......,_................._............-.._ ...................................................... _................. _ ................. i.................. i.................. _ ................. _................. _ .................. f.................. : .................

-2 0 2 4 6 T]UE (SEC) ......... :. ................ i................ ._ ................ ..:.................:................._................._IZIIII I ......... _ ............... _" ............... : ...............

i i ............... _................ _................. _ ................ Stai-==gn._ZZZZk.ZZZZZIZIZZZZI A i i : O5 0 2 4 {SEC} Figure 4d. Modified F-15 SMTD Integrated SOFFT Control Simulation, or=30 °, Propulsion Variables -1 X -2 -3 0 2 4 6 TIdE (SEC) X -2 -4 0 2 4 6 TIdE (SEC) Figure 4e. Modified F-15 SMTD Integrated SOFFT Control Simulation, a=30 °, Control Variables As the pitch angle and the angle-of-attack fall, the aircraft picks up speed from the combined effects of the pitch action reducing the drag and the initial increase in thrust.

The propulsion variables shown in Figure 4d first increase to produce more thrust needed to accelerate, but then fall to lower levels to avoid producing too much acceleration at the new angle-of-attack. Note that the stan margin remains in positive territory most of the time; i.e., it remains higher than its original comfortable level.

Finally, note the coordinated action of the propulsion and pitch controls shown in Figure 4e. Both controls move when the pitch rate command pulse input by the pilot ends at 3 • 74 seconds. On the other hand, when the acceleration pulse ends at 4 seconds, we see a reaction only in the propulsion control. This behaviour seems completely in line with our previous analysis of the feedforward gain matrices. Since the simulated plant and the design plant model are the same, the feedforward control produces the perfect response desired and the feedback control is null because the feedback state error is zero.

Figure 5 shows the simulation results for the second case considered. In this case, the simulated plant is different than the one used in designing the feedforward and feedback control laws. Accordingly, the feedforward control results in a state trajectory which is almost perfect but not exactly the same as the desired response. Therefore, the feedback state vector, the difference between the actual and feedforward states, is not zero. As a result the feedback control law tries to-minimize the error by appropriate action.

i • ..... i ..... ............. :...................... Pilot Inp_ Command axs _" 2 r,.,d -2 2 4 ...... .... ............ ........ :- .......... Pilot bnut Commandrn_ _(x_ .................... _.......

0 ....................

......................................... i.................. ...............................................................

...... ...... : ................... i .............. : ........... -z ............ i ................... _ ...... ....

........... i ......... : ................... i............................. i ............ .............

-6 2 4 T_E (SEC) Figure 5a. Modified F-15 SMTD Integrated SOFFT Control Simulation, Perturbed Plant Pilot Input Commands As seen in Figure 5a, the pilot inputs are the same as the in the first case. The pilot commands an acceleration in the forward speed and a negative pitch rate action. It may be of interest to note that the feedforward control, which (by design) depends only on the input commands, will be the same as the first case since the design plant model has not changed. Recall that the simulated plant model has been perturbed, but the control design models have remained the same.

Figure 5b shows the relevant command model state variables generated in response to the pilot input commands. Since the command model has not changed, the input commands produce the same desired response as the first case.

In Figure 5c, we se¢ some of the airfi'ame variables. As noted earlier, since the control laws have been designed using an erroneous plant model, the state variables are not going to be perfect replicas of the desired response. On close observation, we can see that the forward acceleration is slightly different than the commanded acceleration in Figure 5b.

However, the overall response is quite similar. It is harder to notice the differences in the pitch rate. Comparing the pitch rate and angle-of-attack to the desired response shown in Figure 4c, any offsets are hard to see.

N 0 2 4 2 ................... ............................... ,............. _ ......... Commanded(q+0.2_) ...........

-6 2 4 TM_ ($EC) Modified F-15 SMTD Integrated SOFFT Control Simulation, Perturbed Plant Figure 5b.

Command Model Outputs >., 0 2 4 6 2.5 ..... " ............... Rtc_Rete- :¢ ............... i .................................... . ....................

i i : i i c_ X -2.5 >- -5.0 O_ 2 4 6 .................. i............... !.................. _ ................. i................. i.................. i.................. :................ - ................. !.................. !.................. • .................

-4 -8 >- _________________i_Z_______________!_________________i__.______________i_________________i____ iiiiiiiiill ........ ;....... IIIIIIIIII.".IIIIZIIIIIIIIIIIIIIIIIIIIIIIIII]IIIIIIIIIIIIIIIIIIIIIIIIIIIIIIIII -12 0 2 4 6 2 4 6 ................ - ....................................................... _ ................. _. ................. _............ Spead...V. ................................. i ................

0 2 4 6 "riMe (SEC) Figure 5c. Modified F-15 SMTD Integrated SOFFT Control Simulation, Perturbed Plant Airframe Variables _" -2 >- -4 -6 0 2 4 6 11vE (SEC) : " h_ RotorSpeeclN 2 :......

aD, -2 -4 0 2 4 6 TUvE (SEC1 _CT _" -.1 >- -.2 0 2 4 6 (SEC] .25 .................... ..... ............ Stall Marcjn SMHC : ............

.05 -.05 2 4 "riME(SEC) Modified F-15 SMTD Integrated SOFFT Control Simulation, Perturbed Plant Figure 5d.

Propulsion Variables Figure 5d shows the propulsion variables. Differences from the previous case are small and hard to see.

The feedforward control variables are shown in Figure 5e. Note that these variables are the same as the first case simulations for the same reasons discussed previously; i.e., the pilot input commands are the same, the command model and the plant model used in the feedforward control design are the same as the first case.

F'-'FD Ptop_sion Control ..... ......

3< -2 -4 0 2 4 6 "nk4E (SEC) FFD Pitch Control re- X -2 -4 0 2 4 TIME(SEC) Figure 5e. Modified F-15 SMTD Integrated SOFFT Control Simulation, Perturbed Plant Feedforward Control Variables The feedback control variables are shown in Figure 5£ Note thatthe feedback control law in this case is not nuU and is trying to reduce the errors present between the actual state and the feedforward state corresponding to the desired response. Thus, a I(FA error in the control effectiveness matrix is barely noticed. It should also be noted that the feedback control is able to achieve this response with a control effort equal to approximately one tenth of the feedforward control leveJs. Of course, when plant and measurement noise is present, the feedback control activity will increase noticeably as the high frequency content will increase.

,2 .=.J l =_ --.2 -.4 0 2 4 TME (SEC) .4 ...._ .h......:_ __...... ..... ............................. .............................. FB 'Pitch Con_'ot............................. " ........

.2 ......................... i ................

#=_ -.2 .=_ b=- -- .4 -.6 0 2 4 6 TIE [SEC) Figure 5£ Modified F-15 SMTD Integrated SOFFT Control Simulation, Perturbed Plant Feedback Control Variables 8O The closed-loop modes of the aircratt are shown in Table 15 Table 15. Closed-Loop Equivalent s-plane Eigenvalues REAL IMAGINARY DAMP ING P_TIO _LP+T_ FREQUENCY -0.001793 0.000000 1.00 0.00 -0.016570 0.000000 1.00 0.02 -0.017264 0.000000 1.00 0.02 -0.101969 0.000000 1.00 0.10 -0.108535 0.000000 1.00 0.11 -0.392538 0.000000 1.00 0.39 -0.555280 0.000000 1.00 0.56 -1.986110 -2.050269 0.70 2.85 -1.986110 2.050269 0.70 2.85 -2.289180 -0.513772 0.98 2.35 -2.289180 0.513772 0.98 2.35 -2.359946 -0.092281 1.00 2.36 -2.359946 0.092281 1.00 2.36 -4.524391 -4.666417 0.70 6.50 -4.524391 4.666417 0.70 6.50 -6.218162 -4.595787 0.80 7.73 -6.218162 4.595787 0.80 7.73 -10.174557 0.000000 1.00 10.17 -12.552118 0.000000 1.00 12.55 -56.438641 0.000000 1.00 56.44 -199.904281 0.000000 1.00 199.90 V. CONCLUSIONS The need for integrated or constrained control systems came into greater focus as advanced aircraf_ brought new subsystems with significant coupling between them. These included advanced propulsion subsystems with vectored thrust and new aerodynamic designs of the _e sometimes with new surfaces such as canards and sometimes with fewer control surfaces such as the tailless aircrafL The coupling produced by these subsystem developments only added to the existing need for a general method for designing control laws with arbitrary coupling constraints for decentralized systems.

In this study, we develop an integrated control design methodology which accommodates constraints on the coupling of subsystems or variables. The methodology uses the SOFFT control approach and structure, thus maintaining all the advantages of the SOFFT approach.

The Integrated/Constrained SOFFT Control methodology uses a centralized feedforward control and a constrained feedback control law. The main conclusion of this work is that the approach mentioned takes advantage of the coupling among the various subsystems while maintaining the identity of subsystems for validation purposes and the simplicity of the feedback control law for ease of understanding the system's behaviour in complicated nonlinear scenarios. While the use of a centralized SOFFT feedforward is recommended, it is not a necessity for the methodology. It is possible to use a constrained SOFFT feedforward to accommodate constraints using the algorithm developed for feedback systems or by using other methods. The methodology is formulated in detail and the necessary mathematical development is shown in the previous sections.

The Variable-Gain Output Feedback Control methodology is extended to include equality constraints which can avoid coupling two variables or two subsystems. Although this approach was developed within the SOFFT context, it is an independent method for designing feedbackcontrollaws which can be used with or without any feedforward control system. The Variable-Gain Output Feedback Algorithm is extended to accommodate equality constraints by specifying that any element of the gain matrix be zero.

In a more general setting, the algorithm developed can have nonzero values placed as constraints as well. This allows the optimization of one part of a control law while the other part (which may have been designed and tested under previous circumstances) is leR unchanged. Another application is to produce a known amount of coupling between two variables, such as the aileron-rudder interconnect, by constraining a gain element or block to maintain a specified value while optimizing the remaining control gains. The algorithm is monotonic in the cost, reducing the cost at each iteration. The rate of convergence depends on the particular problem and on the particular constraints placed for a given problem.

Finally, the Integrated SOFFT Control methodology is used to design an integrated flight and propulsion control system for the modified F-15 SMTD aircraft as an example of the approach developed. A centralized SOFFT feedforward control law and a constrained output feedback control law are designed at a 30 degree angle-of-attack flight condition.

Using the SOFFT approach, the command model is selected to produce desirable flying qualities for the aircraft. The response of the aircraft to some pilot input commands are presented.

REFERENCES I. I-Ialyo, N., Direskeneli, H. and D. B. Taylor, "A Stochastic Optimal Feedforward and Feedback Control Methodology for Supe_ty", NASA CR-4471, November 1992 . Halyo, N., "A Stochastic Optimal Feedforward and Feedback Control Methodology for Superagility", ICS FR-689102, Information & Control Systems, Inc., 732 Thimble Shoals Blvd., Newport News, VA, 1989.

. Halyo, N. and J. P,- Broussard, "Investigation, Development, and Application of Optimal feedback Theory. Volume I --A Convergent Algorithm for the Stochastic Infinite-Time Discrete Optimal Output Feedback Problem", NASA CR-3828, August .

Halyo, N. and R. E. Foulkes, "On the Quadratic Sampled-Data Regulator with Unstable Random Disturbances", IEEE S_C Cos. Proc. 1974 Int. Conf. on Sys., Man and Cybern., October 1974.

. Halyo, N., Moerder, D. D., Broussard, J. R. and D. B. Taylor, "A Variable-Gain Output Feedback Control Design Methodology", NASA CR.4226, March 1989.

. Halyo, N., "A Variable-Gain Output Feedback Control Design Approach", Proc. AIAA Guidance Navigation, and Control Conf., Boston, MA, August 1989.

. Halyo, N. and J'. R. Broussard, "A Convergent Algorithm for the Stochastic Infinite- Time Discrete Optimal Output Feedback Problem", Proc. of the 1981 Joint Auto.

Control Conf., Volume 1., Amer. Auto. Control Council, c. 1981, paper WA-1E Charlottesville, VA.

8. Ostroff_ A. J., "High-Alpha Application of Variable-Gain Output Feedback Control", ,l., Guidance, Control and Dynamics, Vol. 15, No. 2, pp. 491-497, March-April 1992.

9. Hueschen, R. M., "The Design, Development, and Flight Testing of a Modem Control- Designed Autoland System", American Control Conference, Boston, MA, June 1985.

I 0. Halyo, N. and A. K. Caglayan, "A Separation Theorem for the Stochastic Sampled-Data LQG Problem", Int. J Control, Volume 23, No. 2, pp. 237=244, February 1976.

11. 1L E. Kalman and R. S. Bucy (1961), Hew Results in Linear Filtering and Prediction Theory", J. Basic Eng., Trans. ASME, Ser. D, 83, pp. 95=108.

12. Kwakemaak, H. and 1L Sivan, Linear Optimal Control Systems, John W'dey & Sons, Inc., New York, 1972.

13.

Broussard, J. R. and C. S. McLean, "An Algorithm for Simultaneous Stabilization Using Decentralized Constant Gain Output Feedback", 1EEE Trans. on Automatic Control, Vol. 38, No. 3, pp. 450-455, March 1993.

14. Etldn, B., Dynamics of Atmospheric Flight, John Wiley & Sons, Inc., New York, 1972.

APPENDIX A INCREMENTAL IMPLEMENTATION WITH CONTROL RATE LIMITS The incremental implementation given in [5], pp.46-50, can be used to implement the SOFFT control laws developed in this work. However, this implementation is intended for systems with position limiters placed on the control values. Here, we want to present an incremental implementation for systems which contain both control position and control rate limiters.

The motivation for using an incremental implementation rather than a standard implementation (which would use total values) remains essentially the same; i.e., avoiding integrator wind-up due to position limiters, eliminating the use of trim values, improving the response to constant disturbances, etc. In the present implementation, we include the effect of control rate fimiters on tracking errors and the resulting introduction of phase shifts or time delays which can produce significant instabilities in the dosed- loop system. The reasons for selecting one particular form of incrementation over another are of a heuristic nature. Here, we simply present the proposed incremental implementation without further comment.

The system we want to control is of the form shown in eq. (81) and (82) or , more generally, as given in [5], eq. (22) and (23). These represent a nonlinear system having a wide operating range over which the system parameters may vary considerably. The details of the linearization are shown in Section II. 1 of [5].

Compute the control position command using the following equations.

u_ = lime {u,a,_ _ +At vk} (A-l) (A-2) where lira R and lira e are functions which limit the control rate and the control position to the specified range of values, respectively. We select these limits to be the same as those set by the plant actuator limiters. In other words, the system actuators do not allow the control position and rate to go above these values. Accordingly, the performance of the closed loop plant is not impacted by the software limit functions placed in the equations above, at least not immediately.

The feedforward control law can be implemented as follows.

zk+_ = _, (Pk) zk + F, (Pk) u,, (A-3) Az, = z, - zk_ 1 (A-4) Au,k = u,k - u,,_, (A-5) Au_ = -K_ (p,) Ax_ - K, (p,) Az k - K.. (Pk) Au,_ (A-6) Axe., = _x (P,) hx_ + F. (p,) hu_ (A-7) =cAp ) Ax; (A-8) y: = y__, + Ay_ (A-9) We do not place any limiters on the feedforward control variables computed in the above equations. However, depending on the particular problem, it may be necessary or desirable to place some limits on the feedforward state, measurement or control variables.

Also note that the total value of the feedforward state vector is not computed although a value is implied by the incremental feedforward state vector which is used to compute the total value (except for the measurement bias valde which will caned out in (A- 11)) of the feedforward measurements. Similarly, the total value of the feedforward control is not computed in this implementation.

The feedback control law is implemented as follows.

(A-IO) Y_ =Yn -Y_ (A-11) A_k =-K=(Pk) AY_ -At K_(pk)[Hy(pk)___ + H_(Pk) u__,] (A-12) - At K, (Pk) Vk-I (A-13) (A-14) _, = u-__ l +At v__ l We do not place any limiters on the feedback state, measurement or control vectors in the equations above although circumstances may necessitate their use in certain problems.

The limiting function is placed on the total control rate and control position vectors shown in equations (A-l) and (A-2). In a sequential presentation of the incremental implementation equations, first (A-2) then (A-l) would follow (A-14).

It is also important to note that the control faltering action of the PIF feedback structure is still present in the implementation given above. Equation (A-12) is simply an algebraic restatement of the more usual form of this equation. As long as no limits are placed on the feedback implementation equations, the filtering action will be present.

APPENDIX B INTEGRATED FLIGHT AND PROPULSION MODEL DERIVATION In this appendix, we show the derivation of the integrated flight and propulsion model used in Section IV.A. as the example problem which is a modified version of the F-15 SMTD aircraft. The derivation is quite straight-forward and is given here for completeness.

The state equations for the airframe and propulsion subsystems are of the form

x, = al x, + B1 ux_ + ";, u. 03-1)

x 2 = A 2 x 2 +B 2 u.2 +B; u=

(B-2)

where the subscript 1 denotes the airframe parameters and the subscript 2 denotes the propulsion parameters. The cross-coupling terms, the last terms in the equations above, are given by where F 8 and Dp are the gross thrust and the drag due to the engine.

The cross-coupling terms defined above can be expressed in terms of the state and control vectors as follows.

03-4)

u u = Enx: + FIzu,.2 + Gnu.

03-5)

U2 2 m E21Xl Since the airframe model includes both the longitudinal and lateral dynamics, the components of u=, namely the angle=of-attack and the sideslip angle, can be expressed in terms of the airframe state alone.

Substituting (B-5) into 03-4), eliminate the cross-coupling term u,,.

03-6)

u,l = (GI2E2_)x_+ E_:x2 + Fnux:

(I3-7)

= EllXl -I.- E12x2 4- Fi2uz2 Now substituting 03-5) and 03-7) for the cross-coupling terms in 03-1) and 03-2) and manipulating, the integrated continuous open-loop system can be found as

03-8)

xe = [B_2E21]xl + A2 x2 + B2 u=2 03-9) Comparing (116) and (117) with 03-8) and 03-9), we note that they have the same form.

Therefore, equating the system matrices provides the desired result.

Form Approved

REPORTDOCUMENTATIONPAGE

OMB No. 0704-0188 Highwly, Suite1204. Adinglon. VA 2220Q.4302.andto_'mOfltceolManagecmmtand uOgeL_ _ m_,ec_(0704,.0 88)._DC 20503.

2. REPORT DATE 3. REPORT TYPE AND DATES COVERED 1. AGENCY USE ONLY (Leave I_iank) July 1996 Contractor Report 5. FUNDING NUMBERS 4. TITLE AND SUBTITLE C NAS1-20185 Integrated Control Using the SOFFT Control Structure WU 505-68-33-01 s. AUTHOr(S) Nesim Halyo 8. PERFORMING ORGANIZATION 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) REPORT NUMBER Information & Control Systems, Inc.

732 Thimble Shoals Blvd., Suite 801 Newport News, VA 23606 10. SPONSORING/MONITORING 9. SPONSORING I MONITORING AGENCY NAME(S) AND ADORESS(ES) AGENCY REPORT NUMBER National Aeronautics and Space Administration NASA CR-4748 Langley Research Center Hampton, VA 23681-0001 11. SUPPLEMENTARY NOTES Langley Technical Monitor. A. J. Ostroff 12b. DISTRIBUTION CODE 121. DISTRIBUTION I AVAILABILITY STATEMENT Unclassified Unlimited Subject Category - 08 13. ABSTRACT (Maudmum 200 words) The need for integrated/constrained control systems has become clearer as advanced aircraft introduced new coupled subsystems such as new propulsion subsystems with thrust vectoring and new aerodynamic designs.

In this study, we develop an integrated control design methodology which accomodates constraints among subsystem variables while using the Stochastic Optimal Feedforward/Feedback Control Technique (SOFFT) thus maintaining all the advantages of the SOFFT approach. The Integrated SOFFT Control methodology uses a centralized feedforward control and a constrained feedback control I zw. The control thus takes advantage of the known coupling among the subsystems while maintaining the idemity of subsystems for validation purposes and the simplicity of the feedback law to understand the system response in complicated nonlinear scenarios.

The Variable-Gain Output Feedback Control methodology (including constant gain output feedback) is extended to accomodate equality constraints. A gain computation algorithm is developed. The designer can set the cross-gains between two variables or subsystems to zero or another value and optimize the remaining gains subject to the constraint. An integrated control law is designed for a modified F-15 SMTD aircraft model with coupled airframe and propulsion subsystems using the Integrated SOFFT Control methodology to produce a set P,# _,',,r;rr_rl _h,;,._,', _,,,',lH';_e" 15. NUMBER OF PAGES 14. SUBJECT TERMS Feedforward, Feedback, SOFFT, Integrated, Control, Optimal, Constrained, Output feedback, Flight & propulsion 16. PRICE CODE A05 20. LIMrrATiON OF ABSTRACT 19. SECURITY CLASSIFICATION 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION OF ABSTRACT OF THIS PAGE OF REPORT Unclassified Unclassified Standard Form 298 (Rev. 2-89) NSN 7540-01-280-5500 Presct_ed by ANSI Std. Z39-18 92 298-102