application of real structured singular values to flight ... - CiteSeerX

APPLICATION OF REAL STRUCTURED SINGULAR VALUES TO FLIGHT CONTROL LAW VALIDATION ISSUES Piero Miotto and James D. Paduano Department of Aeronautics and Astronautics Massachusetts Institute of Technology Cambridge, Massachusetts

ABSTRACT This paper describes flight control law validation issues raised at NASA Dryden, and recasts these problems into the real-µ analysis framework. To solve these problems, we extend the algorithm presented by Dailey to include repeated real perturbations, and describe how it can be used to find both a lower bound for real-µ and detailed direction information about the worst case perturbation.1 The latter information is useful in applications that require proper weighting of the perturbation block structure. An automated weighting procedure is described which effectively gives the relative importance of the real uncertainties being analyzed. Other developments motivated by flight test issues are also described: block structures for analyzing phase margin and transport delay, "robustly guaranteed" gain and phase margins, and techniques for real-time implementation of robustness measures based on flight data. These tools and techniques are demonstrated on X-31 flight control laws. 1.0 INTRODUCTION Recent years have seen the steady maturation of real-µ and mixed-µ as a framework for analyzing the robustness of multivariable control systems, as well as algorithms for computing upper and lower bounds for µ .7 Since the definition of µ succinctly captures practically motivated robustness questions, much of the recent literature has concentrated on methods for computing tight bounds while reducing computation time.7,8,13 The primary application of the methodology is often considered to be design: once algorithms are in place to reliably and efficiently compute µ , they are folded into a synthesis procedure which attempts to minimize its maximum value. The application considered in this paper is fundamentally different. In the flight test and control law validation environment, robustness characterization is an end in itself: given a complex flight control system and nonlinear model of the flight dynamics, determine "what can go wrong" in flight, during maneuvers, or at the next envelope expansion point. The problems

which must be dealt with in this environment often involve delays, nonlinearities, actuator rate and/or position saturation, mode switching pathologies, handling qualities and pilot/cockpit interactions, and complex aerodynamic/aeroelastic effects. The goal of this paper is to begin to address some of these problems, by recasting them in the real-µ framework. Our approach is to develop methods to answer specific questions raised during the analysis of the flight control laws for the X-31 Enhanced Fighter Maneuverability Demonstrator. These methods form building blocks for analysis of more complex robustness questions, which will be addressed in future work. The X-31 lateral-directional flight control law will be used to motivate the analyses and demonstrate the procedures. Specifically, we address the following questions: - How should the inputs to a given delta-block structure be weighted, so that the parameters which are most important to the control system's stability are made apparent? - How can phase margin and transport delay be included in the robustness problem? - How can robustness to several aerodynamic parameters be characterized in a way which is useful to flight test engineers? - How can a measure of multivariable robustness be derived during a flight test and presented in the control room? Admittedly, these problems constitute only a narrow subset of the issues faced during flight test and control law validation. However, these relatively simple questions are important application issues which must be addressed before even the most rudimentary implementation of structured singular value analysis is attempted. The paper is organized as follows: the definition of the robustness problem is first given, and Dailey's algorithm for computing real-µ is described and extended to the repeated-real case. The issue of

1 American Institute of Aeronautics and Astronautics

weighting is then discussed, and an illustrative example is used to highlight the importance of iteratively reweighting for detailed robustness analysis. Next, block structures are developed that allow phase and delay to be included as part of the real-µ problem. These block structures become part of a second example, where we use the concept of "robustly guaranteed" gain and phase margin as a bridge between classical robustness measures and the modern techniques used here. Finally, we briefly describe the methods used to include flight test information in the analysis problem, and show how an experimental µ plot might be useful as part of flight test envelope expansion procedures. 2.0 NOTATION AND DEFINITIONS The notation used in this paper is fairly standard. Superscript * denotes the complex conjugate transpose of a complex matrix M , a k × k identity matrix is denoted Ik , and the largest singular value is denoted by σ (M ). The first step in a real-µ analysis is to transform the problem into the interconnection structure of Fig. 1, in which M ( jω ) represents the nominal closed-loop dynamics, and all perturbations are captured in the structured uncertainty ∆ . For this paper, the underlying block structure of the uncertainties in ∆ is defined as follows: at each frequency ω we have the matrix n n M ( jω ) ∈C × . An integer m r ≤ n specifies the number of real uncertainty blocks in ∆ . The block structure K (mr ) is a set of positive integers:

(

K = k1 , k 2 ,L k mr

)

∑ ki = n

The above definition, if applied to our set of allowable perturbations XK , can be simplified. In this case the 2-norm ∆ = σ (∆ ) is identical to the norm max δ i , which is more easily interpreted in i applications. Since the size of ∆ is often of interest, it −1 will be convenient to define k µ ≡ µ K . Thus the size of the smallest ∆ (where size is defined in terms of the norms above) which destabilizes the system in Fig. 1 is, by definition, k µ . The following statements follow in a straightforward manner: - The system in Fig. 1 is stable for all ∆ in XK such that max δ i < k µ . This inequality defines a hypercube ofi perturbations for which the system is guaranteed to remain stable. - If a weighting matrix W ( jω ) is introduced as in Fig. 2, the resulting system is stable for all ∆ ∈ XK with ∆ < 1 if and only if:

[

]

µ K W ( jω )M ( jω ) < 1

for ω ∈ (0 ,∞ ) .

- If W is a diagonal frequency independent weighting matrix,

(

W = diag w1 , w 2 , L w mr

),

then the system in Fig. 2 is stable for ∆ in XK such that δi < kµ . wi

This inequality defines an elongated hypercube of guaranteed stable perturbations. 3.0 STABILITY ROBUSTNESS ANALYSIS WITH ITERATIVE WEIGHTING

.

The set of all allowable perturbations is then defined as: XK =

1 k1 , δ2 I k2

]

−1

with µ K ( M ) = 0 if no ∆ ∈ XK solves det (I − ∆M ) = 0 .

max

mr

{∆: ∆ = block diag(δ I

[

i = 1Lmr

which specifies the dimensions of the perturbation blocks. These dimensions must be compatible with the dimension of M ( jω ):

i =1

  µ K ( M ) =  min σ (∆ ) : det( I − ∆M ) = 0  ∆∈X K 

)

}

,Lδ m r Ik m , δ i ∈ R r

The size of the smallest destabilizing perturbation in XK is characterized by calculating µ K ( M ) at each frequency, where µ K ( M ) is defined below. Definition 1. The structured singular value µ K ( M ) of a matrix M ∈C n× n with respect to a block structure K (mr ) is defined as:

3.1 Algorithm for Computing Real µ Several algorithms have been proposed for the calculation of upper and lower bounds of µ ; we will adopt the approach outlined by Dailey.1,7,15,17 Although this algorithm is not as efficient and general as other approaches, it provides valuable insights into the robustness scenario, which we will translate into procedures for iterative weighting. The main assumption that is made in this algorithm is that the worst case perturbation at a given frequency, whose size determines µ , is in the vicinity of one of the

2 American Institute of Aeronautics and Astronautics

corners of the largest stable hypercube in the uncertain parameter space. What we mean by vicinity is that all but two elements of the worst-case ∆ are equal to 1 µ ; the remaining two elements are less than or equal to 1 µ . Many examples have supported the validity of this assumption, especially in the frequency region were µ is maximum. Dailey gives a full description of the algorithm; here we will point out its main characteristics, to motivate our use of weightings.1 We first describe the case of distinct real uncertainties, which is Dailey's case. We then extend the analysis to include repeated real uncertainties. 3.1.1 Distinct Real Uncertainties. In this case all uncertain elements have multiplicity 1. The set of allowable perturbations is then defined as:

{

}

XK = ∆: ∆ = diag(δ1 , δ 2 ,L ,δ m r ), δ i ∈R

To direct our search in the region of one of the corners of the hypercube, we fix all but two elements of the ∆ block at initial values of ± kg (all combinations of the sign of each element must be checked.) The values of the remaining two elements, which we designate δ1 and δ 2 , can be computed by solving a quadratic equation. If δ1 and δ 2 are complex, or if their norm is greater than k g , then no solution exists with ∆ = kg ; we therefore increase k g and repeat. On the other hand if the norm of (δ1 , δ2 ) is smaller than k g , then a solution with ∆ = kg does exist; in this case we decrease k g and repeat. The search thus converges toward the minimum value of k g for which a solution exist with ∆ = kg , and this gives us a lower bound on µ:

separate direction does one 'switch faces' and iterate in the new direction. The main advantage of this algorithm is that together with the value of µ we also know which single uncertain element is actually causing the minimum k g . This information is particularly valuable when we want to weight the variables to find the 'most voluminous' stable hypercube. A weighting procedure based on this information is described in Section 3.2. 3.1.2 Repeated Real Uncertainties. The k g search (in the vicinity of each corner) required in the case of a repeated-real block structure is more complicated than the one described above. In this case, we still fix all but two elements and perform a search, but the solution for a given k g is not given by a simple quadratic equation. Instead, a subsearch for the (δ1 , δ2 ) with minimum norm must be conducted for each value of k g . Otherwise the procedure is unchanged. Let us describe the subsearch. Suppose that the current direction of our search is defined by a diagonal n D × n D matrix D with diagonal elements equal to +1 or -1 defining the direction of the search. The remaining two uncertainties form a two–block structure of diagonal uncertainties whose multiplicities are k1 and k 2 :  Ik δ ∆' =  10 1 

We are looking for the minimum norm ∆' such that: ∆'  det  I −  0  

( 1)

− − µ K = k µ ≥ max k g 1

This search is repeated for all combinations of 'all but two elements fixed', and the maximum lower bound obtained is in practice an excellent lower bound on µ K . Because of the combinatorial nature of the procedure (the number of searches conducted is n(n − 1)2 n−3 ), practical use of the algorithm is currently limited to a maximum of about 9 uncertainties. Normally this procedure is applied to a transfer function matrix M ( jω ) over a range of frequencies. At most frequencies the norm of (δ1 , δ2 ) converges to the tightest lower bound during the first search, because one simply chooses to search in the direction that worked at the previous frequency. The remaining searches are then unnecessary; as soon as the maximum value of k g is found, one simply verifies that there are no other directions for which a solution exists with ∆ = kg . Only when a solution actually exists in a

0  , n ∆' = k1 + k2 Ik 2 δ 2 

0   M11 kg D  M21

M12   0 M 22   =

Where k g is the current estimate of k µ and M has been suitably partitioned. Rearranging this determinant:  ∆' det  I −  0  

0   In ∆' kg In D   0

M12   M 22   = 0 ,

0   M11 D M 21

we have:  I n − ∆' Q11 − ∆' Q12  det  ∆' Q In D − kg Q22  = 0 ,  −k g 21 where the definitions of Qij are obvious. Since det In D − kg Q22 is different from zero except under pathological circumstances, the matrix inversion lemma allows us to convert the above condition to:

[

]

(

det  I n ∆' − ∆' Q11 − ∆' Q12 I n D − k g Q22  Now, if we let:

3 American Institute of Aeronautics and Astronautics

)

−1

k g Q21  = 0 . 

(

( )

P kg = Q11 + Q12 I n D − k g Q22

)

−1

kg Q21 ,

and also partition P appropriately, we can write:  0  P11 P12    Ik δ det  I n ∆' −  1 1   = 0 . 0 I  k 2 δ2  P21 P22    The condition to be met with minimum norm of (δ1 , δ2 ) is then:  I n − δ1 P11 det  1 P  −δ 2 21

−δ1 P12  In 2 − δ 2 P22  = 0 .

Again we apply the matrix inversion lemma, which gives:

[

]

det In 1 − δ1 P11 ⋅

(

det  I n − δ2  P22 + P21 In − δ 1P11  2 1 

[

)

 δ1 P12  = 0  

−1

]

As before, det In 1 − δ1 P11 is not equal to zero, so we have the following result: Given a value for k g and a guess at the size of δ1 , the value of δ 2 which makes det(I − ∆M) = 0 is given by: δ 2 = min

λ i ∈R

1

{(

λ i B δ1 , kg

,

W 0 = diag(1,1,1,1,1,1) after running the algorithm we find a certain value of the µ 0 . We then know that stability is guaranteed in the following n-dimensional box:

where

(

(

)

)

−1

δ1 P12 .

 w  S0 = δi : max δ i < 0i  i 1Lm µ0  = r 

Here we have used the fact that

(

)

nB

det I n B − δ 2 B = ∏ [1 − δ 2 λ i {B}], i=1

where n B = k1 +n D . Clearly the minimum δ 2 to perturb the determinant to zero is determined by the maximum real eigenvalue of B δ 1 , kg -- the subsearch for the value of δ1 which minimizes max δ1 , δ 2 can thus be conducted.

(

)

In the previous section we described an algorithm for calculating a lower bound of µ which is based on determining the specific uncertain element which is actually causing the maximum µ . It is reasonable then to argue that by reducing the weighting on this element we will allow larger stability bounds on the other elements, just as in our 2 × 2 example. Let us say that we start with a certain weighting matrix:

)}

B δ 1 , kg = P22 + P21 I n1 − δ1 P11

with equal weighting on these two parameters, standard µ -analysis will not uncover this fact. From this figure we can see how reweighting the parameters in the ∆ block using a real diagonal weighting matrix W helps to improve our knowledge of the stable space. As the relative weighting on δ 2 goes down, the guaranteed box becomes elongated in the δ1 direction. In the twodimensional case this continues until the corner of the box strikes the stability boundary; in multi-dimensional cases a different edge of the stable hypercube usually strikes the boundary; weighting on the new edge is then necessary. Thus reweighting gives both a more complete picture of the robustness interrelationships, and in general enlarges the guaranteed stable region, because the union of any two guaranteed regions is also guaranteed stable.

(

)

3.2 Iterative Weighting Procedure When analyzing several uncertainties in an engineering environment it is of tremendous interest to determine which uncertain elements dominate the robustness measure, which elements are interrelated, and which elements do not greatly affect the stability of the system. Weightings are one way to address these questions. Fig. 3 illustrates why weightings are important. In this simple example the stability of the system is less sensitive to uncertainty in δ1 . However,

We also know that the element that is causing our maximum µ is for example the one related to W 0 (2 ,2 ). For the next weighting matrix we choose: W1 = diag(1, 0. 75,1,1,1,1) We then solve for µ1 ; the associated guaranteed stable region is:  w  S1 = δi : max δ i < 1i  . i 1Lm µ1  = r  In many cases µ1 < µ0 , and the new guaranteed stable region is more voluminous. The stability boundaries on all the elements other then ∆ (2, 2) are larger then before. The bound on ∆ (2, 2) is tighter then before, because we reduced the weighting on this element.

4 American Institute of Aeronautics and Astronautics

As we continue changing weights, the elements that never strike the stability boundary represent uncertainties that tend not to affect the stability of the system. At the end of this process the overall guaranteed stable region is: S = (S0 ∪ S1 ∪ S2 L) From Fig. 3 we can see how the union of all the stability regions can provide extra information about the shape of the stability boundary. In the following example we will show how after only five reweightings we gain a good deal of information about the robustness picture. 3.3 Example I: Iterative Weighting to Isolate Important Lateral-Directional Aerodynamic Parameters of the X-31 In this example we use iterative weighting to determine the relative importance of several lateraldirectional aerodynamic parameters to the robustness of a linearized X-31 lateral-directional control system. Fig. 4 is a block diagram of the control system; for this example we analyze a trimmed level flight condition at a Mach number of 0.6 and altitude of 20,000 ft, 70% quasi-tailless (see Section 4.1 for a description of the quasi-tailless configuration). When analyzing the robustness of a flight control law it is important to incorporate the requirements for flying and handling qualities.4 Lateral directional flying qualities are partially specified based on closed loop pole locations, and these specifications form boundaries which can easily be incorporated into the robustness analysis. The requirements that we will consider are those of a fighter (Class IV) in flight Category A. Our robustness analysis will reflect the desire to maintain level 2 flying qualities, in the face of variations in the stability derivatives of the aircraft. The specifications are: − Spiral mode: min time to double 8 sec (pole to the left of 0.0866); − Dutch mode: min ζ d = 0.02; min ζ d ω nd = 0.05; min ω nd = 0.4; − Roll mode: max roll time constant 1.4 sec (pole to the left of 0.7143). Standard analysis indicates that the two modes that most greatly affect the robustness of the X-31 in the lateral-directional axes are the spiral mode and the Dutch roll mode. For this reason we incorporate only

the requirements for these two modes. During the stability test we need to find the condition at which the spiral mode moves to the right of 0.0866 and the Dutch roll mode moves to the right of a line passing through zero at an angle of 91.2 deg. These requirements can be implemented in our robustness stability analysis by simply calculating det (I − ∆M (s)) (from the definition of µ ) along the following contour in the s plane (see Fig. 5): 1  β + jβ − tan cos −1 (ζ ( d) s =  0. 0866 

(

)

for

β = (0, ∞ )

for

β =0

From Fig. 5 and the above equations it is clear that the requirement for the Dutch roll mode is not much more stringent the stability requirement; β is almost equal to ω in the frequency region of interest. On the other hand, the requirement on the spiral mode is very important to the analysis because the nominal closed loop pole of the spiral mode is slightly unstable. Normally this condition would violate the assumptions of our robustness analysis framework, which require that the poles of the nominal system be to the left of the jω -axis, along which one normally computes µ . But because the handling qualities requirements allow the spiral mode to be unstable, we can simply 'indent' the contour a finite amount at ω = 0 , so that there are no nominal poles to the right of the contour along which we compute µ . This approach must be treated with caution; in general a perturbation ∆ might cause the spiral pole to split off the real axis and become part of a complex conjugate pair - the contour chosen would then be invalid. In this example, the perturbation required to change the spiral mode dynamics to this extent are far outside the range of interest, and the analysis gives meaningful results. The following are the elements of the A matrix that have been chosen as uncertain. Each element can be related to two aerodynamic derivatives; the first element in parentheses is the aerodynamic derivative that most strongly affects the corresponding A element.

(

)

(

)

A(1,1) ⇒ f C Lp , CNp

A(1, 2) ⇒ f (C Lr , CNr ) A(1, 3) ⇒ f C Lβ , CNβ

A(2,2 ) ⇒ f (CNr , C Lr )

( ) A(3, 3) ⇒ f (C )

A(2, 3) ⇒ f C Nβ ,C Lβ Yβ

The technique of Morton and McAfoos was used to create a ∆ -block corresponding to these perturbations.5 The initial weighting is chosen so that 100% variations in each element are given equal weighting, then the

5 American Institute of Aeronautics and Astronautics

weighting procedure in Section 3.2 is applied. Table I lists the combined percent uncertainty that is admissible in each of the matrix elements. Notice that the initial weighting allows perturbations of all elements of equal magnitude. After a series of reweightings, the algorithm concludes that the two most important elements are the A(1,1) element and the A(1, 3) element: if these elements are known within 30% accuracy, then all of the other aerodynamics need only be known to an accuracy of about 70% (row 5 of Table I). Physically these most important elements correspond to C Lp and C Lβ , both of which have a strong influence on the spiral stability of the aircraft. Throughout the analysis, the frequency corresponding to maximum µ is zero. Thus the spiral mode handling qualities requirement sets the sensitivity of this particular system. Table II shows the worst case perturbation vector for each weighting, as well as the parameter which determines µ , and thus is the next variable to be weighted.

uncertainty in the system, then the single-loop gain margin is computed as gm = 1+ δ m . Weighting ∆ A(1,1)

The algorithm and weighting procedure described above is applicable to the general class of problems described in Section 2.0. In applications, however, an interesting step is often that of creating the ∆ -block structure to be analyzed, based on commonly asked engineering questions. For instance, classical measures of robustness (gain margin and phase margin) are still common tools for assessing the stability of flight control systems. Although singular values and structured singular values are more general measures of robustness, their interpretation is sometimes difficult; the intuition that engineers have developed for phase and gain margins is still valuable. Furthermore, the concept of delay margin is a very important variant on phase margin -- transport delays are often a primary concern during flight system development, because they can change as the flight code grows.9,14 For these reasons, we would like to be able to incorporate gain changes, phase changes, and delays into our 'toolbox' of available perturbations to the flight system. After describing block structures for these types of perturbations, a useful way to present the resulting analysis for engineering interpretation is presented in Section 4.1. It is straight-forward to incorporate gain margin concepts into a robustness problem. The interconnection of Fig. 6 shows how we have chosen to do this using a multiplicative uncertainty in the feedback path. If the uncertain gain δ m is the only

∆ A(1,3)

∆ A(2,1)

∆ A(2,2)

∆ A(2,3)

1

44.30

44.30

44.30

44.30

44.30

44.30

2

32.67

49.02

49.02

49.02

49.02

49.02

3

23.42

52.70

52.70

52.70

52.70

52.70

4

27.21

61.24

40.83

61.24

61.24

61.24

5

30.60

68.86

30.60

68.86

68.86

68.86

6

21.81

73.61

32.71

73.61

73.61

73.61

Table I. Percent admissible variations for each weighting. Allowable perturbations in this table are unweighted. δA(1,1)

δA(1,2)

δA(1,3)

δA(2,1)

δA(2,2)

δA(2,3)

I6

0.8740*

0.5914

13.0885

-0.1700

-1.4952

0.0829

W(1,1)=0.6

0.6447*

0.6544

14.4812

-0.1881

-1.6543

0.0917

W(1,1)=0.36

0.4621

0.7036

15.5702*

-0.2023

-1.7787

0.0986

W(1,1)=0.36

0.5369

0.8175

12.0614*

-0.2350

-2.0668

0.1146

0.6038*

0.9193

9.0419

-0.2643

-2.3240

0.1288

0.4303*

0.9827

9.6650

-0.2825

-2.4842

0.1377

Weighting

4.0 PHASE AND DELAY AS REAL UNCERTAINTIES

∆ A(1,2)

Changes

W(3,3)=0.6

W(1,1)=0.36 W(3,3)=0.36

W(1,1)=0.216 W(3,3)=0.36

Table II. Worst case (unweighted) perturbation vector for each weighting. * indicates next channel to be weighted.

Although this direct relationship exists between gain margin and a real uncertain gain, no such relationship exists for phase margin. Instead, we will use an nth order Pade' approximation of a delay to incorporate phase lag into the system, and thus define phase as a real uncertainty. We know that the Laplace transform of a pure delay τ is: L {f (t − τ )}= e − τs F(s) . An nth order Pade' approximation provides a way to incorporate delay (or phase change at a specific

6 American Institute of Aeronautics and Astronautics

frequency) without subsequently changing the gain of the system: e − τs =

1 − p1 s + p2 s 2 K pn sn 1 + p1 s + p2 s 2 K pn sn

In order to incorporate τ into a real-µ perturbation structure, we start with a 1st order Pade' and then do some block-diagram manipulations to isolate τs from the rest of the block diagram. These manipulations are shown in Fig. 7, and the resulting block diagram can easily be incorporated into, for instance, a Simulink representation of the flight system ( s = jω is incorporated into M ( jω ) for analysis). To create a phase margin block diagram, we simply realize that the phase lag introduced by the Pade' approximation is a function of frequency. So to incorporate a pure phase lag which is independent of frequency we need a frequency-dependent time delay e − τ ( ω ) s , where τ (ω )s = jφ . This results in the block diagram in Fig. 8. In many application a first order Pade' may not adequately describe the delay or phase uncertainty. Specifically, in cases where more than 90 degrees of phase lag are required, the first order Pade' approximation will not be adequate. A second order or higher order Pade' approximation may be incorporated using an approach similar to that described above. For example, for a 2nd order Pade' we have: 2

e − τs

τ τ 2 s+ s 2 12 ≅ τ τ2 2 1+ s + s 2 12 1−

*

= 1+

k1 τs k τs + 1 * τs − a1 τs − a1

k1 = i 6

3

a1 = −3 + i 3 In this case the phase uncertainty in the robustness problem takes the form of a pair of repeated real uncertainties. Fig. 9 (a) shows the block diagram that describes the second order Pade' phase uncertainty. The matrices T and Q are: τs T =  0 

0 τs

0 1 a1 *  Q = 0 1 a1  1 1 

−k1 a1  * * − k1 a1  1 

The block diagram of Fig. 9 (b) is in the form of a standard repeated-real stability robustness problem. As before, it is simple to convert to delay uncertainty by

letting jφ = τs , and incorporating ω into M ( jω ) where necessary. Once gain and phase perturbations can be incorporated into a system block diagram and manipulated into the standard ∆ -block form, one can look at simultaneous guaranteed gain and phase margin variations for multi-loop systems using real µ . By separating gain and phase into separate real perturbations, several advantages over complex uncertainties are gained: First, one can scale gain and phase separately to reflect the relative importance of each. Also, using the algorithm described above, it is possible to consider only gain and phase variations in directions that are considered important. For instance, if one is primarily interested in the effects of phase lag, then phase lead perturbations can be excluded from the µ -search of Section 3.1. Another advantage of the block structures presented here is that time delay robustness and delay margin can now also be analyzed directly and unconservatively (see McRuer et al for a related approach).14 Finally, specific robustness questions can be answered directly in terms of standard engineering measures; this is the topic of Example II. 4.1 Example II: Robustly Guaranteed Phase Margins In this Section, we utilize the phase margin block structure introduced in the previous section, and also introduce the concept of a 'robustly guaranteed' margin, which is simply a phase margin or gain margin which is guaranteed to hold in the face of other plant parameter variations. An X-31 quasi-tailless example illustrates the concept. A current flight program at NASA Dryden is the X31 'quasi-tailless' configuration, in which the rudder channel is redesigned to simulate a configuration in which the tail has been removed. Flying the X-31 without a vertical tail is possible because of the thrustvectoring available on the aircraft; the benefits incurred (in drag, radar cross section, and high angle of attack maneuverability) by removing the tail make it an attractive option to pursue. The quasi-tailless configuration is a conservative approach to testing the performance without the tail, because the tail can be 'brought back' at any time, by switching back to the already-validated flight control laws. Additionally, one can phase in 'taillessness' incrementally, using changes in the control laws instead of hardware, and fly the aircraft with, for instance, 25%, 50%, 75% and 100% of the tail 'removed'. These quasi-tailless configurations are denoted in percent quasi-tailless, where 100% quasitailless refers to the tail completely 'removed'.

7 American Institute of Aeronautics and Astronautics

The aerodynamic derivatives of the X-31 will change significantly in the quasi-tailless configuration. The derivatives that are most important are those related to the size of the tail, and those related to the lateral directional control power of thrust vectoring: C Lβ , C Nβ , CNr , CLδ TV , and CNδ TV . These derivatives in turn factor almost directly into the state space system matrix elements A(1, 3) , A(2, 3) , A(2, 2) , B(1, 3) , and B(2,3) . Using the weighting procedure described in the Section 3.2 and also looking at SISO and MIMO gain and phase margins, we have identified phase margin in the thrust vectoring channel as the most direct measure of lateral stability in the quasi-tailless configuration (clearly, reducing the authority of the rudder increases the importance of thrust vectoring for stability). The question that we pose is the following: what is the phase margin in the TV channel for different percent uncertainties in the most important aerodynamics derivatives? In other words, what phase margin can we guarantee in the presence of 30%, 40%, and 50% uncertainty in the derivatives listed above? In Fig. 10 is shown the Simulink block diagram utilized for this example. The uncertainties in the state space model are incorporated into the robustness problem using the method of Morton and McAfoos.5 The five uncertainties in the plant, and the phase uncertainty in the TV channel, are apparent in this figure. In Fig. 11 are shown the SISO and MIMO (i.e. 'robust') phase margins in the TV channel as functions of the percent tail reduction. As expected, the SISO phase margin is reduced from the nominal value of 75 deg to about 40 degrees at 100% quasi-tailless. In the nominal (0% tailless) configuration, the MIMO robust and SISO phase margin are the same, which means that the phase margin is not strongly affected by the size variations in the derivatives that we have chosen. As the percentage tailless increases, the MIMO robust phase margin becomes much smaller than the SISO phase margin, because the latter does not take into account plant parameter variations at all. The conclusions one reaches from these plots are as follows: if a 45 degree phase margin is required for flight safety, the SISO phase margin suggests that one can fly to about 80% quasi-tailless. On the other hand, the MIMO robust phase margin indicates that, with 30% percent uncertainty in the uncertain elements, one should not fly above 63% quasi-tailless. With 40% and 50% uncertainty in the aerodynamic parameters, one is confident to fly only up to 57% and 52% quasi-tailless respectively. These results corroborate results at NASA Dryden obtained through more conventional engineering analysis of the quasi-tailless configuration.

5.0 IMPLEMENTATION OF ROBUSTNESS MEASURES DURING FLIGHT TEST Having discussed a technique for analyzing the generalized robustness problem, and given methods to apply the procedure to answer specific questions, we turn now to the problem of implementing multivariable robustness measures during flight test. We address two issues in this context. The first is the question of how to incorporate experimental data into the problem; here we treat only uncertainties that can be reflected external to the measured plant dynamics. The second issue is what specifically to display in the control room to provide easily interpreted information to the flight test engineers. The question of what to include in the deltablock structure is also of interest; for our example we choose simultaneous phase and gain margins, because single-loop gain and phase margins are currently used by NASA Dryden. Thus multivariable margins provide a smooth transition from currently used measures of system robustness. The procedure shown here is a relatively straightforward extension of the technique implemented by Burken, who applied complex structured singular values to open-loop transfer functions measured in flight.6 Procedures for determining open-loop transfer functions in flight are described by Burken, and will not be repeated here. It suffices to give a brief review of the motivation and background of the problem. During an envelope-expansion type flight test, a central goal is to determine whether the behavior of the system is enough like that predicted to allow the next flight condition to be confidently flown. One technique currently used by NASA Dryden to assess how well the overall system is behaving is in-flight phase and gain margin determination. Using either injected signals or pilot inputs, the system's open-loop transfer function is deduced using spectral analysis techniques. This experimental transfer function is then used to determine the actual gain and phase margins of the control system. As systems become more complicated and multivariable, it is of interest to generalize this procedure to a multivariable robustness evaluation procedure. Robustness can be defined in the general framework described above, but one must work out how to incorporate experimental data into the analysis procedure. The plant transfer function is represented by the frequency domain transfer function G ( jω ), and the controller by the transfer function K ( jω ) . We will assume that the controller transfer function is perfectly known and that an estimate of the plant transfer function

8 American Institute of Aeronautics and Astronautics

has been obtained during flight.6 Fig. 12 shows the TITO form for the resulting stability robustness problem: u  P11 r  = P21

P12  v  P22  y 

y = Gr ,

for which the transfer function from v to u can be expressed as follows:

[

M = P21G (I − P11G ) P12 + P22 −1

].

In this formulation the matrix P represents the part of the transfer function M that is perfectly known, (i.e. a manipulated version of the controller K ), while G is the estimated transfer function of the aircraft plus actuators and sensors (from measurable inputs to measurable outputs). Using this equation, one can translate G ( jω ) into M ( jω ) which is the matrix transfer function required for the robustness analysis. Once this experimental M ( jω ) is found, all of the analytical procedures described above can be applied. 5.1 Example III: Experimental Multivariable Gain and Phase Margins The Simulink block diagram of the linearized X-31 lateral-directional flight control system is presented in Fig. 13. We have inserted gain and phase uncertainties at the plant input, but any other set of uncertainties in the control systems is possible; uncertain delays, phases, and gains can be placed anywhere in the Simulink block diagram. We are again analyzing the quasi-tailless configuration; thus we are interested in determining from flight data the robustness of the aileron and thrust vectoring channels. Suppose that the requirements for gain and phase margin are defined as follows: Aileron

Rudder

TV

Gain Margin [dB]

3.0

0.8

3.0

Phase Margin [deg]

17.0

6.0

17.0

The gain and phase margins on the rudder channel have been intentionally reduced because this is the channel used to destabilize the aircraft and simulate the tailless condition. The other values are reasonable combined multivariable gain and phase margins based on simulation studies. The question is now: are the above values of gain and phase margin simultaneously guaranteed in a given actual flight condition? Gain and phase margins at a given point in the envelope are just

one of many aspects that must be verified before moving to the next point. Thus it is important to have a near-real time measure that can be quickly interpreted. A simple way to check robustness is through a properly weightedµ -plot. If we choose the weighting such that ∆ = 1 corresponds to the gain and phase boundaries shown above, then we know from Section 2.0 that stability is guaranteed within the given level of uncertainty if the real µ -plot is less than one (recall that for the real diagonal uncertainty we are interested in, ∆ = max δi -- thus if any of the i weighted ∆ -block parameters exceed there limits, ∆ will exceed 1). Fig. 14 is a plot of the lower and upper bounds of the real-µ calculated using an 'experimental' G ( jω ) from a nonlinear simulation, implemented in the framework described above. Note that either the lower bound or the upper bound provide an excellent measure of the actual proximity of the system to the minimum robustness level set by the weighting, and that the flight test engineer need only check the proximity of the plot to 1 to get a quick picture of the robustness situation. From the µ -plot we can see that µ is less then 1. This means that the requested gain and phase margins are fulfilled in this flight condition. (The µ -plot at low frequency, i.e. below 1 rad/sec is not reliable because of the frequency sweep used to obtain G ( jω ).) We can see that µ has a peak at about 3.5 rad/sec, which is the underdamped Dutch roll mode. 6.0 SUMMARY & CONCLUSIONS We have described a set of developments which allow us to apply recent multivariable robustness analysis measures to flight test engineering questions. The first such development was the extension of Dailey's algorithm for computing real µ , which provides important practical information about the robustness situation, to include repeated real uncertainties. We also illustrated the importance of weightings, and described a procedure for automatically reweighting that is tailored to sensitivity and robustness analysis, as opposed to design. In addition, we showed how to cast phase and delay uncertainty as real structured uncertainty, and how to use such uncertainties in ways which provide easily interpreted information about flight safety. Finally, we demonstrated the procedures using X-31 flight control laws, and demonstrated how to include flight test data into the analysis. Clearly useful analysis can be done using the structured singular value framework. The examples presented here invariably corroborated previous results obtained through conventional means, which require

9 American Institute of Aeronautics and Astronautics

more exhaustive approaches. More work is needed to address the plethora of flight safety, flight dynamics, implementation, and nonlinear issues which arise during flight control law validation, but the framework developed here shows promise due to its versatility and intuitive appeal.

[10]

Mukhopadhyay, V. and Newsom, J. R., "Application of Matrix Singular Value Properties for Evaluating Gain and Phase Margins of Multiloop Systems," AIAA-821574, August 1982.

[11]

Doyle J. C., "Analysis of Feedback Systems with Structured Uncertainties," IEE Proceedings, Vol. 129, Part D, No. 6, November 1982, pp. 242-250.

[12]

Paduano, J.D., and Downing D.R., “Sensitivity Analysis of Digital Flight Control Systems Using Singular-Value Concepts,” Journal of Guidance, Control ,and Dynamics, Vol. 12, No. 3., May 1984, p. 297.

[13]

Sideris, A, and Peña, R. S., "Fast Computation of the Multivariable Stability Margin for Real Interrelated Uncertain Parameters", Proc. American Control Conference, 1988, pp. 14831488.

[14]

McRuer, D. T., Myers, T. T., and Thompson, P. M., “Literal Singular-Value-Based Flight Control System Design Techniques,” Journal of Guidance, Control, and Dynamics, Vol. 12, No. 6, Nov-Dec 1989, pp. 913-919.

[15]

De Gaston, R.R.E. and Safonov, M., “Exact Calculation of the Multiloop Stability Margin,” IEEE Trans. on Automatic Control, Vol. 33, No. 2, pp. 156-171, February 1988.

[16]

Fan, M.K.H. and Tits, A.L., “Characterization and Efficient Computation of the Structured Singular Value,” IEEE Trans. on Automatic Control, Vol. AC-31, No. 8, pp. 734-743, August 1986.

[17]

Sanchez Pena, R.S. and Sideris, A., “A General Program to Compute the Multivariable Stability Margin for Systems with Parametric Uncertainty,” Proc. of the American Control Conference, Atlanta, Georgia, 1988.

ACKNOWLEDGMENTS This work was supported by NASA cooperative agreement NCC 2-4001, John Burken, technical monitor. Special thanks go to John Burken, Bob Clarke, and Joe Gera for helpful discussion, and to Lothar Juraschka and Pat Stoliker, for technical assistance with the nonlinear simulation and Simulink modeling of the X-31. REFERENCES [1]

Dailey R. L., "A New Algorithm for the Real Structured Singular Value”, Proc. American Control Conference, June 1990, p. 3036.

[2]

Young P. M., Newlin, M. P. Doyle, J. C. , "Let's Get Real", Electrical Engineering, 11681, California Institute of Technology, Pasadena, CA 91125.

[3]

Balas, G. J., Doyle, J. C., Glover K., Packard A., and Smith R., µ -Analysis and Synthesis Toolbox, MuSyn, Inc., April, 1991.

[4]

Military Standard, MIL-F-8785C, 5 November 1980

[5]

Morton, B. G. and McAfoos, R. M., “A µ test for robustness analysis of a real parameter variation problem”, Proc. American Control Conference, 1985, pp. 135-138, Boston.

[6]

Burken, J. J., "Flight-Determined Stability Analysis of Multiple-Input-Multiple-Output Control Systems," NASA TM-4416, November 1992.

[7]

Young, P. M. and Doyle, J. C. (1990), “Computation of µ with real and complex uncertainties.” Proc. of the 29th IEEE Conf. on Decision and Control, pp.1230-1235.

[8]

Doyle, J. C., “Structured uncertainty in control system design,” IEEE Conf. on Decision and Control, Ft. Lauderdale, 1985.

[9]

McRuer, D. T., Ashkenas, I., and Graham, D., Aircraft Dynamics and Automatic Control, Princeton University Press, 1973.

10 American Institute of Aeronautics and Astronautics