Aerospace control system design with robust fixed order compensators

Aerospace Science and Technology, 1997, no 6, 391-404

Aerospace* Control System Design with Robust Fixed Order Compensators H. Buschek *, A. J. Calise ** (*) Bodenseewerk Gertitetechnik GmbH, P. 0. Box 10 11 53, D-88641 if,fberlingen, German)). (“‘) Georgia Institute of Technology, AeT+ospaceEngineering, Atlanta, GA 30332-01~0, USA. Manuscript

Buschek H., Calise A. J., Aerospace

Abstract

received

September

Science and Technology,

27, 1996, revised version January 21, 1997.

1997, no 6, 391-404.

A novel approach to synthesize robust controllers of constrained dimensions is presented. A homotopy algorithm is used to solve an H, control problem in which the dimension of the controller is fixed a prioui. This technique is incorporated into a mixed real/complex b-synthesis procedure resulting in controllers which decrease conservatism and reduce controller complexity. The capabilities of the new method are demonstrated on an attitude controller for a satellite example and on a flight control system for a hypersonic vehicle model. Several fixed order mixed ,Y controllers are designed and the results illustrate the feasibility of the developed technique.

Keywords: Aerospace control - Robust control - Fixed order compensators

- ReaUcomplex

p-synthesis

- Hypersonics

I - INTRODUCTION Over the past decade modern robust control theory has revolutionized multivariable controller design. H, and b-synthesis techniques consider multiple uncertainty sources at different locations in the system, external disturbances, as well as performance specifications when designing for robust performance [ l]-[3]. While these techniques greatly simplify multivariable controller synthesis, they are accompanied by the inherent disadvantage that the resulting compensator is of the same o;der as the system it is designed for, the so-called generalized plant. Frequency dependent weights have to be included in the design framework in order to achieve

the desired performance characteristics and to account for the structure in the uncertainty. Thus, the order of the generalized plant is increased resulting in high order controllers. When implemented, large order controllers can create time delays which may be undesirable. One solution to this problem is to use model order reduction on the controller realization. This technique, though, does not consider the properties of the closed-loop system when reducing the order of the controller and therefore robustness properties are not guaranteed. Another approach to avoid controllers of high dimension is to constrain the order of the controller a priovi in the design process. The first application of fixed order dynamic compensation to robust control

This research has been supported by NASA’s Langley Research Center under grant NAG-l-1451 and by the Army grant DAAH 04-93-6-002. A preliminary version of MATLABT” software to perform the D,G-K Iteration procedure Dr. P. M. Young, Laboratory for Information and Decision Systems, Massachusetts Institute of Technology. Aerospace

Science and Technology,

1270.9638,

97/06/O

Elsevier,

Paris

Research Office under was made available by

392

was developed in Ref. (41 with the introduction of the optimal projection equations. This approach involves the solution of two modified Riccati equations and two modified Lyapunov equations, all coupled through an optimal projection operator. The optimal projection approach is extended in Ref. [S] to a mixed HzIH, problem where an Hz overbound on performance is optimized while an H, bound is enforced simultaneously. It was shown in Ref. [6] that if either a controller or observer canonical form description is imposed on the compensator structure, the number of free parameters is reduced to its minimal number. Augmenting the compensator to the system, the problem is converted to a static gain output feedback formulation. In Ref. [7] a differential game approach was employed to derive the necessary conditions for such a formulation and a conjugate gradient algorithm was used for the solution. Recently, homotopy methods have been utilized to synthesize fixed order Hz, H,, and mixed Hz/H, controllers [8]. If multiple uncertainty sources are present in the system, H, controllers are conservative since they do not account for the structure in the uncertainty. b-synthesis alleviates this problem but is still conservative in that only complex representations of uncertainty models are considered. For real parameter uncertainty this representation can be conservative and may unnecessarily sacrifice robust performance. Therefore, developing methods of design for variations in real parameters is currently an active research area. One of the first approaches to account for both real parameter variations and unstructured uncertainty is given in Ref. [9] where robust stabilizability conditions and a controller synthesis procedure are derived for SISO systems. A method to introduce real parameter uncertainty as additional noise inputs and additional weights is presented in Ref. [lo]. The approach is based on the mixed Hz/H, formulation in Ref. [5] and accounts for the parameter uncertainty via the H, bound. The notion of positive realness and a bilinear sector transform are used in Refs. [l l] and [12] to approach the problem of real parameter uncertainty. The D-scales of complex p-synthesis are replaced by the so-called Generalized Popov Multiplier matrix capturing both real and complex uncertainty effects. The problem is reformulated as a Bilinear Matrix Inequality Feasibility problem in Ref. [ 131 for which no global solution presently exists. Another approach using absolute stability theory to introduce real parameter uncertainty is described in Refs. [14] and [IS]. Based on the mixed HdHa framework in Ref. [5], robust controllers are designed by optimizing an Hz overbound while an analysis test based on a Popov stability multiplier is enforced to account for real uncertainty. Extensions towards the inclusion of monotonic and odd monotonic nonlinearities in order to reduce conservatism even further are described in Ref. [ 141. Another approach to introduce real parameter uncertainty employs a

H. Buschek, A. J. Calise

differential game formulation where uncertain system parameters together with external disturbances and initial states try to maximize a performance index [ 161. The resulting controller provides internal stability and robustness towards real parametric uncertainty in a given set. In Ref. [17] the so-called G-scales matrices are introduced in addition to the D-scales of complex p-synthesis to refine the upper bound on the structured singular value p in case of mixed real/complex uncertainties. A Linear Matrix Inequality (LMI) approach is chosen in Ref. [ 181 to compute this upper bound while in Ref. [19] standard optimization techniques are employed in conjunction with Branch and Bound schemes. More recent results extend the analysis to the synthesis problem by using a D,G-K iteration [20]. In this paper, a new approach to p-synthesis is presented by combining a fixed order controller design technique with the real/complex p-synthesis procedure from Ref. [20]. The resulting p controllers possess the inherent advantages that they are of fixed order and are robust to mixed real/complex uncertainties. The problem formulation is given in Section 2 together with some details on the numerical solution procedure. In Section 3, some key features of the mixed real/complex p-synthesis method are presented, and the integration of the fixed order algorithm is described. An attitude controller for a flexible satellite in Section 4 and a flight control system for a hypersonic vehicle in Section 5 illustrate the capabilities of the method. Conclusions are presented in Section 6.

II - FIXED

ORDER

II.1 - Problem

CONTROLLER

DESIGN

formulation

For a standard control problem, the generalized plant is given by 2 = Az + &w z = ClX y =

Czx

+ Bzu

+ Dlyu +

D21w

(1) (4

+

D22u

(3)

where x E R” is the state vector, w E R”” is the disturbance vector, u E ‘R”” is the control vector, 23 E 7-z”” is the performance vector, and y E RnY is the observation vector. It is assumed that l (A; Br, Cr) is stabilizable and detectable; l (A, B2; C2) is stabilizable and detectable; l DQ has full column rank; l Dzl has full row rank.

Aerospace Control

System Design with Robust Fixed Order Compensators

In controller canonical form [6], the compensator is defined as

kc = PO& + NO?&- NOy

(4)

UC = -Px,

(5)

u = -Hz,

The augmented system is defined by:

5X-i&, P+ +[-,‘t’,,,]w

(6)

r

0

1

...

NO

-NO&

I

'

-

=A%+BIw+B~~

(12)

z = [Cl 012 + [II12 = c1n:+ll12u

O] G (13)

g = [O I] Ir: = c, z fj,=-

diag {P,“, . . . : Pi’,)

0

B2

+

where x, E Rnc and u, E RnY. P and H are free-parameter matrices, and P” and No are fixed matrices of zeros and ones determined by the choice of controllability indices vi as follows: P” = block

393

(14)

g=-Gy

01

Eqs. (12)-(15) define a static gain output feedback problem where the compensator is represented by a minimal number of free parameters in the design matrix, G. The closed-loop system is given by

No = block

diag { [0 . . .O

l]TX,,, }

ci = (iI - &GC2)

i = 1, . . . , ny

z + B,w

=iiiE+ld?w

(16)

(9)

The controllability following condition:

indices

ci vi = nc

vi i=

must l,...,ny

satisfy

z=(C~-&~G~~)Z=&

the

(10)

Figure 1 shows the structure of such a controller. With this formulation, the compensator states can be absorbed into the generalized plant. Let

In the H, problem, an overbound y on loop transfer function performance outputs z T,,

(17)

the objective is to minimize the m-norm of the closedfrom disturbance inputs w to given by

= @I

- A)-%

(18)

In Ref. [7], a differential game formulation together with a worst case disturbance model was used to derive the fixed order H, optimization problem g&

{J,(G)

= tr{Qm~~T}}

(19)

where G, = {G E R(TLu+ny)x7Lc : a is stable and llTzwllm < r} and tr denotes the trace of a matrix. &CCis the positive semidefinite solution of

jTQ, Fig.

1. - Compensator

1997, no 6

in controller

canonical

form.

+ Qmii + c’c

+

r-2Qm&jTQm

=

o.

GQ

H. Buschek, A. J. Calise

394

In order to obtain the H,-optimal Lagrangian is defined as

compensator, the

C(Qm,L, G) = tr{Q&BT + (jTQ, + Q&i + Y2Qocl~8’Q&}

+ c-?~c!? (21)

where L represents the Lagrangian multiplier. Matrix gradients are taken to determine the first order necessary conditions for an H,-optimal fixed order controller:

dC

= (ii + r-28i?TQm)L

8QCC

0 (24 dL

z

=

AT&cc+

Q,A

+ y-“QmBBTQw

+ tiTC = 0

(23)

A conjugate gradient method was used in Ref. [7] to solve the fixed order H, problem. However, a shortcoming of conjugate gradient algorithms is that convergence slows down near the optimum. Also, a starting guess is required for the compensator gain matrix which stabilizes the the closed-loop system. In this study, a homotopy method is employed to design fixed order H, controllers. This method greatly simplifies the problem of finding an initial stabilizing controller. Additionally, the prediction/correction scheme used in a homotopy algorithm allows for a naturally automated procedure to synthesize fixed order controllers. II.2 - Homotopy

algorithm

In this study, a homotopy method is employed to calculate the fixed order controller design. For details regarding the use of homotopy methods, the reader should consult Refs. [21] and [22]. The particular equations arising from the implementation of the fixed order formulation in the homotopy framework are provided in Ref. [Sj. Homotopy methods begin with a “simplified version” of the original problem which has a known or easily calculated solution. This starting guess is then

gradually deformed until the solution of the original problem is obtained. In the homotopy algorithm employed in this paper, an initial guess for the compensator gain matrix Go is obtained by a full order, low authority Hz design. It has been observed numerically that order reduction techniques - although not always reliable - work best for low authority LQG controllers [21]. However, if the open-loop system is unstable, order reduction techniques are not reliable even for low authority H2 designs. Thus, it may be required to shift individual poles of the system into the left half plane in order to obtain a reducible H2 controller. This controller is then reduced to the desired order using an order reduction scheme with balancing transformation [23] and subsequently transformed into controller canonical form [24]. The philosophy of imposing a canonical form on the compensator structure minimizes the number of free parameters but can also lead to numerical ill-conditioning of the problem [21]. A balancing transformation [23] which does not affect the controller characteristics relaxes the strict structure in the P” and No matrices in Eqs. (7)-(9) and improves the conditioning of the problem considerably. In the Hz problem, the homotopy transforms the gain Go of the low control authority H2 design for a stable system to the gain Gz of a full authority Hz design for a possibly unstable system. This is done by gradually deforming the weights on control cost and measurement noise as well as the system A-matrix while proceeding along the homotopy path until the desired problem formulation is recovered. The designed fixed order Hz controller can serve as starting guess in the H, problem. Accordingly, a second homotopy is appended to perform the H, design. The upper bound for llTrW [loo, y, is reduced from an initial value (greater than the m-norm of the H2 design) towards its minimal value for which a controller exists such that llTtw lloo < y. The entire procedure is illustrated in Figure 2. For an Hz design, the procedure is stopped after the first homotopy path Go --f G2, while a second homotopy reduces y in an H, design. In principle, G2 + G, it is possible to use a direct homotopy from GO to G, while simultaneously deforming the weights and y. However, the transition low-to-full authority and/or stable-to-unstable open-loop system always has to be completed in order to obtain the desired plant, whereas the minimum value of y that can be achieved is usually not known beforehand. Thus, the proposed two step procedure is more feasible for practical purposes. III - A NEW APPROACH

TO /L-SYNTHESIS

HI.1 - Mixed real/complex

p-synthesis

An in depth discussion of mixed real/complex p analysis and synthesis using D,G-K iteration is Aerospace

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Aerospace Control System Design with Robust Fixed Order Compensators

395

where ,LLis the structured singular value, KS is the set of all real-rational, proper controllers that nominally stabilize P, and M(P, K) is the linear fractional transformation of the plant P and the controller K: M(P, K) = P,, + Pz,K(I

- PYuK)-lPYw.

(27)

PZW is the plant transfer function from disturbance inputs w to performance outputs Z, P,, is the plant transfer function from control inputs u to performance outputs Z, etc. (see Fig. 3). Unfortunately, the structured singular value cannot be calculated directly. In order to formulate an upper bound on p(M), two sets of block diagonal frequency dependent scaling matrices D and G are defined: D = { block diag WL dmc

.> QrL,+m,,

4

I, m,+n,+mc)

lkm,+m,+l

) .

. ;

:

Di E CkzXkz, det(Di)

f 0, di E C, di # 0} (as)

G ={ block diag (Gl,...jGmy;Oli7,LI+1,. Fig. 2. ~ Homotopy controllers.

procedure

for

designing

fixed

order

H,

presenred in Refs. [17], [19] and [20]. Hence, only some key features are reviewed in this paper. For a more thorough description of the developed methodology the reader is referred to Ref. [25]. Consider the robust performance problem depicted in Figure 3. The definition of the structured singular value ~1depends upon the underlying block structure of the uncertainties. The following types of uncertainties are considered: possibly repeated real scalars S’, possibly repeated complex scalars SC,and full complex blocks AC. The set of allowable perturbations can be defined as

. ~h,~+,,+,,)

Gi = G; E C”zx”‘}

(2%

where 0 denotes a zero matrix and G* the complex conjugate transpose of G. In addition to the D-scales of complex h-synthesis, the G-scales take into account the phase information available for real uncertainites permitting a refinement of the upper bound on ,LLin the mixed case. The refined upper bound on b is then given as [ 171 I*(M)

5 inf inf DED OER GtG 8>0

D”;-’

e

- jG)

(I+

G’)-‘)

L

__.--______-.---___-----~~~~~~~.

The non-negative integers m,, m,, and mc specify the number of uncertainty blocks of each type while the positive integers ki, i = 1; . j m, + m, + mc, define the dimensions of each block (i.e., the multiplicity of real or complex scalars should they be repeated, or the size of a full complex block, respectively). In p-synthesis, it is desired to design a controller K achieving

Fig. 3. ~ Framework 1997, no 6

:

for robust

control

design.

< I}

d

(30)

H. Buschek, A. J. Calise

396

where a(.) denotes the maximum singular value. /? represents the upper bound on the structured singular value and is obtained by an infimization over D, G, and p subject to the inequality constraint a(. . .) 5 1. The p-analysis problem in Eq. (30) can be combined with Eq. (26) to form the mixed p-synthesis problem: inf {P(W) : l? < l} inf sup inf 3(W)GR KEKs wee D(o;)EP G(U)EG P(w)>0

(31)

where

D(w)M(p,

IT=6 ((

K)D-l(w) PO-4

(I+c’(w))-“).

_ jG(w)

(32)

As in complex p-synthesis, the problem of finding optimal D(w), G(w) and p(w) for a given controller K is quasi-convex. Real-rational transfer matrices D(s) and G(s) are fitted to D(w) and jG(w), and augmented to the plant in such a way that the resulting interconnection of D(s), G(s), and M(s) is stable. This involves state space factorizations as described in Ref. [20]. Holding D, G and ,L?fixed, the problem of finding K is reduced to a standard H, problem, which is convex in the full order case. The overall procedure, however, is not convex. This leads to an -iterative approach denoted D,G-K iteration [20] where alternately optimal full order H, controllers for constant scaling matrices and optimal scalings for a constant controller are designed until achievable robust performance cannot be improved anymore. III.2 - Designing mixed p controllers

of fixed order

As shown in the previous section, H, controller design is a subproblem in mixed b-synthesis using D, G-K iteration. The augmentation of the transfer matrices D(s) and G(s) to the generalized plant increases the dimension of the problem and subsequently the dimension of the full order controller. By replacing the full order H, controller design in the D,G-K iteration procedure with the fixed order technique described earlier, p controllers of significantly lower order can be synthesized. A critical step in the fixed order design technique is obtaining the initial guess for the fixed order compensator in order to start the homotopy procedure. Since a full order mixed p controller is usually designed first to determine the lowest possible upper bound on mixed ~1which can be achieved, an attempt can be made to reduce this full order design to the desired order using an order reduction scheme. If this reduced order controller stabilizes the closedloop system, it is used as starting guess for the fixed

order algorithm. Since the reduced order compensator already represents an H, design, the homotopy procedure is entered at the position of “Gz” in Figure 2 and the algorithm is used to improve over the reduced order controller. However, the D- and G-scales have been optimized for a full order design and may not be representative for a lower order controller. This may result in a possibly conservative design and additional D,G-K iterations with fixed order controllers may need to be performed to obtain the sub-optimal design. If order reduction of the full order mixed p controller results in a compensator which is not internally stabilizing, the entire fixed order design procedure beginning with an Hz compensator as shown in Figure 2 needs to be performed. The Hz design can be obtained either for the augmented system from the full order design which contains D- and Gscales, or for the original generalized plant. In the first iteration, the homotopy algorithm starts with the H2 controller as initial guess. In subsequent D,G-K iterations, the previous fixed order controller is used to start the code. An alternative to starting with an Hz controller (when order reduction of the b design to the desired order fails) is to reduce the b controller to a higher order for which the reduced controller is internally stabilizing. As an intermediate result, this reduced p controller serves as a starting guess for a fixed order design as described above. Once the fixed order design is optimized, an attempt can be made to further reduce this fixed order controller to the desired order or, if necessary, to another intermediate higher order design. Similarly, it can happen that order reduction succeeds in providing an internally stabilizing controller which is of lower order than desired. This controller can also serve as starting guess for a higher fixed order design. Additional states can be augmented in order to achieve the desired controller order while a zero block at the corresponding location in the output matrix maintains the transfer function characteristics of the initial stabilizing lower order controller. Another possibility to obtain a low order starting guess is to design a p controller with constant Dscales. This provides initial information about the uncertainty structure without increasing the number of states over a full order H, design. Obviously, a variety of different approaches can be used to start the fixed order mixed p procedure. In the ideal case it is expected that all approaches achieve the same level of robust performance, no matter which starting guess procedure for the fixed order D,G-K iteration is used. However, since fixed order controller design has not been proven to be convex, different initial guesses may result in different local optima. (In the context of this paper, the phrase “fixed order” always refers to an order smaller than the full order case.) Also, numerical difficulties associated with the homotopy algorithm may prevent a design which is close to the optimum, especially when multiple fixed Aerospace

Science

and Technology

Aerospace Control

397

System Design with Robust Fixed Order Compensators

order D,G-K iterations are performed. Thus, starting with a reduced order mixed p design is preferred since it has been observed that only a few additional iterations are required to obtain a sub-optimal fixed order b controller. In case of an Hz design based starting guess, using the augmented plant can have advantages since less D,G-K iterations are required than when starting with the generalized plant and no a priori available information on the uncertainties. Unfortunately, there is no guarantee that the globally optimal solution can be found due to the non-convexity of the overall problem. However, just like in complex p-synthesis it has been observed that the employed iterative approach yields excellent results for most practical purposes. IV - ATTITUDE CONTROLLER A FLEXIBLE SATELLITE

variables are y = [zi ii J z1 &IT. The purpose of the attitude control system is to maintain satellite pointing accuracy during a station keeping maneuver. The system specifications considered in this design are 1) to maintain a 0.02 deg (3.5 . 10W4 rad ) pointing accuracy for a 1.0 in-lbf step disturbance torque at the plant input, and 2) to maintain this pointing accuracy in the presence of a 25 % variation in the structural frequency due to uncertainty in the stiffness matrix. The uncertainty in the spring stiffness k is represented by a scalar perturbation to the nominal model. This perturbation can be rearranged to obtain input and output quantities da and ea. The matrices Ba, CA, and DA can be chosen from the uncertain model Li= Az + Bu + Bada ea = C,x + Dau

FOR

A simple example to demonstrate the features of the fixed order mixed p algorithm is a robust attitude control system design for a flexible spacecraft which is taken from Ref. [26]. Focusing on the yaw axis only, the model dynamics are

Mz+Dx+Kz=nu

(33)

where z = [z~zZ]~ and z1 is the yaw angle displacement in radians, z2 is the modal displacement used to represent the flexible dynamics, and u is the control torque in inch-pounds (in-lbf). The coefficient matrices are

(38)

Y= Cz+Du da = @.a

If a scaling has been applied so that the values of a range between -1 and 1, Eq. (38) represents the set of all possible models. It was shown in Ref. [26] that for this example a 25% variation in the structural frequency is represented by a stiffness variation of 0.053296 < k- 5 0.154953 where k is the only non-zero element m Ic. Therefore, the nominal value of k for robust control design was selected to be ka = 0.104124 and the uncertainty weight is w, = 0.050829

M==

K=

[ 2:: 0 [0

24F.1]

zJ = [:

O.O9(i696]

which normalizes the uncertainty, i.e. 61; E [- 1, 11. According to the framework in Eq. 38, the uncertainty matrices are

O.O,(:,,,]

I3 = [:

I-

(34)

Augmenting the model by the integral of z1 and rewriting it in first order form leads to the system .k=Az+Bu

(35)

where z = [z’ iT / ~1 dtlT and

02X2 A=

-M-lK

[1 01

I2

-J!&l2) 01x2

Ba = [0

0

ml2

c* = [O 1 0

mz2

0

O]

(40)

(41) (42)

n = SI,

(43)

where ml2 and rn22 are the (1,2) and (2,2) elements of -M-l, respectively. The weights reflecting the performance specifications are selected to be 02x1

O2x1

0

(36)

WT = 1 W, = diag [400

I

(37)

where I denotes the identity matrix and 0 the zero matrix of appropriate dimensions. The measured 1997. no 6

OIT

Dn=O

(44) 1 201

w, = 0.1 B=

(39)

w, = 1oP .1s

(45) (46)

(47) where W, is the weight on the torque disturbance at the plant input, IV, is the performance weight on the measured variables y, W, is the weight on the control u, and W, is the sensor noise on the measured

H. Buschek, A. J. Calise

398

++-

Fig. 4. - Generalized

variables y.

plant

for satellite

Table 1. - Peak values of upper fixed order p controllers, satellite of states for full order controllers).

: ‘.“” ‘..,.

example.

I.

The generalized plant is depicted in

p bounds for full, reduced, and example (in parentheses: number

Figure 4.

In a first design step, both full order complex and mixed ,u controllers were synthesized. In the complex design the uncertainty in the stiffness was considered frequency dependent whereas mixed bsynthesis accounted for the real nature of the stiffness variation. The peak value of the (scaled) upper p bound of the complex design was p = 0.923. The mixed p controller could improve over this result and achieved p = 0.768. Controller complexity, however, increased from 11 states of the complex controller to 17 states of the mixed design. Using various order reduction schemes, the lowest order for which a reduced mixed p controller was internally stabilizing was seven. The upper b value of the reduced order design was p = 1.091 indicating a loss in robust performance. Using the reduced controller as starting guess for the fixed order design, full order performance could be fully recovered and the fixed 7tk order controller achieved also b = 0.768. Since even this 7th order controller could not be reduced any further without loosing internal stability, an alternative was to design a complex p controller with constant D-scales. This resulted in a 5th order complex p controller achieving p = 0.945. Combining this controller with the augmented generalized plant from the full order mixed p design containing D- and G-scales as starting guess for the fixed order procedure, a fixed 5th order mixed p controller was obtained which could also recover the robust performance level of the full order mixed b design by achieving p = 0.776. The p values of all designs are summarized in Table 1. Figure 5 illustrates the the upper p bounds of the closedloop systems with the full and fixed order controllers implemented. The improvement of the mixed over the complex p designs is obvious, and it can be observed how the fixed order controllers recover the full order performance. Figure 6 shows the yaw angle displacement for a 1.0 in-lbf step disturbance torque for the full order complex p design and the fixed 5th order mixed p design. Both designs satisfy specifications 1) and 2). The mixed b controller, however, reduces the amplitude of the displacement and achieves a slightly shorter settling time indicating improved nominal

full

order

complex

(11 states)

10"

16 frequency

Fig.

5. - Upper

p bounds

[radk]

for flexible

satellite

example.

6 Bx

-0.510

20

40

60

Fig. 6. - Yaw angle displacement 5th order mixed p designs, 525%

80 time [s]

100

120

for full order complex variation in structural

140

$0

p and fixed frequency.

performance while maintaining the same level of robustness as the complex p design. The results of the flexible satellite example illustrate the benefits of the controller design procedure. The mixed p controllers reduce conservatism in robust control design but tend to be of relatively large order. The fixed 7th order controller synthesized with the new method is able to improve over a corresponding reduced order design. Moreover, a fixed 5th order Aerospace

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Aerospace Control System Design with Robust Fixed Order Compensators

399

controller could be designed whereas a reduced 5th order controller was not internally stabilizing.

V - FLIGHT CONTROL SYSTEM A HYPERSONIC VEHICLE

FOR Fig.

Hypersonic atmospheric flight as it is envisioned for future single-stage-to-orbit vehicles (SSTO) will require a synthesis of all major aerospace disciplines to an extend never experienced before. Significant coupling effects between aerodynamic, structural, and propulsive characteristics are to be expected when the aerospace vehicle transitions from subsonic via supersonic to hypersonic speeds up to Mach 25. As a consequence, performance, guidance, and control properties will be affected considerably. Designing a control system for this type of vehicle becomes a multidisciplinary effort in which a variety of interactions between individual subsystems and components has to be taken into account. In hypersonic airbreathing propulsion, the entire vehicle aerodynamic configuration must be considered as part of the propulsion system. Before reaching the inlet the external flow will be compressed along the forebody, and after leaving the combustor the aftbody is utilized as an external nozzle. Changing the angle of attack for control purposes alters these flow fields and results in variations in thrust vector magnitude and direction influencing stability and control of the vehicle. The need for a low structural weight mass fraction introduces relatively low structural vibration frequencies. Significant elastic-rigid body mode interactions are likely to occur imposing additional requirements on the flight control system. Additionally, bending of the forebody influences the flow conditions at the inlet which propagates through the engine and together with the elastic deformation of the aftbody again affects the thrust vector. Determination of the elastic mode shapes will be highly uncertain. This section illustrates a control study for a typical hypersonic vehicle employing the developed controller synthesis procedure. V.l - Hypersonic

vehicle model

The hypersonic vehicle model used in this study is the so-called Winged-Cone Configuration described in Ref. [27]. Main characteristics of this vehicle are an axisymmetric conical forebody, a cylindrical engine nacelle section with engine modules all around the body, and a cone frustrum engine nozzle section, see Figure 7. In order to carry out control studies, a five state linear model of the longitudinal dynamics is used representing flight conditions for an accelerated flight through Mach 8 at approximately 86000 ft. State and control variables are: 1997, no 6

7. - Hypersonic

x=

Winged-Cone

-If-

velocity (ftisec)

a

angle of attack (deg)

4 Q

=

pitch rate (deg/sec)

(48)

pitch attitude (deg)

-hLI=

Accelerator

altitude (ft) symmetric elevon (deg) fuel equivalence ratio (-)

=

1

(49)

The fuel equivalence ratio is defined as 77=

mi mazr

(50)

stoichiometric

where tif and r&,, are fuel and air mass flow rates, respectively. Since the linear model is defined for a trimmed, accelerated flight condition, it should be noted that the state and control variables are perturbation quantities representing deviations from a climbing and accelerating flight condition. This model represents pure rigid body dynamics and therefore does not account for any aeroelastic effects. Also, the propulsion system model used for the Winged-Cone Configuration does not include sensitivity to angle of attack variations. The interconnection structure for controller design is shown in Figure 8. The design is carried out as a velocity and altitude command tracking system. Velocity error, altitude error, pitch rate and pitch attitude are the measurements fed back to the controller. Control actuator dynamics are represented by first order filters with 30 rad/sec bandwidth for elevon and 100 rad/sec bandwidth for fuel equivalence ratio. Dryden turbulence models are used to introduce atmospheric disturbances” The turbulence spectrum is defined by the weights F, for longitudinal and F, for vertical wind gusts [28]. In order to minimize control effort and to reduce the sensitivity of the control response to atmospheric disturbances weights are imposed on control deflections and actuator rates: w,, = 3;

w,,

= 3

(51)

W& = 5,

ws, = 1.

(52)

H. Buschek, A. J. Calise

400

dedot

detadot

I I

I ’

vc

H

deta

wvc

commands hc-

Whc

-

he

44 Wnoise

noise

K 1

I

controller

Fig, 8. - Interconnection

structure

for

controller

design.

To guarantee natural responses in velocity and altitude, command tracking is pursued using a model following approach. The velocity command is passed through a first order filter, where the time constant is selected to result in a 5% settling time of approximately 60 seconds:

Since the velocity signal has to be integrated one more time to obtain altitude, the altitude command is shaped by a second order transfer function f-4 =

WV = 3,

wh = 1.

(55)

A measurement noise weight Wnoise = 1o-41‘$

w,,, = *.

whc

transient response consistent with the vehicle’s natural dynamics:

59 + 2&&s + w;

(54)

where a damping ratio of 5 = 0.8 and a desired settling time of 80 seconds result in a natural frequency of W n = 0.0468radlsec. With this approach constant weights on the error signals velocity and altitude suffice to ensure a small steady state error, and a

is employed for all the fedback variables. The noise model is not intended to represent realistic sensor data but has to be included because the application of II, theory to output feedback problems requires that all measurement signals be corrupted by noise. Aerospace vehicles in hypersonic flight regimes will typically utilize scramjet propulsion systems which are highly integrated into the airframe. This results in an increased sensitivity to variations in angle of attack [29]. The most important impact of these propulsive perturbations is on the pitching moment leading to significant control surface deflections to stabilize the vehicle [30]. This phenomenon is addressed Aerospace

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Aerospace Control System Design with Robust Fixed Order Compensators

as parametric uncertainty in the pitching moment sensitivity to angle of attack variations, cMa. This uncertainty can be modeled in accordance with the framework in Eq. (38) and is labeled “uncertainty 1” in Figure 8. Significant coupling between the elastic and rigid body modes is expected for this vehicle type which has to be considered in the design of a robust flight control system. Moreover, fuselage bending affects propulsion system performance via perturbed inflow conditions which in turn influences the rigid body flight dynamics or even excites the elastic modes [3 11. Since the model used in this study comprises only rigid body dynamics and no aeroelastic information on the Winged-Cone Accelerator was available, a simple yet concise method had to be developed to introduce aeroelastic effects as uncertainty into the rigid body behavior. Base:d on elementary theory in structural dynamics [32], a second order transfer function was derived exhibiting elastic deformations of a long, slender, uniform body caused by a force P(s) suddenly applied at a certain location of the body, z+:

* B&; s) =

(fl;(qJ

(9 + 25;wis + w”) Mi p(s)

(57)

Bi(z, s) is the angle by which the deformed body is deflected from its original shape due to the ith elastic mode, 4;(x) is the mode shape function of the ith elastic mode, & and wi are the corresponding damping ratio and natural frequency, respectively, and Mi is the modal mass (for more details the reader is referred to [32] or any other book on structural dynamics). In the case of the Winged-Cone Vehicle structural excitement is caused by lift increments due to elevon deflections which are represented by P(s). The first three fuselage bending modes are considered and are given by WI = 2.95 Hz, w2 = 5.72 Hz, and w3 = 7.74 Hz. The damping ratio for structural vibration modes is typically small and was chosen to be 5; = 0.01 for all three frequencies. The body deflection angle 6’(~! s) can also be interpreted as change in the angle of attack &X(X, s) due to elastic deformation. As mentioned above, angle of attack perturbations affect the rigid body dynamics via the propulsion system, thus the transfer function given in Eq. (57) is evaluated at the location of the inlet entrance. The frequency response of the flexible mode function in Eq. (57) together with the rigid body response is illustrated in Figure 9. The resulting estimation of the angle of attack variation due to flexible body motion excited by elevon deflections is then fed as uncertainty into the rigid body model. The uncertainty model is labeled “uncertainty 2” in Figure 8. A more thorough description of this model is provided in Ref. [33]. 1997, no 6

401

10" frequency

Fig. 9. - Angle of attack responses

to elevon

input.

In addition to uncertainty models representing typical hypersonic effects for an integrated flight control design, uncertainty in control effectiveness is also accounted for. Both elevon and fuel equivalence ratio control channels are considered. Variations in control effectiveness are modeled as parametric uncertainties in pitching moment sensitivity to elevon deflections, CM&e, and in thrust sensitivity to fuel flow variations, CT&,. These parameter uncertainties are also modeled according to the framework in Eq. (38) and are labeled “uncertainties 3, 4” in Figure 8. V.2 - Analysis of achievable robust performance In order to design controllers for robust performance, the uncertainty models are implemented in the general synthesis framework. For propulsion system perturbations, 25% uncertainty in C~~ was introduced. The effects of aeroelastic deformations are modeled according to the transfer function in Eq. (57). Uncertainty levels in control effectiveness were chosen to be 10% in both CM.& and cTSv. Hence, three real parameter uncertainties (ScnJcy,SC~MS~, ScTGrl)and one complex uncertainty (aeroelastic body bending) are incorporated into the system. The inclusion of the uncertainty models leads to a robust performance problem with structured real/complex uncertainty. In this application only mixed h controllers are considered. The full order design resulted in a 47th order compensator and achieved a peak value of 1.223 of the mixed upper h bound. Employing the balanced order reduction scheme a reduced 5th order controller could be obtained which was internally stabilizing. However, its peak value of 9.228 of the mixed upper p bound indicates a significant loss in robust performance as compared to the full order design. According to the design procedure outlined in Section 3, the reduced order controller was used as starting guess for the developed algorithm. Two additional fixed order D,G-K iterations sufficed to

H. Buschek, A. J. Calise

402

decrease the mixed upper ,U bound to 1.330. results are summarized in Table 2. Table

2. - /I values

for full,

reduced

and fixed

order

The

controller.

Full order (47 states)

5” order reduced

5” order fixed

1.223

9.228

1.330

P

Implementing the different mixed p controllers, time simulations were conducted for simultaneous step commands in velocity and altitude of V, = 100 ft/sec and h, = 1000 ft while encountering longitudinal and vertical atmospheric turbulence. The time simulations were carried out using the linear model at Mach 8 which suffices at the initial stage of the present design. However, the closed loop system was perturbed by a worst case, real-rational, stable perturbation A,(s) reflecting the effects of worst case uncertainty and disturbances within the given bounds. This perturbation was obtained by searching for the peak value of the lower p bound over frequency and constructing the corresponding perturbation such that I - MA, = 0 at this frequency [34]. AP (s) has the same block structure as the uncertainty. For the full order mixed p controller, AP (s) is given by A pert = [-0.32191 B pert = [0

c pert

!

-0.8207

D pert =

0

0 0

- 0.7269

=

0

Fig. 10. designs.

10

*o

- Velocity

30

40

50 time [sl

responses

for

60

the full

70

and 5th

80

order

90

100

mixed

p

(58) 0

0]

(59)

i I 0 -0.7269 0 0

0 -0.8207 0 0

,I

0 0 0.8207 0

-200’0

I 10

20

30

40

50 time [s]

60

70

80

90

100

(60)

Fig. 11. designs.

0 0 0.8iO7 I (61)

This worst case perturbation is implemented in place of the A’s in Figure 8 in order to carry out the simulations. The same perturbation is also used for the fixed order controllers to provide a consistent simulation environment. Velocity and altitude histories of the different controllers are shown in Figures 10 and 11. The velocity responses of the full and the fixed 5th order designs are very similar. However, the loss in robust performance for the reduced 5th order controller is evident in Figure 10 where the response exhibits a significant oscillation. The altitude responses in Figure 11 are similar for all designs.

- Altitude

histories

for

the full

and 5t”

order

mixed

p

The effect of atmospheric turbulence is demonstrated on the control histories in elevon and fuel equivalence ratio of the fixed 5th order controller in Figures 12 and 13. Even though the control rates were penalized to reduce sensitivity to turbulence, atmospheric disturbances dominate the responses. Nevertheless, the significant oscillations in the equivalence ratio response of the reduced 5th order compensator in Figure 14 are not exclusively due to turbulence influences but also confirm the loss in robust performance which was already observed in Figure 10. The results illustrate that the fixed order controller meets the performance requirements for tracking commanded velocity and altitude. Moreover, the controller demonstrates robustness to the impact of atmospheric turbulence and a worst case disturbance model. However, due to the linear nature of the controller synthesis procedure, these results are only Aerospace

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Aerospace Control System Design with Robust Fixed Order Compensators

403

valid in the vicinity of the given flight condition at Mach 8. For different flight conditions gain scheduled controllers need to be designed. VI - CONCLUSIONS

-1.51

0

10

20

30

40

50 time [s]

60

70

80

90

100

Fig. 12. - Eleven control history, fixed 5’” order controller

A new method to synthesize fixed order robust controllers for mixed real/complex uncertainties has been presented. A design procedure for fixed order H, controllers using a homotopy algorithm is incorporated into a mixed p-synthesis framework to provide robustness to both real and complex uncertainties. The benefit of the new method is its ability to constrain the order of the controller a p&vi in the design process. This becomes an extremely valuable advantage in the face of substantially growing system dimensions due to the scaling matrices which are augmented to the system in the mixed bsynthesis procedure. Design examples for a flexible satellite model and a hypersonic vehicle demonstrate the relevance of the method to typical aerospace applications. The fixed order mixed h controllers synthesized with the presented technique are of considerably lower order, and closely approximate the robust performance measures of the full order designs. They are shown to improve over the achievable robust performance levels of corresponding reduced order controllers. Moreover, internally stabilizing fixed order controllers have been designed where reduced order compensators only resulted in unstable closed-loop systems. REFERENCES

-0.11 0

,o

20

SO

40

50 me

60

-A*

70

80

90

[s]

Fig. 13. - Equivalence ratio control history, fixed 5t” order controller. 14

-0 2

Fig.

0

14.

10

-

controller. 1997, no 6

20

30

40

50 t,me [s]

60

70

80

90

100

Equivalence ratio control history, reduced gf” order

[l] Doyle J. C., Glover K., Khargonekar P. P., Francis B. A. - State-Space Solutions to Standard Hz and H, Control Problems. IEEE Transactions on Automatic Control, August 1989, 34, No. 8, 832-846. [2] Stein G., Doyle J. C. - Beyond Singular Values and Loop Shapes. Journal of Guidance, Control, and Dynamics, Jan./Feb. 1991, 14, No. I, 5-16. [3] Packard A., Doyle J. C. - The Complex Structured Singular Value. Automatica, 1993, 29, No. 1, 71-109. [4] Hyland D. C., Bernstein D. S. - The Optimal Projection Equations for Fixed-Order Dynamic Compensation. IEEE Transactions on Automatic Control, Nov. 1984, AC-29, No. 11, 1034-1037. [5] Bernstein D. S., Haddad W. M. - LQG Control with an H, Performance Bound: A Riccati Equation Approach. IEEE Transactions on Automatic Control, 34, No. 3, March 1989, 293-305. [6] Kramer F. S.; Calise A. J. - Fixed-Order Dynamic Compensation for Multivariable Linear Systems. Journal of Guidance, Control and Dynamics, 11, No. 1, January 1988, 80-85. [7] Sweriduk G. D., Calise A. J. - A Differential Game Approach to the Mixed Hz/H, Design Problem. AIAA Paper 94-3660, Proceedings of the 1994 AIAA Guidance, Navigation and Control Conference, Scottsdale, AZ, Aug. 1994, 1072-1082.

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[S] Whorton M., Buschek H., Calise A. J. - Homotopy Algorithm for Fixed Order Mixed Hz/H, Design. Journal of Guidance, Control, and Dynamics, 19, No. 6, Nov./Dee. 1996, 1262-1269. [9] Wei K., Yedavalli R. K. - Robust Stabilizability for Linear Systems with Both Parameter Variation and Unstructured Uncertainty. IEEE Transactions on Automatic Control, Feb. 1989, 34, No. 2, 149-156. [lo] Yeh H.-H., Banda S. S., Heise S. A., Bartlett A. C. - Robust Control Design with Real-Parameter Uncertainty and Unmodeled Dynamics. JournaZ of Guidance, Control and Dynamics, NovJDec. 1990, 13, No. 6, 1117-1125. [II] Safonov M. G., Chiang R. Y. - Real/Complex K,Synthesis without Curve Fitting. Control and Dynamic Systems, 1993, 56, 303-324. [12] Safonov M. G., Ly J. H., Chiang R. Y. - P-Synthesis Robust Control: What’s Wrong and how to Fix It?. Proceedings of the 1993 IEEE Conference on Aerospace Control Systems, West Lake Village, CA, May 1993, 563-568. [13] Safonov M. G., Goh K. C., Ly J. H. - Control System Synthesis via Bilinear Matrix Inequalities. Proceedings of the 1994 American Control Conference, Baltimore, MD, June 1994, 45-49. [ 141 Haddad W. M., How J. P., Hall S. R., Bernstein D. S. - Extensions of Mixed-b Bounds to Monotonic and Odd Monotonic Nonlinearities Using Absolute Stability Theory. International Journal of Control, 1994, 60, No. 5, 905-951. [15] How J. P., Haddad W. M., Hall S. R. - Application of Popov Controller Synthesis to Benchmark Problems with Real Parameter Uncertainty. Journal of Guidance, Control, and Dynamics, July/August 1994, 17, No. 4, 759-768. [16] Hong S., Speyer J. L. - Robust Game Theoretic Synthesis in the Presence of Uncertain Initial States. Journal of Guidance, Control, and Dynamics July/August 1995, 18, No. 4, 792-801. [17] Fan M. K., Tits A. L., Doyle J. C. - Robustness in the Presence of Mixed Parametric Uncertainty and Unmodeled Dynamics. IEEE Transactions on Automatic Control, January 1991, 36, No. 1, 25-38. [18] Fan M. K. H., Nekooie B. - An Interior Point Method for Solving Linear Matrix Inequality Problems. SIAM Journal on Control and Optimization, to appear. [19] Young P. M., Newlin M. P., Doyle J. C. - p Analysis with Real Parametric Uncertainty. Proceedings of the 30th IEEE Conference on Decision and Control, Brighton, England, December 1991, 1251-1256. [20] Young P. M. - Controller Design with Mixed Uncertainties. Proceedings of the 1994 American Control Conference, Baltimore, MD, June 1994, 23332337.

H. Buschek, A. J. Calise

[21] Collins E. G., Davis L. D., Richter S. - Design of Reduced Order, Hz Optimal Controllers using a Homotopy Algorithm. Proceedings of the 1993 American Control Conference, San Francisco, CA, June 1993, 2658-2662. [22] Richter S. L., DeCarlo R. A. - Continuation Methods: Theory and Applications. IEEE Transactions on Circuits and Systems,June 1983, CAS30, No. 6, 347352. [23] Chiang R. Y., Safonov M. G. -Robust Control Toolbox, User’s Guide. The MathWorks Inc., Natick, MA, 1992. [24] Luenberger D. G. - Canonical Forms for Linear Multivariable Systems. IEEE Transactions on Automatic Control, June 1967, AC-12, No. 6, 290-293. [25] Buschek H. - Synthesis of Fixed Order Controllers with Robustness to Mixed Real/Complex Uncertainties. Ph. D. Thesis, School of Aerospace Engineering, Georgia Institute of Technology, February 1995. [26] Calise A. J., E. V. Byrns Jr. - Parameter Sensitivity Reduction in Fixed-Order Dynamic Compensation”. Journal of Guidance, Control and Dynamics, March/April 1992, 15, No. 2, 440-447. ]27 Shaughnessy J. D. et al. - Hypersonic Vehicle Simulation Model: Winged-Cone Configuration. NASA TM 102610, November 1990. w3 Turner R. E., Hill C. K. - Terrestrial Environment (Climatic) Criteria Guidelines for Use in Aerospace Vehicle Development, 1982 Revision, NASA TM 82473, June 1982. ]29 1 Walton J. - Performance Sensitivity of Hypersonic Vehicles to Changes in Angle of Attack and Dynamic Pressure. AIAA Paper 89-2463, 2Sh Joint Propulsion Conference, Monterey, CA, July 1989. [30 1 Calise A. J., Buschek H. - Research in Robust Control for Hypersonic Vehicles. Progress Report No. 1 to NASA Langley Research Center, Contract No. NAG1-1451, November 1992. [31] Raney D. L., McMinn J. D., Pototzky A. S., Wooley C. L. - Impact of Aeroelasticity on Propulsion and Longitudinal Flight Dynamics of an AirBreathing Hypersonic Vehicle. AIAA Paper 93-1367, Proceedings of the 34’h Structures, Structural Dynamics and Materials Conference, La Jolla, CA, April 1993, 628-637. [32] Paz M. - Structural Dynamics, Theory and Computation, Yd edition. Van Nostrand Reinhold, New York, 1991, 437-460. [33] Buschek H., Calise A. J. - Robust Control of Hypersonic Vehicles Considering Propulsive and Aeroelastic Effects. AIAA Paper 93-3762, Proceedings of the 1993 AIAA Guidance, Navigation and Control Conference, Monterey, CA, August 1993, 550-560. [34] Balas G. J. et al. - p-Analysis and Synthesis Toolbox, User’s Guide. The MathWorks Inc., Natick, MA, 1991.

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