SIGN OF ROBU W I T H

Abstract

This paper presents a design approach that injects new procedures into the LQ design method to solve its per- formance and robustness problems. Performance of the LQ systems is enhanced by augmenting the orig- inal state spaces and robustness is improved by in- corporating principal-region-based multivariable sta- bility margins in the optimization algorithm. The de- sign approach combines the merits of time-domain and frequency-domain techniques and has been success- fully applied to a Multi-Input Multi-Output (MIMO) flight control system of a statically unstable fighter airplane.

I. Introduction

The LQ optimal control theory introduced by Kalman [1]-[3] has attracted much attention due to the guar- anteed control optimality with respect to a quadratic performance measure. As opposed to the classical con- trol theory that is an iterative design process based on some graphical design tools (;.e., root locus, Bode di- agram, Nyquist diagram, Nichols chart, etc) , the LQ design process can be automated through an array of analytical procedures , greatly easing the design efforts.

For a linear finite-dimensional time-invariant dynamic system described by

i = Ax+ Bu,

IST LINEAR-QUADRATIC CONTROLLERS DY N A M IC CO M P EN S A T 0 RS

Won-Zon Chen* Naval Postgraduate School

Monterey, California

Nhan Levan** University of California Los Angeles, California

where * denotes the complex conjugate transpose, Kal- man showed that the optimal feedback control law that minimizes the cost functional is

where P is a nxn positive definite matrix solution from the Steady State Riccati Equation (SSRE),

Furthermore, if the state variables of the system can not be all measured and the sensor measurement is

y = c x ,

where C is an lxn matrix, then a Kalman filter that takes the form of

2 = A2 + Bu + L ( y - C 2 ) ,

can be designed and, from which, the estimated state vector 2 can substitute the true state vector in (3).

As briefly described above, the LQ design method al- ways yields a feedback controller of constant gains for full state feedback cases and an additional Kalman filter if the full state is not available. However, this LQ controller's structure contradicts to the common practice of using dynamic compensators such as inte- grators, lead-lag filters, lag-lead filters, notch filters, etc to shape system responses over frequency bands where different requirements are imposed. Hence, lack of a mechanism in the LQ design method to address frequency-dependent requirements for feedback con- trol systems can obviously limit its attainable perfor- mance for some applications.

where x is the nxl state vector, u is the mxl input vec- tor , A and B are nxn and nxm matrices respectively, and for a quadratic cost functional

03

J = 1 x*(t)Qz(t ) + u*(t)Ru(t)dt ,

* Associate Professor, Electrical & Computer Engineering Dept . , Senior Member AIAA.

** Professor , Electrical Engineering Department,

This paper is declared a work of the U.S. Govern- ment and is not subject to copyright protection in the United States.

26

P A + A*P - PBR-'B*P + Q = 0.

“Robustness” in feedback control means that a control system can maintain stability and good performance in the presence of model uncertainties or model errors. This is of paramount importance in practice because the math model used for control design is often differ- ent from the real plant dynamics due to sources such as plant nonlinearities, truncation of high order dy- namics, multiple operation configurations, etc. For the LQ design method, robustness is not explicitly mea- sured in the cost functional. Unfortunately, it has been shown that the LQ design with full state feed- back can not guarantee robustness to any degree for non-uniformly perturbed model uncertainties [4]. As for the cases of output feedback with a Kalman filter, the problem of robustness is generally worse [5].

Numerous methods have been proposed in the past attempting to solve the performance and/or the ro- bustness problems of the LQ design method. These methods can generally be categorized into two groups. The first group totally abandoned the time-domain LQ design approach and reverted to the frequency- domain methods by generalizing the design proce- dures for Single Input Single Output (SISO) systems to MIMO systems. Examples of them are character- istic loci method [6],[7], diagonal dominance method [8], singular value loop shaping method [9], and H,- optimization method [lo]-[14]. On the other hand, the second group tried to improve the LQ method through reformulation of the problem and/or injection of new procedures into the theory. Loop transfer recovery method 191 ,[15] and frequency-shaped cost functional method [16],[17] are examples of this group.

In this paper, a design approach that can be catego- rized into the second group is presented. The LQ per- formance is enhanced by augmenting the state space in order to allow the use of more general cost functionals. Robustness is guaranteed by specifying multivariable stability margins using the concept of principal region in Nyquist plot. Optimal controller is then searched in the enlarged state space constrained by the required robustness, resulting in robust LQ controllers with dy- namic compensators.

In Section 11, multivariable stability margins are first defined. The motivation and general model for state augmentation are given in Section 111. Four theorems are also given to show the conditions by which state- augmented systems can be checked for controllability and observability. In Sections VI and V, the main problems of constrained optimization for state feed- back and output feedback cases are formulated and algorithms are developed, respectively. Application of the new design method to an MIMO flight control sys- tem of a statically unstable fighter airplane is demon- strated in Section VI. A summary in Section VI1 con-

2 7

11. Formulation of Multivariable Robustness

For SISO systems, robustness is generally measured by the gain and phase margins that measure the distance from the loop transfer function to the critical point (-1 + j 0 ) in the s-plane. For MIMO systems, the use of stability sector due to Safanov [18] and the use of minimum singular value of ( I + L ( s ) - l ) due to Doyle 1191, where L(s ) is the loop transfer matrix, have been suggested. In this paper, the MIMO robustness will be defined by using the concept of “principal region” introduced by Postlethwaite [20].

The principal region of a feedback control system can be found by decomposing the loop transfer matrix into the forms

where U ( s ) is unitary, HR(s) and H L ( s ) are positive semi-definite Hermitian matrices admitting the same eigenvalues. The eigenvalues of H R ( s ) or H L ( s ) are defined as principal values and the arguments of the eigenvalues of U ( s ) are defined as principal phases. It is noticed that the principal values are also the singular values. Hence, a t each s the maximum and minimum principal values, u,,, and urnin, and the maximum and minimum principal phases, and &an, form a curvilinear rectangle in the s-plane. If these curvi- linear rectangles are constructed for values of s around the Nyquist-D contour, a principal region can then be outlined as exemplified in Figure 1.

PRI NC 1 PAL REG I ON

Fig. 1: Principal Region and Multivariable Stability Margins

Postlethwaite showed that the spectrum of L ( s ) was contained in the principal region. As suggested by the multivariable Nyquist stability criterion [7], the degree of closed-loop stability can be estimated by the short- est distance from the principal region to the critical point ( - l + j O ) . Therefore, the definitions of SISO gain margin, gain reduction tolerance (GRT), and phase margin are extended to MIMO systems as also illus- trated in Figure 1. The advantage of using MIMO stability margins as the robustness measure for con- trol law designs is that an accurate knowledge of the model uncertainties is not required. It should also be noticed that this robustness measure is less conserva- tive than using the singular values alone as the phase information is used. The MIMO stability margins aa defined above reduce to the classical gain and phase margins for SISO systems.

The use of state augmentation is motivated by the observation that any linear plant and its linear dy- namic feedback controller can be modelled as a state- augmented system with constant feedback gains. The general effect of state augmentation is to artificially enlarge the state space of the original plant to where n is the dimension of the original state and q is the dimension of the augmented state. This added freedom by state augmentation can then create a pos- sibility for finding a better optimal controller than pos- sibly can by the standard LQ method. For this paper, a general model for state-augmented system is defined hereinafter as

( 9 ) = (;c :) ( : > + ( 3 ) u ,

and

(9)

Proof of this theorem can be obtained by showing that the uncontrollable subspace for the state-augmented system cannot be trivial if the uncontrollable subspace of the original plant is not trivial.

: For a decoupled state-augmented sys- tem ( i . e . , DC = 0 ) , if there are no common poles be- tween A and E , then a necessary and sufficient condi- tion for the state-augmented system to be controllable is that the pairs ( A , B ) and ( E , F ) be both control- lable.

Proof of this theorem is also omitted here as the results can be easily obtained by transforming the system into a Jordan canonical form. Treatment for the more gen- eral case where DC # 0 is given in the next theorem.

: If the original plant is controllable and there are no common poles between A and E. Then, the state-augmented system is controllable if and only if the pair ( E , F +TB) is controllable, where T is the unique solution of the Lyapunov equation:

T A - ET+ DC = 0.

Proof: Define a new state vector

z"=z+Tz.

Then,

Z=i+TX = Ez" + ( F + TB)u + (TA - ET + DC)X (12) = E2 + ( F + T B ) u .

where ( z z ) * is the (n+q)xl augmented state vector and ( y z ) * is the ( p + q)xl output vector. The fol- lowing four theorems give the relationship of control- lability and observability between the original system and the state-augmented systems.

heorem 1: A state-augmented system is uncontrol- lable if the original system is uncontrollable. Equiva- lently, for a controllable state-augmented system the original plant must be controllable.

We note that such a unique T exists for arbitrary DC and for matrices A and E that have no common poles. Combining (11) and (12), a decoupled system is ob- tained

The rest of the proof simply follows the results of The- orem 2. Q.E.D.

heorem 4: The state-augmented system is observ- able if and only if the original system is observable.

Proof: The observability matrix for the state-augme- nted system is

28

With some row operations we obtain

C 0 CA 0

CAn-l 0 0 4

Hence, the observability matrix has full rank ( n + q) if and only if the matrix

( " ) CAn-l

has full rank of n. Q.E.D.

IV. Constrained Optimization for State Feedback

Having defined state augmentation and multivariable robustness, it is now ready to define the main prob- lem of robustness constrained optimization for feed- back control of state-augmented systems. Discussion in this section is restricted to the full state feedback

1.

I-

For a linear plant whose state is augmented to

find the optimal feedback control law

, - r . I

2 9

5 . t . the cost functional,

is minimized, subject to the following multivari- able robustness constraints:

GM L Go GRT 5 GRTo

'$M 2 '$0,

where Go, GRTo, and '$0 are given.

The constrained optimization problem defined by (14) to (17) can be solved by a parameter optimization method in which the key procedure is to iterate con- troller feedback gains from an initial guess until the op- timal solution, subject to the constraints, is obtained. The main steps for the parameter optimization proce- dure are :

(1) Solve the SSRE as if there were no robustness constraint for the initial value of ( G H )

(2) For a given value of ( G H )

a. Evaluate the cost functional in (16)

b. Evaluate the robustness constraints in (17)

H ) for next (3) Based on 2a and 2b, modify ( G iteration

(4) Iterate over steps (2) and (3) until an optimd solution is found.

The algorithms developed for the evaluation of cost functional and multivariable stability margins are gi- ven below.

Evaluation of the Cost Functional

The state and input variables in (16) are first substi- tuted by the solution from (14) for a linear feedback control law, u = -(Ga:+Hz), with an initial condition of ( 20 zo )*. This yields

where A is the closed-loop system matrix,

If A is stable, then by Lyapunov's theorem

where C is the unique solution of the following Lya- punov equation

A*C+CA+(I - ( G H ) * ) (§*

I * ( - ( G I€))=''

Since xo and zo are arbitrary, an equivalent cost func- tional is the norm 1 1 . 11 of C .

Evaluation of the Multivariable Stability Margins

The loop transfer function matrix for the state-augme- nted system at the plant input position with a feedback gain matrix ( G H ) is

L(s ) = ( I + H ( s I - E)- 'F)- l (H(sI - E)- lD

+ G)(sI - A)- lB (22a)

* (SI - A) - lB . (226)

=(G + H ( s I - E + FH)- l (D + FG))

Both equations can be used to calculate L(s ) , although (2%) is easier than ( 2 2 ~ ) as i t uses two fewer matrix inversions. From Figure 1, it is found that the bound- aries of a principal region are always defined by the corner points of the curvilinear rectangles. For the left side of the boundary which is used for determin- ing MIMO stability margins, a convex curve is traced out if these corner points are maximum principal gain (urn,,), and a concave curve is traced out if the corner points are minimum principal gain (amin). Based on this observation, the MIMO gain margin and the GRT can be determined by the following algorithm:

(1) Find frequency w1 s . t . dmax = T , with a corre- sponding r m a x l and g m i n l ,

(2) Find frequency w z s.t . (bmin = T , with a corre- sponding cr,,,, and urnin,,

(3) Then, GM = l/max(umazl, a m a x 2 ) , and GRT = l/min(uminlj urnin,),

and the phase margin can be determined by the fol- lowing algorithm:

(1) Find frequency w1 s.t . amax = 1, with a corre- sponding &inl

(2) Find frequency w z s.t. amin = 1, with a corre- sponding dminz,

(3) Then, d~ = min(dmin1, dmina) - 71.

V. Constrained Optimization for Output Feedback

For the cases where the full state of the original plant cannot all be measured, a direct approach to feedback control design is to insert an observer in the feedback loop for estimating the unmeasured state variables. Without loss of generality, we can define the state- augmented system with output feedback as

(;) = (;; i) (:) + ($I) (23)

and

where x1 is the n lx l state vector which can be mea- sured, xz is the nzxl state vector which cannot be measured, and nl + n2 = n. The minimum-order ob- server for estimating x2 is of order n2 and in the form of

where KO is the observer gain to be chosen. define the estimation error as

If we

e = xz - xz,

then the resulted closed-loop system dynamics is

30

= A o ( ; ) ,

closed-loop system matrix with observer dynam- . It is clear from (27) that if the matrix (A22 - A12) is stable, the estimation error will be expo- tially diminishing with time and the observer is

id to be “convergent”. Such a IC0 always exists if e (A12, A22) pair is observable.

ilar to (20) for the full state feedback cases, the uation of cost functional for the output feedback

ases can be formulated to

where GO satisfies the Lyapunov equation

Hence, an equivalent cost functional is ((Coli, and ((CO(( is greater than IICII. This can be seen by partitioning

for which it can be proved that C1 = C. The eval- uation of multivariable stability margins for output feedback can follow the same algorithm for full state feedback with a new loop transfer matrix

31

We note that this loop transfer matrix reduces to (22a) with G2 = 0 and (GI 0) = G . Finally, the last issue is the selection of initial IC,. This can be done by solving

(33)

where &E and RE are chosen to be q2B2B; with a large scalar q and I , respectively. This ensures that the initial KO gives a stable and robust observer.

Now, the parameter optimization algorithm is modi- fied for output feedback cases in the following:

(1) Solve the §§RE for the initial value of (G H ) and (32) for the initial value of KO.

(2) For given values of (G H ) and KO

a. Evaluate the cost functional in (28)

b. Evaluate the gain margin, GRT, and phase margin for the loop transfer matrix in (31)

(3) Based on (2a) and (2b), modify (G N) and IC0

for next iteration

(4) Iterate over steps (2) and(3) until an optimal solution is found.

VI. An MIMO Example

The example used here to illustrate the new design method and demonstrate its effectiveness is a lineariz- ed model for the pitch axis dynamics of F-20, a single- engine agile fighter airplane designed by Northrop in early 1980’s. Two modifications are made to the origi- nal airplane such that a more complex control problem is created: 1) the center of gravity is arbitrarily moved aft to 35 percents of the wing chord, resulting in a

highly unstable airplane, and 2) the trailing-edge flaps are modelled as an active continuous pitch control sur- face as opposed to the one in the real aircraft that can only be scheduled to a few discrete positions. The lin- earized pitch-axis model for the modified F-20 at the flight condition of .5 Mach number and 10,000 feet al- titude is

This is because better turbulence suppression can be achieved with higher loop gains. For using quadratic cost functionals, this loop-shaping effect can also be achieved by introducing frequency-dependent weight- ing factors and such a cost functional is shown below

(i) = (2.7055 -.834 -;89 .996 0 :) (9) (34)

where the three state variables are angle of attack (a) , pitch rate ( q ) , and pitch angle (e ) , and the two control inputs are horizontal tail position (Sh) and trailing- edge flap position (Sf). This unstable F,20 aircraft has three poles at -2.52, 0, and 0.79. It is also as- sumed that only two state variables, angle of attack and pitch angle, are measured and the pitch rate has to be estimated.

The design problem on hand now is how to design a “fuselage-pitch-pointing” (FPP) control system, as il- lustrated in Figure 2, with “good turbulence responses and robustness”. For this design task, designers who follow the original LQ method would choose a quadra- tic cost functional as exemplified in below:

J = ~ m ( 1 5 0 a z + q 2 + 5 0 ( O - a ) 2 + S ~ + . 0 4 6 ~ ) d t , (35)

where a large weighting factor of 50 is assigned to flight path angle, y = (e - a ) , in order to suppress any devi- ation from the initial flight trajectory. The weighting factors for the other terms are chosen so as to induce a good closed-loop stability at nominal flight conditions. However, turbulence suppression and robustness are not addressed in (35).

V- V V

Notation is somewhat abused in (36) where time and frequency scales are mixed together. Nevertheless, the intent is to show that a first-order high-pass filter with frequency responses, as shown in Figure 3, is included in this cost functional to relax the control penalties for frequencies lower than 30 rad/sec. Therefore, higher control activities in this frequency band are resulted to suppress turbulence more effectively.

Amp (A) - db

0.

-10 W - rad/sec

-20

-30

Fig. 3: Frequency-Dependent Input Weighting Factor

Gupta [16] ,[17] first showed that a frequency-depend- ent quadratic cost functional can be accommodated by a modified LQ design procedure. Following Gupta’s procedure for the cost functional in (36) results in a state-augmented system of two more state variables (z1, a)

-.834 .996 0 0 0

0 0 0 -30 0 0 0 0 0 -30

Fig. 2: FPP Control Mode / -.2 -.093\

To achieve better turbulence responses, this would nor- mally be done by inserting lead-lag and lag-lead com- pensators in the feedback loops such that higher loop gains can be obtained for low frequencies while keeping the same level of loop gains for high frequencies.

-11.4 2.03 +[ :o 30 : J (;;) (37)

32

15.4 3.0 5.2 -.006 .032 36.9 -1.69 -37.4 .085 .528

with a “frequency-independent” cost functional I<= (

oi)

J = I ( C Y (I 6 z1 z2 6 j S f ) and a minimum-order observer - - Jo ‘

200 0 -50 0 0 0 0 CY 0 1 0 0 0 0 1; = -87.gP- 3 3 0 2 . 9 ~ - 4331.68 - 3.83bh - 5 . 5 ~ ~ ~

q = p + 3 7 . 9 5 ~ ~ + 49.266. (41)

This optimal feedback controller with output feedback is depicted in Figure 4. It is noticed that there are three dynamic compensators incorporated in the con- troller, of which one is for pitch rate estimation and the other two are due to state augmentation for im- proved disturbance attenuation. The feedforward gain of 0.141 is calculated to balance out the relative effec- tiveness of the two control surfaces.

Time histories of aircraft responses to a step input are shown in Figure 5. As required by the FPP mode, very little flight path motion is present relatively to the motion of pitch angle and angle of attack. Also, the estimated pitch rate follows the true pitch rate

0 0 0 - 1 0 1 0 0 0 0 -.04 0 .04

* dt. (38)

This demonstrates that more general cost functionals can be accommodated through state augmentation for performance enhancement. As for ensuring some min- imum level of robustness, the following multivariable stability margins, as defined in Section 11, are chosen:

GRT 5 33% near perfectly.

The aircraft responses to turbulences are examined by a discrete gust of 62.5 feet/sec. Figure 6 demonstrates that aircraft’s sensitivity to external disturbance is re- duced by nearly 50 percents through state augmenta- tion.

GM 1 2 0 d b (39) $M 2 60 degrees.

Thus far, a constrained optimization problem has been completely formulated for this MIMO example. Using the algorithm in Section V results in a gain matrix

n

n I I - 3 3 0 3 ~ - 43328 = + 31.9a + 49.20 I I

36.9 -37. -1.69 .085 5 3

Lastly, the reason and the effectiveness of using ro- bustness constraints for this example is illustrated in Figure 7. As shown, the LQ design can yield a con- troller with its principal region almost encompassing the criticai point (-1 + j O ) . With the robustness con- straints as chosen in (39) a fairly comfortable cushion between the principal region and the critical point is obtained.

VII. Summarv

A design approach that injects new procedures into the LQ design method have been presented in this pa- per. With the use of the new design approach LQ controller's performance is enhanced by augmenting the state space and robustness is guaranteed to meet the specified multivariable stability margins. This de- sign approach is particularly useful as it is direct and matches well with the formulation and design objec- tives of real-life control problems. An MIMO flight control problem for an unstable aircraft has been used to illustrate the new design approach and d.emonstrate its effectiveness.

---_ LQ Design _._____. LQ Design w i th State Augmentation

0.1

- M

5? .05 I 6

0.0 Y I I I I 0 1 2 3 4

0.5

h V

$ .25 E?

I U

0.0 0 1 2 3 4

0 1 2 3 4

Time - (sec)

Fig. 6: Improvement in Disturbance Attenuation

.06

.I4 h M .g .02 u

0.0

-.02 0 1 2 3 4

ii o.2 D 0.1 m 7

0.0

-0.1 0 1 2 3 4

1 .0

0.5

0 0.0 s -0.5

-1.0

h bo

0 1 2 3 4 2.0

- 5 0 g 1.0 e a e:

0.0 0 1 2 3 4

Time - (sec)

Fig. 5: FPP of Robust LQ Controller

NO ROBUSTNESS CON S'I'It A I N T S 0 U TP UT FEEDB A CI<

WITH ROBUSTNESS CJ 0 N S T R A I N ED 0 P T.

I111

1 c I

Fig. 7: Robustness Comparison

34

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