Yedavalli Ieee Taes 1989

I. NOMENCLATURE

Robust Control Design for Aerospace Ap p I icatio ns

RAMA K. YEDAVALLI, Senior Member, IEEE The Ohio State Univelsity

The aspect of time domain control design for stability

robustness of linear systems with structured uncertainty is

addressed. Upper bounds on the linear perturbation of an asymptotically stable linear system are obtaimd to maintain

stability by us@ the structural information of the uncertainty. A quantitative measure called stability robustness index, is

introduced and this measure is used to design controllers for

robust stability. The proposed state feedback control design

algorithm can be used for a given set of perturbatiom, to select the range of control effort for which the system is stability

robust and conversely it can be used, for a given control effort, to determine the range of the size of tolerable perturbation. The

algorithm is illustrated with the help of exanples from aircraft

control and large space structure control problems.

Manuscript received February 1, 1987.

IEEE Log No. 27570.

This work was supported in part by NASA Grant NAG-1-578 and by United States Air Force Contract E33615-84-K-3606.

Authors address: Dept. of Aeronautical and Astronautical Engineering, The Ohio State University, 330 CAE Bldg., 2036 Neil Ave., Columbus, OH 43210.

0018-9251/89/0500-0314 $1.00 @ 1989 IEEE

RQ 4 Belongs to. A[.]

Real vector space of dimensions a.

Eigenvalues of the matrix [.I. _. 4 . 1 [.Is I(.)I

Singular value of the matrix [.I. Symmetric part of a matrix [.I. Modulus of the entry (.).

{ A([~l[.lT>) l**

[.I,,, V For all.

Modulus matrix = matrix with modulus entries.

II. INTRODUCTION

The aspect of robust control design for linear systems subject to parameter uncertainty has been an active topic of research in recent years. The published literature on the robustness of linear systems can be viewed mainly from two perspectives, namely: 1) frequency domain analysis, and 2) time domain analysis. The main direction of research in frequency domain has been to extend and generalize the well-known classical, single input, single output treatment to the case of multiple input, multiple output systems, using the singular value decomposition [l, 21. In the case of frequency domain results, the perturbations are mainly viewed in terms of gain and phase changes [3, 41 and as such are more suited to treat unstructured perturbations. The time domain treatment, on the other hand, is more amenable to the consideration of structured perturbations in the form of parameter variations and nonlinearities. This paper treats the stability robustness analysis and design from the time domain viewpoint.

One factor which clearly influences the analysis and design of robust controllers is the characterization of perturbation. Assuming the nominal system to be stable, the perturbations can be viewed to take different forms like linear, nonlinear, time invariant, time varying, structured, and unstructured. Structured perturbations are those for which bounds on the individual elements of the perturbation matrix are known (or derived) whereas unstructured perturbations are those for which only a norm bound on the perturbation matrix is known (or derived). We focus our attention on linear structured (possibly time varying) perturbations as affecting the nominal system.

Under this perspective, some researchers have presented analysis and design procedures for tolerable perturbations for robust stability. Kantor and Andres [5] present an algorithm to determine tolerable perturbations in the frequency domain using M matrix analysis. In the time domain, Horisberger and Belanger [6] present an algorithm to determine an output feedback control gain that yields the largest possible tolerable perturbation such that the closed loop system is stable, but no explicit bounds are given. Zheng

314 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 25, NO. 3 MAY 1989

[7] presents a procedure to find the stability regions as a function of parameters but considers only time invariant perturbations, and no synthesis procedure is given. Eslami and Russell [SI also address the same problem, but no explicit bounds are obtained. Chang and Peng [9], Patel and Toda [lo], Patel, Toda, and Sridhar [ l l ] and Hinrichsen and Pritchard [12] give bounds on the norm of the perturbation matrix (which amounts to the radius of a sphere in the parameter space and thus can be categorized as an unstructured perturbation case). Here the emphasis is on utilizing the structure of the perturbation matrix and thus the aim is to give bounds on the individual elements of the perturbation matrix rather than on the norm of the matrix. This analysis, which is at the elemental level, is termed structured uncertainty analysis. Leitmann [13], Hollot and Barmish [15] and their bibliographies discuss the role of the structure of the uncertainty in the robust stabilization of uncertain systems using Lyapunov theory but no explicit bounds on the elements are obtained.

Within this framework, the author has developed bounds on the elemental perturbations (treating the variations as independent variations) [16-201. The philosophy behind the design procedure proposed here is to utilize these bounds in a design formulation and give an algorithm to synthesize controllers for robust stability. Towards this direction, a quantitative measure called stability robustness index is introduced and based on this index a design algorithm is presented by which one can pick a controller that possesses good stability robustness property. The algorithm, for a given size of perturbation, can be used to select the range of control gain for which the system is stability robust or alternatively, for a given control gain, can be used to determine the range of the size of allowable perturbations for stability. Two aircraft control examples [4, 211 are considered and constant gain linear state and state estimate feedback control laws are presented to attain stability robustness for given perturbations. Similarly, the vibration suppression problem of a large space structure (US) model with modal data uncertainty is considered [22] and the various design implications based on this example are discussed.

The paper is organized as follows. In Section I11 the results of [16-201 are briefly reviewed. Section IV illustrates the application of the proposed design to two aircraft control examples and an LSS control example and discusses the different design implications. Finally, Section V offers some concluding remarks.

Ill. PERTURBATION BOUNDS FOR ROBUST STAB I L ITY

Briefly reviewed here are the upper bounds developed by the author in [16-201 for linear uncertain

systems to maintain stability assuming structured uncertainty.

Consider the following linear dynamical system

i ( t ) = A ( t ) ~ ( t ) = [A0 + E ( t ) ] ~ ( t ) (1) where ~ ( t ) -+ R is the state vector, A0 is the n x n nominally stable matrix and E( t ) is the error matrix, whose elements are such that

A A E;j ( f ) < ~ ; j = IE;j(t)l and c =Max ~ ; j . (2) 1.1

Thus E is the magnitude of the maximum deviation expected in the entries of Ao.

By taking advantage of the structural information of the nominal as well as perturbation matrices, improved measures of stability robustness are presented in [16-201 as follows.

The system of (1) is asymptotically stable if

or < ps (3b)

(3c)

(W

for all i , j = 1, ..., n where P satisfies APo + A,TP + 21, = 0

and A

Ue;j = ; j / (thus 0 5 U,;j 5 1).

It may be noted that U, can be formed even if one knows only the ratio E ~ , / E instead of knowing E ; , (and E ) separately. One suitable choice for the ratio is

ueij = i j / E = lAoi j I / IAoi j lmax

for all i, j for which

REMARK 1. maximum modulus deviations expected in the individual elements of the nominal matrix Ao. If we denote the matrix A as the matrix formed with c ; j , then clearly A is the majorant matrix of the actual error matrix E ( t ) , where by majorant matrix we mean the matrix with as its elements. In other words, it is the matrix formed with the maximum modulus deviations, ci j . It may be noted that U, is simply the matrix formed by normalizing the elements of A (i.e., ~ i , ) with respect to the maximum of ~ ; j (i.e., E ) . For example,

# 0. From (2), it is seen that cij are the

A = EU, (absolute variation). (4)

Thus 6;j here are the absolute variations in Ao;j. Alternatively, one can express A in terms of percentage variations with respect to the entries of Ao;j. Then one can write

A = 6Aom (relative (or percentage) variation) (5)

YEDAVI: ROBUST CONTROL DESIGN 315

where Aomij = JAoijI for all those i , j in which variation is expected and Aomij = 0 for all those i , j in which there is no variation expected and 6;j are the maximum relative variations with respect to the nominal value of Aoij and 6 = Maxi,j6;j. Clearly, one can then get a bound on 6 for robust stability as

0. 0.-

0. 0.

0. 0. I 0. 0. B = where P is the same as in (3).

Using the concept of state transformation, it has been shown in [19] that it is possible to further improve the bounds on E. For structured perturbation, it is possible to get higher bound even with the use of a diagonal transformation. This result can be stated as follows:

- M = diag[ml,mZ,.. ., m,]. (6)

The system matrices for the drone lateral attitude control system considered in [4] are given by

I -0.0853 -0.0001 -0.9994 0.0414 O.oo00 0.1862 -46.8600 -2.7570 0.3896 O.oo00 -124.3OOO 128.6ooo -0.4248 -0.0622 -0.0671 O.oo00 -8.7920 -20.4600 I 0.000 1.oooO 0.0523 O.oo00 O.oo00 O.oo00 A = I 0.000 0 . m 0 . m 0 . m - 2 0 . m 0 . m 1 L 0.000 0 . m 0 . m 0 . m 0 . m -20.mJ

(sa)

L:: ::I The system of (1) is stable if With a linear state feedback control gain

or

where

and

I -215.1000 4.6650 7.8950 233,u)oo -6.7080 2.5540 -231.5000 -3.7230 7.4530 -213.5000 2.5540 -6.8690 i j < ps ueij = p,+u C ' I .. (7a) G = [ m. maxi,j 12ueijI

(&>

the closed loop system matrix 2 = A + BG is made asymptotically stable.

Now assuming the element A21 to be the uncertain parameter (having a nominal value = -46.86) we get the stability robustness bound on this parameter (using

< P: (7b)

the U, matrix as Ue21 = 1 and U,ij = Ofor all other (7c) i , j ) , as

~ 2 1 = 2.43. (9)

However, using the transformation - + + 21, = 0 and 0, is formed (7d) M = diag[0.005 1 1 1 1 11 (10)

we get the bound on A21 as such that e . . i

ir,,, = -fl

where

p & = 573.46 (11)

which is clearly a significant improvement.

IV. TIME DOMAIN CONTROL DESIGN FOR ROBUST STABILITY

The foregoing discussion in Section I11 is basically and

E A . = ijmu; 2 0 = M-lAoM. (7e) concerned with the analysis of stability robustness for linear systems. No effort was made to synthesize a controller to achieve stability robustness. In this section, we address this design aspect from a systematic algorithmic point of view. The philosophy behind the proposed procedure is to make use of the perturbation bounds developed in the Previous

(Note that in general 0, # R- 'U,W) .

EXAMPLE 1. Application to the drone example /4]:


section in a design formulation and give an algorithm to synthesize controllers for robust stability. Towards this direction, a quantitative measure called stability (relative variation) (17) robustness index is introduced and based on this index

Alternately, we can write

A A = 6,Am

AB = 6hB, a design algorithm is presented by which one can pick a controller that possesses good stability robustness property. The algorithm, for given size of perturbation can be used to select the range of control gain for which the system is stability robust or alternatively, for a given control gain, can be used to determine the range of the size of allowable perturbations for stability. In this attempt, we first consider the case of full state feedback controllers and then investigate the use of state estimate feedback controllers.

A. Linear State Feedback Control Design Using Perturbation Bound Analysis

Consider the linear, time invariant system described by

X = A x + B u

y = cx where x is n x 1 state vector, the control U is m x 1 and output y (the variables we wish to control) is k x 1. The matrix triple (A, B, C) is assumed to be completely controllable and observable. Let the control law be given by

U = Gx. (13)

Now let A A and AB be the perturbation matrices formed by the maximum modulus deviations expected in the individual elements of matrices A and B, respectively. Then one can write

A A = ,Uea (Absolute variation) (14)

AB = fbUeb

where E , is the maximum of all the elements in A A and q, is the maximum of all elements in AB. Then the total perturbation in the linear closed loop system matrix of (12) with nominal control U = Gx is given by

A = A A + ABG, + eauca + bUcbGm. (15) Assuming the ratio q,/& = F is known, we can

extend the main result of (3) to the linear state feedback control system of (12), (13) and obtain the following results. Result 1. The perturbed linear system is stable for all perturbations bounded by E , and Eb if

and q, < 71-1 where P ( A + BG) + ( A + BG)TP + 21, = 0 (16b)

and ( A + BG) is an asymptotically stable matrix.

. . .

where Amij = IAijI and Bmij = IBijI for all those i, j in which variation is expected and Amij = 0, Bmij = 0 for all those i, j in which there is no variation expected. For this situation, assuming 6 b / 6 , = 6 is known, we get the following bound on 6, for robust stability. Result 2. The perturbed linear system is stable for all relative (or percentage) perturbations bounded by 6, and db if

and 66 < 61-1, where P satisfies (16b). Note that the above expressions can be suitably

modified only if either A A or AB is present.

REMARK 2. If we suppose A A = 0, AB = 0 and expect some control gain perturbations AG, where we can write

AG =egUeg (19a) then stability is assured if

(19b) 1

~max[PmBm Uegls = P g fg <

In this context, pg can be regarded as a gain margin.

B. Stability Robustness Index and Control Design A Igo r i t h m

We now define, as a measure of stability robustness, an index called stability robustness index ,& as follows.

Case a). Left-hand side (LHS) of (16) or (18) is known (i.e., checking stability for given perturbation range). For this case

(20a) A

b R = p - f a (or 1-11 - 6,).

Case b). LHS of (16) or (18) is not known (is., specifying the bound). For this case

b R A 1 - 1 (or p r ) . (20b)

It is clear from the expressions for ,U in (16), the error matrix in (15) and &R in (20) that these quantities depend on the control gain G and as the gain G is varied / ~SR changes. In order to plot the relationship between &R and the gain G , we need a scalar quantitative measure of G . For this we can either use

Jm = IlGlls = gmax(G) P a )

or 112 112

J,, = [ L m ( u T u ) d i ] = [ L m x T G T G x d t ] (21b)

YEDAVALLI: ROBUST CONTROL DESIGN 317

where J,, denotes a measure of nominal control effort. We use (21b) here.

The variation of AR with the control effort J,, is very much dependent on the perturbation matrices and on the behavior of the Lyapunov solution, which cannot be described analytically in a straightforward way. Assuming stability robustness is the only design objective, the design algorithm basically consists of picking a control gain that maximizes stability robustness ( /~SR) . Specifically the algorithm involves determining the index / ~ S R and the control effort J,, for different values of the control gain G and plotting these curves. These design curves can then be used to pick a gain that achieves a high ,&R. The algorithm thus provides a simple constant gain state feedback control law that is robust from a stability point of view.

The linear control gain G of (13) can, of course, be determined in many different ways. In this section, we assume the control gain G to be given by the standard linear quadratic regulator algorithm. Accordingly, we determine G as

where K satisfies the Riccati equation.

R, K A + A T K - K B - B T K + G = O (22b)

P c

for a given symmetric positive semidefinite matrix a and Ro = I,,,. Thus p, serves as the design variable.

In other words, in the proposed procedure, we determine the gain by some nominal means and then investigate the robustness of the closed loop system by checking if the gain makes the index ~ S R positive for a given ea (or e b ) (Case a) situation) or by determining / ~ S R = p for a given control gain (Case b) situation).

C. State Feedback Control for a Vertical Takeoff and Landing Aircraft

A widely used vertical takeoff and landing (VTOL) aircraft control problem with varying flight conditions is considered and a stability robust controller is synthesized using the proposed methodology. This example illustrates the design methodology for Case a) situation while the next application is given for Case b) situation.

The linearized model of the VTOL aircraft in the vertical plane is described by

i = ( A + AA)x + ( B + AB)u, ~ ( 0 ) = XO. (23)

control vector U -+ R2 are given by

x1 + horizontal velocity (knots)

x2 -+ vertical velocity (knots)

x3 -+ pitch rate (degreeheconds)

x4 -+ pitch angle (degrees)

u1 + collective pitch control

u2 + longitudinal cyclic pitch control.

Essentially, control is achieved by varying the angle of attack with respect to air of the rotor blades. The collective control u1 is mainly used for controlling the motion of the aircraft vertically up and down. Control u2 is basically used to control the horizontal velocity of the helicopter.

In [21], the mathematical model is presented assuming the nominal airspeed to be 135 knots. For this nominal case ( A A and AB = 0) the matrices A and B are given by

I -0.0366 0.0271 0.01% -0.4555 0.0482 -1.01 0.0024 -4.0208 0.1002 0.3681 -0.707 1.4200 (24a) 0 0 1 0 A = [ r 0.4422 0.17611

3.5446 -7.5922 = i.49 1

We take the initial condition to be

~ ~ ( 0 ) = [ O S 0.15 0 -0.051. (244

As the airspeed changes significant changes take place in the elements ~ 3 2 , UM, and b21. Let us consider, for illustration purposes, the following range of variations in these parameters.

Case 1.

0.3545 5 Z32 = 0.3681 5 0.3817 1.31 5 21% = 1.42 5 1.53 3.39 5 521 = 3.544 5 3.702

i.e.,

lAa32I = 0.0136; lAu%l = 0.11; [Ab211 = 0.157.

(25)

Case 2.

IA~321 = 0.041; lA~,l = 0.332; [Ab211 = 0.47.

(26)

Case 3.

IAa321 0.068, I A u ~ ~ = 0.553; lab211 = 0.78.

The components of the state vector x + R4 and the (27)


1 I 1 I I I I

x u Fig. 1. Variation of AR with nomina1 control effort J,,(AA # 0,

AB # 0).

Case 4.

IAa32) = 0.1363; IAa341 = 1.106, JAb211 = 1.5674.

(29)

In other words, the matrices A A and A B are known for these five cases.

For designing the nominal state feedback control that stabilizes the nominal closed loop system, for this example, we employ the standard Riccati equation

R,' K A + ATK - KB-BTK +e = 0 (30)

Pc

where Q and Ro are (n x n), and (m x m) symmetric positive definite matrices and p c is a scalar variable used for designing the control gain G .

For this case, the nominal closed loop system matrix is given by

2 = A + BG, G = -R,'BTK/pc (31) and 2 is asymptotically stable. For each case using

0.01 0

- - -

0 2 3 2 c M

fig. 2. Variation of ,&R with nominal control effort J,,(AA = 0, or AB = 0).

(1) From these plots, it may be observed that as the parameter perturbation range is increased, the range of control effort for stability robustness is decreased.

(2) For a given set of parameter perturbations, there is a unique control effort for which AR is maximum. Obviously this is the control effort (and the corresponding control gain) that we seek, assuming practical constraints are satisfied.

(3) It is to be noted that for all these cases, the maximum AR occurs almost at the same control effort. However, it is also to be kept in mind that the range of variations considered in Cases 2 through 5 are simply some multiples of the range of Case 1.

Now let us consider the cases when parameter perturbation is in only one of the matrices, A or B. Accordingly, we consider the following ranges.

Case 6. lAa32l = 0.3018; IAU,~ = 1.300; lAb211 = 0.

Case 7. lAa32I = 0.1366; ~ A u M ~ = 0.106; lAb211 = 0. (32)

Case 8. IAa321 = 0; ~ A u M ~ = 0; 1Ab21) = 2.5671.

Case 9.

lAa32l = 0; ~ A u M ~ = 0; [Ab211 = 1.5674. Fig. 2 corresponds to these cases. It may be

observed from these plots that, as before, it turns out that the smaller the size of the perturbation, the more is the control effort range. But consider Cases 6 and 9. From plots corresponding to these two cases, it can be seen that the control range (for stability) for perturbations in A is larger than the control range for perturbations in B indicating that the variations in matrix B are more critical from the stability robustness point of view. REMARK 3. by (3a)-(3d) assume independent variations in each

and Ro = 12 and pc as design variables, the standard optimal LQ regulator control gain and the corresponding control effort J,, of (21b) are computed and / ~ S R is calculated. The plots of AR versus Jcn for these five cases are shown in Fig. 1.

In interpreting these plots, it is to be recalled that the region of control effort for stability robustness is the region in which / ~ S R > 0.

It may be noted that the bounds given


of the entries of the perturbation matrix and thus when applied to an application where the perturbation matrix elements are dependent on each other as in the case of the VTOL example, the bounds may turn out to be somewhat conservative. Attempts are being made to reduce the conservatism of the bounds by further exploiting the dependency among the uncertain parameters in [24, 251.

D. Extension to Linear Stochastic Systems with State Estimate Feedback

We now extend the above treatment to the case of linear stochastic systems. Let us consider a continuous, linear, time invariant system described by

f ( t ) = Ax(t) + Bu(t) + Dw(t), x(0) = xo (33a) Y ( t > = Cx@> (33b) z( t ) = M x ( t ) + v( t ) (33c)

where the state vector x is n x 1, the control U is m x 1, the external disturbance w is q x 1, the output y (the variables we wish to control) is k x 1, and the measurement vector z is 1 x n. The initial condition x ( 0 ) is assumed to be a zero-mean, Gaussian random vector with variance XO, i.e.,

E[x(O)] = 0; E[x(0)xT(O)] = I,. (3%) Similarly, the process noise w(t) and the measurement noise v(t) are assumed to zero-mean white-noise processes with Gaussian distributions having constant covariances, W and V, respectively, i.e.,

E[w(t)] = E[v(t)] = 0 (3%)

where pe is a scalar greater than zero and V = p e V o and d is the dirac delta function and E is the expectation operator.

as a function of the measurements, where the state estimator has the following structure

The state'x(t) of the stochastic system is estimated

i ( t ) = Ai?(t) + Bu(t) + G i ( t )

2 ( t ) = z ( t ) - M q t )

(Ma)

P b ) where

is called the measurement residual. For the minimum variance requirement, the estimator of (34) is the standard Kalman filter.

We also assume that the matrix pairs [A, B ] and [A, D] are completely controllable, and the pairs [A, C] and [A,M] are completely observable.

consider the control law given by For this case of a linear stochastic system, we

(353) 1

Pc

= ~ i ? = - R ; ~ B T K ~

where

4 = A X + BU + &(z - M a ) , f(0) = 0 (35b) = (A + BG - & ~ ) a + &Z (35c)

and P and K satisfy the algebraic matrix Riccati equations

R-

Pc K A + A ~ K - KBLB~K +Q = o (3%)

V-' BA= + AP - B M ~ L M B + D W D ~ = 0. (35f) Pe

The nominal closed loop system is given by

where 2, = A + BG - &M and the closed loop system matrix

(37)

is asymptotically stable. Letting AA, AB, AC, A M , and A D be the

maximum modulus derivations in the system matrices A, B, C , M , and D, respectively, we can write the total error matrix of the closed loop system as

and writing A A = caUea, AB = CbUeb, AM = c,Ue,, ..., etc. and knowing the ratios ca/cb etc., one can get the stability robustness condition in the same manner as the equations given by (16).

E. Application to the Drone Lateral Attitude Control Problem

The linearized model of the lateral attitude control problem of a drone aircraft, with perturbations in the plant parameters is given by

f = ( A + AA)x + Bu, ~ ( 0 ) = no. (39) The components of the state vector x --$ R6 and the control vector U + R2 are given by

X T = [P ,4 ,$ ,61 /20 ,~2 /20]

uT = [ u I u ~ ] , u1 = elevon command = 61, (40)

u2 = rudder command = 62.


0 0

m

0 0 - 0 0

m

W 3

= 0

N

0

-

0 0

0

\

I 00 -1.00 0:OO 1 : O O 2 : O O 3 : O O 4 ; CO

1 OG I R Y O C I

Fig. 3. Variation of bound p with control weighting pc with full state linear state feedback for drone example.

0 0

LQR ;h =0 -2.00 -1.00 0. 00 1.00 2.00 3.00

LOG ( R H O C I

Fig. 4. Variation of bound p with control weighting pc with state estimate feedback for drone example.

The matrices A and B are given by (&)-(8b). We assume that the parameters with nonzero

nominal values in the A matrix are subject to perturbations and thus we take the Ue, matrix as Ueaii = IA;jI/IAijlmax. Accordingly, the matrix U,, is given by

U,, =

0.0007 O.oo00 0.0078 0.0003 O.oo00 0.0014

0.3644 0.0214 0.0030 O.oo00 0.9666 Loo00

0.0033 0.0005 0.0005 O.oo00 0.0684 0.1591

0 . m 0 . m o.oo00 o.oo00 0 . m o.oo00

0 . m 0 . m 0 . m o.oo00 0.1555 o.oo00

O.oo00 O.oo00 O.oo00 O.oo00 O.oo00 0.1555

The linear state feedback control gain is determined using the Riccati based equations of (35). For a given control gain (is., given pc), the bound

p is calculated. Since e, is not known, in this case the stability robustness index ,&R is simply given by ~ S R = p. The plot of p with the design variable pc is given in Fig. 3.

higher the control effort (lesser the pc), the higher is the tolerable perturbation for robust stability.

We now extend the algorithm to the stochastic controller case using

From the plot, it is seen that, for this problem, the

1 0 0 0 0 0 (42) I. w = 12; vo = 12; M = [ 0 0 0 1 0 0 The plot of p versus p, with pe as a parameter and

the comparison with the pure state feedback case is given in Fig. 4.

REMARK 4. p with state estimate feedback is lower than the one with pure state feedback.

REMARK 5. For a given p,, the bound p is higher as the measurement noise covariance is decreased, i.e., as p, is decreased. This appears to be reasonable, because this means that p becomes higher with better or more accurate measurements.

From this plot, it is seen that the bound

F. Robust Linear State Feedback Control for Large Space Structures

In this section we apply the robust control design methodology presented in the previous section to modal systems (which arise in large space structure (LSS) control problems). We specifically consider the LSS model with vibration suppression of the flexible modes as the control objective. We seek a linear state feedback control that achieves a reasonable tradeoff between the nominal performance and stability robustness by accommodating the modal uncertainty structure into the design procedure. Towards this direction, the fact that the modal data uncertainty increases with mode number is incorporated in the characterization of uncertainty in LSS model parameters and this uncertainty structure is used to obtain upper bounds for robust stability which are in turn used to get a robust controller.

LSS Model and Nominal Control Design. the standard state space description of an LSS evaluation model with n elastic modes:

Consider

f = AX + Bu, ~ ( 0 ) = XO, x + Rn=2N u + R m (43a)

~ = C X , y + R k (43b)


where

A A = 6,

X T = [x;,x:,x; ,..., x;] ;

* I 0 0 282 282 0 0 383 3@3

A = block diag.[. . .Ai;. . .I,

xi = [ 51 (434

The performance index for vibration supression problems may be written as

which can be written in the form

J = 1- [yTQy + pCuTu] dt [xTCTQCx + p,uTu] dt = J,, + pcJ,

= 1- ( a b ) where the matrix C of (43) is given by

C = block diag.[. . . Ci . . .] (453) and

ci = [; ;] Let the nominal control law be designed by minimizing the performance index of (Ma) which results in

u = G x (&) where

The closed loop system matrix

Z = ( A + B G ) (47) is asymptotically stable. In the nominal design situation, an appropriate value for pe (and hence G) is determined such that a reasonable tradeoff between Jy and J, is obtained. However, in U S models, the parameters of the plant matrix A, namely the modal frequencies and modal damping, as well as the parameters of the control distribution matrix B , namely the mode shape slopes at actuator locations, are known to be uncertain. It is also known that the uncertainty in these parameters tends to increase with an increase in mode number. Thus with variations A A and A B in the matrices A and B of (43), the

nominal control G of (46) cannot guarantee stability of the closed loop system. Thus one needs to design a control gain G that guarantees stability for a given range of perturbations A A and AB. This is done using the design procedure given in the previous section. In other words, the control design algorithm for robust stability consists of picking a control gain (i.e., pc) that achieves a positive AR (for Case a)) or a high value of AR (for Case b)).

The design algorithm involves determining the index &R and the costs J,, and J, for different values of the design parameter pe andd plotting these curves. The algorithm thus provides a simple constant gain state feedback control law (using the standard optimal LQ regulator format) that is robust from the stability point of view.

Next we present a specific characterization of uncertainty for LSS models and use the above methodology to design a controller for the Purdue model [23] of a two-dimensional US. Characterization of Parameter Uncertainty in LSS Models and Application to the Purdue Model. In U S models having the structure given by (43) the uncertainty in the modal parameters such as modal frequencies dampings and mode shape slopes at actuator locations tend to increase with an increase in mode number. One way of modeling this information in the uncertainty structure is given in the following (specifically, we employ the relative variation format of (17))

1":i where 8i indicate the nominal entries corresponding to the ith mode. We assume 6, = 6b which are not known.

With the above proposed uncertainty structure, we apply the robust control design methodology of previous sections to the Purdue model [23]. The model used consists of the first five elastic modes. The numerical values of the model are given in [23]. To conserve space, the model is not reproduced here.


ROBUST CONTROL GAlN DETERMlNATION FOR LSS MOOEL .000251 , I . . , , I , , , 7 I

BOUND VS CONTROL E F F O R T (Dh O;DB=O1 ,11030 , , , , , , , I , ,

.00020

.00015-

r 0.0001 " " " " " "

0. .Ol .02 .03 .04 .05 .06 .07 .08 .09 .IO . I I .I2 CONTROL EFFORT

Fig. 5. Variation of bound with control effort A A # 0, AB # 0.

W 3

1 -

BOUND V S CONTROL E F F O R T IDA=O.DE 01 I . 0 , , , . , . , , , ,

O,gl U. 8 0.6 O . ' t

CONTROL EFFORT

Fig. 6. Variation of bound with control effort for U S Model ( A A = 0, AB # 0).

Since 6, (and 6 b ) are not known, the present design corresponds to Case b) in which case we pick a control gain that gives high ,&R = p r . The plot of p r versus pc is given in Fig. 5. The robust control gain is the gain corresponding to pe = 0.12.

Figs. 6 and 7 present the variation of p, with control effort (i.e., pc) assuming A A = 0 and AB = 0, respectively. From these plots it can be concluded that the control effort range available for guaranteed stability for mode shape ( A A = 0, AB # 0) variation is limited in comparison to the range available for modal frequency variation. Thus mode shape (slopes at actuator location) variations are more critical from a control point of view than modal frequency variations.

V. CONCLUSIONS

The main theme of the paper has been to analyze and synthesize controllers for robust stability for linear time invariant systems subject to linear time varying structured (elemental) perturbations. First the analysis of robustness is considered. Then the aspect of control design is addressed. In this regard, first the case of linear state feedback control is considered. The linear

,0000' ' ' ' ' ' ' ' ' ' ' 0.0 20. 40. 60. 80. 100. 120. 140. 160. 180. 200

CONTROL EFFORT

Fig. 7. Variation of bound with control effort for U S model (AA # 0, AB = 0).

state feedback control is determined by nominal means based on the Riccati equation and the bounds achieved by this control law are computed. The effect of state estimation in the control law (for stochastic systems) on the bounds is illustrated by comparing it with the pure state feedback case. Finally the special nature of "modal systems" (as in the LSS control example) is incorporated in the uncertainty structure and a linear state feedback control utilizing this special structure is developed. Thus the utility of perturbation bound analysis in synthesizing useful controllers for different aerospace applications is established. More research is underway to incorporate both stability robustness as well as performance robustness into the design procedure.

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Rama K. Yedavalli (M7MM86) received his B.S. and M.S. degrees from the Indian Institute of Science, Bangalore, India, and the Ph.D. degree from Purdue University, Lafayette, Ind., in 1981.

He was an Assistant Professor at the Stevens Institute of Technology from 1981 to 1985 and an Associate Professor at the University of Toledo, Ohio, from 1985 to 1987. In September 1987, he joined the Department of Aeronautical and Astronautical Engineering at The Ohio State University, Columbus, Ohio, where he is currently an Associate Professor. Dr. Yedavallis research interests include robustness and sensitivity issues in linear uncertain dynamical systems, model reduction, dynamics and control of flexible structures with applications to aircraft, spacecraft, and robotics control.

Dr. Yedavalli is a senior member of IEEE and a member of AIAA, Sigma Xi, and an associate member of ASME. He serves as a reviewer for many journals, conferences, and the National Science Foundation. He organized and chaired and co-chaired sessions in conferences such as CDC, ACC and AIAA GNC. He is a member of the IFAC Working Group on Robust Control and a member of the IEEE Working Group on Linear Multivariable Systems. He was a consultant to the Lawrence Livermore National Laboratory for two years.

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