Summary-Nonlinear systems of the form 8(t) =g[x(t);t]+u(t), where x(t), u(t), and g[x(t); t] are n vectors, are examined in this paper. It is shown that if ;Ix(t)I/ = v'x12(i) ...
196
T R A N S A CI TE IEOE. h r S
O X ACU OTNOTM RO A TL I C
July
Time-, Fuel-, and Energy-Optimal Control of Nonlinear Norm-Invariant Systems" L. FALBi,
Summary-Nonlinear systems of the form 8(t)= g [ x ( t ) ; t ] + u ( t ) , where x ( t ) , u(t), and g [ x ( t ) ;t ] are n vectors, are examined in this paper. It is shown that if ;Ix(t)I/= v'x12(i) . . * +xn2(t) is constant along trajectories of the homogeneous system k ( t ) = g [ x ( t ) ; 11 and if the control u(t) is constrained to lie within a sphere of radius X , i.e., ;lu(t)lIS X ,for all t, then the control u*(tj = - ~ x ( t ) / l l x ( t ) l drives ~ any initial state 5 to 0 in minimum time and with minimum fuel, where the consumed fuel is measured by .f:i'u!t)l!dt. Moreover, for a given response time T , the control act) = -I'eJx(t)/~llx(t);l drives to 0 andminimizestheenergymeasured by $ j : ! l ~ ( t ) 1 1 ~ d lThe . theory is applied to the problemof reducing the angular velocities of a tumbling asymmetrical spacebody to zero.
+
I. ISTRODUCTION
T
HERE ,\RE a number of papers [1]-[14]in the controlliteraturedevotedtothetime-optimal control of linear systems. There are a few papers [l], [15]-[20] ontheminimum-fuelcontrol of linear systems,andsomeothers [ a ] , [16], [21]-[25] dealing with the minimum-energy control of linear systems. I t is well known that the time-optimal control can be obtainedasan explicitfunction of the state forlinear second-order systems [SI-[lo], [12]-[14] and for some linearthird-ordersystems [12]- [14] bymeans of switching curves and switching surfaces. I n a recent paper [ l ] , the authors have shown t h a t analytic expressions for the optimal control, where the control vector is constrained to lie within a sphere, can be obtained as a function of the state, for a varietl- of performance criteria, provided the s ~ ~ s t eismlinear and self-adjoint. In this paper,which represents a generalizationof the results in Athans, et al., [ l ] ,the problems of minimurntime, minimum-fuel, and minimum-energy control are examined for a class of nonlinear systems, called norm
AND
R. T. LAACOSS$
time and with minimum consumed fuel. It mill also be shown t h a t if the control vector is oppositely directed from the state vector, then any initial stateis reduced t o zerowithminimum energ).. Themagnitude of the energy-optimal control can be found from the norm of theinitialstateandfromthe desiredterminaltime. Thus, the optimal controls are obtained as simple functions of the state. A physical system which falls in the class of norminvariantsystems is an asymmetricalbodyinspace [ X ] , [17]. I t is shown in Section VI1 that, for such a body,thetime-optimalangular ve1ocit)- controlis a simple closed form function of the angular momenta. 11.
SOME ~ I X T H E M A T I C - ~FORMvL-XS L
Let v and w be n vectors with components v i and wi, respectively, for i = 1, 2, . . , n. The scalarproduct of the vectors v and w is n
(v, w ) =
DaW'i.
(1)
r=l
T h e Euclidean norm of v is
i.e., the norm llvi: is the Euclidean length of v. For any vector x(t)
T h e Schwarz inequality states that is often a consequence of the conservation of momen( x ( t ) , Y ( t ) ! ; I !.x(f)l, *IlY(t>:l. (4 tum.Thecontrol u ( t ) isconstrainedto lie within a sphere of radius 211. I t will be shown that, if the control Xnother form of the Schwarz inequality states that vector attains its maximum magnitude and if the control vector is oppositely directed from the state vector, then Tf(t)lz(t)dt]2 an!- initial stateis reduced to the zero state i n minimum
1
[So
[
[S,
* Received December 12, 1962; revised manuscript rereil-ed 5 oJ ' f : ( t ) d t ] . '/z?(t)dl] (4-a) March 25, 1963. t Lincoln Laboratory,hIassachusettsInstitute of Technologl-, Lesmrrton. 1Iass.. operated n.ith support . - from the r.S. .%rnmy, Savg., and Force. wheref(t) and h ( t ) are bounded functions i n the interval t Deoartment of Electrical Enrineerine. Universitv of California. [0, T ] . The above notions will be used in the sequel. BerkeleJ', Calif., and consultant to" Lincoln Laboratoiy.
xacck&&* 197
A t h a m , et al.: Optimal Control of
1_
111.
Twogeneralclasses of systemswhicharenorminvariant are described in Section VI11 of this paper.
S T a T E J i E N T OF T H E P R O B L E M
Consider a nonlinear system of the form di(t) = gi[xl(t),
. . . , x,,(t); t ]
+
2 4 4 )
I\,'.
(5)
X(t)
=
+ u(0.
g [ x ( t ) ;tl
(6)
The vector x ( t ) is the state of the system at time f.; the vector u ( t ) is the control vector at time t. I t is assumed that the control vector u ( t ) is restricted to lie within a sphere of radius X ; in other words
I!u(t)lI5 M
for all t
IVIIXIMUM-TIME PROBLEM
In this Section it will be shown that the control
. . , n, which shall be written in the vector
for i = 1, 2,
S O L C T I O N OF T H E
2 0.
is the time-optimal control, where x * ( t ) is the solution of (6) with u = u * , i e . , x * ( t ) is the solution of
(7)
T h e following lemma is the crucial point of the demonstration. I t is also assumed that the unforced (or homogeneous) Lemma: Let G ( t ) be any control satisfying the conpart of the system (6), i e . , the system straint ( 7 ) and let % ( t ) be the solution of (6) which cor% ( t ) = g[x(t); t ] ; x(0) # 0 (8) responds to G ( t ) . Then d
has the property Ilx(t)li = Ilx(0)lI
for all t 2
o
-M 5 - 1]%(1)11 5 M . dt
(9)
Proof: Using (3) and (6) i.e., the normof the state vector is constant along trajectories of the system (8). Using ( 3 ) one concludes t h a t d W ) , W ) - ( g [ W ) ;4 , - Il%(t)lj = (9) is equivalent to dt Il?i(t,l' ( g [ x ( O ;tl, x ( t ) ) = 0 ( 10)
I!%(f)ll
f o r all x ( t ) and a12 t 2 0 . Of course, it is tacitly assumed that the homogeneous system (8) has a solution, for all t 2 0 and any x(O), such that (9) or (10) holds. The three problems to be solved are: 1) The Xinimzrm-Tiwze Problem. Find the control u , subject to the constraint (7), as a function of the state x ( t ) , which drives the solution of (6) from any initial state c to 0 in minimum-time. 2) The Minimzrm-Fuel Problem. Find the control u, subject to the constraint ( i ) ,as a function of the state x ( t ) , which drives the solution of (6) from any initial state I; t o 0 and which minimizes
w >
(14)
+ ( WI l w, fl (l 0 )
because of (10). Using the Schwarz inequality (4),
Hence, since
i t follows t h a t
or where k is a positive constant of proportionality and [0, T ] is the control interval. It must be noted thatfor somephysicalsystems (11) representsthe fuel consumed. The terminal time T is not fixed. 3 ) The Jlinimunz-Energy Problem. Find the control u, subject to the constraint ( i ) ,as a function of the state x(t), which drives the solution of (6) from any initial state t o 0 and which minimizes
Q.E.D. Kotethat used, then
if thecontrol
u * = -Jfx*(t):Ilx*(t)ll
is
in minimun1 time.
> Ilri
(30)
k Mdt
=
kMt*
(38)
0
which means that the time-optin1al control u* given by (31) requires fuel F* F* = k M t *
=
kI E:/.
(39)
Thus, F*
will drive an initial state states %/ given by
(3 7)
0
But the right-hand side of (37) is the fuel E consumed in the interval [0, T ] by ii. In particular, if the control (31) is used one finds
(28)
i n n1inimum time. Itis also easy toshow that the control
kllf'l 5
5P
(40)
for an\- ii. T h e conclusion is that the time-optimal control u*, i n addition to requiring the least time to reach 0 from E, consumesthesmallestamount of fuelpossible. Also, observe that the control u * , given by (31). is afueloptimal control for target sets of final states xf given by (28), and the control(29) is fuel-optimal for sets of final states 4 given by (30).
I t should be emphasized that the control u* simzt1taneousZy the minimumtime ami the minimum-fuel solution. For cases where the response time is of no importance, it will be demonstrated that the control? = - M x ( i ) j ~ ~ x ( yields f)~~
.
u=-aLy
0
ijg(t)))’
P=F*=kll2;(1time and t* = llt~l/X and z ( t ) is the solution of
To seethis,substitute
’
which
dt
= 0,
.
T Jlx(0ll
i>t*,
d
where
g(t>
\
( 5 1)
!It!)
’m/=--* T
dt
(5.2)
Hence,
t h a ta t t =
which implies
-a
(43)
Let
T , ( I g ( T ) l (= O .
I? be the e.nerg?*required by ir(f).Clearly, 2T
t o >-ield (Ig(t):l =
If Z(7)
Iltll x ( 0
g ( t ) = g[x(tj; I ] . -. -
(41)
ir into (15) toobtain I’ - .;,%(/)I! =
Z ( t ) is the solution of (6) corresponding to G ( t ) , ;.e., g ( t ) satisfies the equation
If the control u ( t )= G ( t ) , then
< a < 1M
requires fuel
In(49),
ll2;li - at.
(44)
then
li t*. T h e fuel consumed by,6 in the interval [O, by
(56)
(47)
i] is P given Therefore,
(57)
I t follows that there exist many fuel-optimalcontrols. Each fuel-optimal control requires a different response time. turns It out that the time-optimal control requires the least amount of fuel and as such,it is convenient to use in aitsystem. But VI.
and
!,!m!! is required that
S O L U T I O N OF T H E k I I X I M L X - E S E R G Y PROBLEM
* The constant 01 may be replaced by a time-varying function a ( t ) provided that 0 < a ( t ) < X for all t.
1! 2 ; 1 1
2 -
J
tI/Li(T)l!d7.
( 5 8)
0
ij2(Tj1:= 0. Therefore,
(5 9)
In this Section, it will be shown that the control
stated in Section I1I : t h a t is to say, thatgiven a desired response time T , the control (49) drives the system (6) from the initial state t to 0 in time T and minimizes
-
But,from(4-a) obtains
[using f ( t ) = I ’ u ( t ) / !and g ( t ) = l ] one
(62)
200
IEEE TRAATSACTIONS O N A U T O M A T I C C O A Y T R O L
and, using (60), that
Thetorquevector Therefore. if
Hence, from (54), it follows t h a t
B 2 E.
(64
Equation (64) means that any control ii(t) requires a t least as much energy as G ( t ) ; it follows that the control G ( t ) is anenergy-optimalcontrol,providedthatthe dissipated energy is measured by (50) i n the physical system.
JldY
~ ( t is ) thecontrolvector
in (67).
then the s?-stem (67) belongs i n the class of nonlinear systems for which t h e t h e o q of optimum control has been de\Feloped in the main part of the paper. I t follows that the torque vector
i.e., the torque vector with components
\.'II. \'ELOCITT COKTROL OF AX ASTMSIETRICXL BODY A physical system whose equations of motion are of the type discussed in the paper is a bod>- in space [ 2 i ] . This body may be a satelliteorotherspace vehicle. Define three principal axes 1, 2, 3 which pass through the center of gravity of the rigid body.Let 11,I?, I 3 be the three momentsof inertia and let sl(t),x?(/),s 3 ( t ) be the three angular velocities about the axes 1, 2. 3, respectively. I t is well known [26], [ 2 7 ] that the angular velocities r l ( t ) ,r z ( t ) , x 3 ( t ) are the solutions of the so-called Euler Equations
will force an>- initial angular momenta, and thus any initial angular velocities, t o zero in minimum time. This correspondstothestopping,intheshortest possible time, of any tumbling motion of the space body. Equation (70) meansthatthetorquevector T attains its maximum magnitude -11 and that the torque vector T is oppositely directed to the angular momentum vector y. Kote that each component of the torque vecI12l(t) = (I2 - 1 3 ) X ? ( t ) X & ) tor at timet is a function on1)- of the angular momenta I?*i.2(t)= ( 1 3 - 1 1 ) z 3 ( L ) x 1 ( f ) Ty(t) (65.) at time t . Thus, the torque is determined as a simple function only of the present state (i.e., angular momen13k:3(f) = (Il- I p ) ~ ~ l ( t ) ~ p~ (~ t()t ) , i tum) of the body. From the physicalpointof view the torque vector where ~ ~ ( t~ )2 (, t ) 7, 3 ( t ) are the three torque components ma!. be generated using reaction wheels or reaction jets. of the torque vector ~ ( t ) If. I l # I ? # I 3 , then the body For the space body the minimum-fuel velocity control has no axis of symmetry and is calledas?-mmetrical. Velocity control of partiall). s)-mmetric bodies has been problem is strongly dependent upon the ph?-sical orientation of t h e reaction jets on the bod!-. T h e necessary discussed by the authors elsewhere [ I ] . [14]. physical arrangementof such jets on the body whichwill Define three new variables y l j t ) , y z ( t ) . y 3 ( t ) by result i n a torque vector satisfying (69) is a subject of Yl(t) = I I X l ( t ) \ future investigations. y2(t) = I?.X?(t) (66) \'III. Two CLASSESOF NORY-~KVARIXXT SYSTEMS ~ 3 ( f )= 1 3 : ~ 3 ( t I)
+ + +
1.
Then yl(t), >(t), and y 3 ( t ) are the components of the angular momentum vector ~ ( t )T. h e y , ( t ) variables are the solutions of the system of equations
The tJ-pes of s)-stems considered in this paper were described by the vector differential equation
+ u(t).
X ( f ) = g [ x ( t ) ;t ]
(72)
The firstassumptionwas on thecontrolvector u(t), i.e., I [ u ( t ) :S[ J I . The second assumption was on the system itself, i.e.? ( g [ x ( t ) ;t l ,
do)= 0
(73)
for all x ( t ) and all t 2 0 . Two classes of systems which satisf)- (73) are presented below: 1) ,411 linear sl-stems of the form The system of equations (67) is nonlinear. I t is cas?- to show t h a t , if 7 1 ( t ) = p ( t ) = T 3 ( f ) = O , the11
+ u(f)
x(t> = A(i)x(f)
(74)
with 1; u(t)ll 5 X and
A(t) = - A'(t)
(75)
where A’(t) is the transpose of the matrix A ( t ) . Equation (75) means that A ( t ) is a skew-symmetric matrix. This class of systems, called self-adjoint, has been examined in Athans, et al., [l]. 2) -411 nonlinear systems of the form
x(t)
=
B ( x (tt))x, ( t )
+ u(t)
= f[Xl(t),
sz(t), f ] X 2 ( t )
&(t) = - f [ X l ( t ) ,
+ u&) +
x2(t), t ] X l ( t )
Sp(f)
&(t) = - s,(t)
+ U(f)
1
(79)
with (ii)
Such nonlinear systems are described by a skew-symmetric matrix B ( x , t ) operating upon the statex ( t ) . For example, the second-order system
&(t)
*l(t) =
(76)
with Iju(t),j,stem described by the equations
For the system (79) one finds
ztz(t)
is such a nonlinear system. The above two classes are not exhaustive.
IX. DISCUSSION OF THE RESULTS
The control which will make Ilx(t)l( decrease as fast as possible is
u(t)
=
- sgn ~ ( t ) .
(82)
Substituting (82) into (81) one finds It has been shown that for a class of nonlinear systems, called norm-invariant, the optimal control can be obtained as a simple explicit function of the state. Forminimum-timeandminimum-fuelcontrol of I t is well known [13], [14], [21], [22] that the control norm-invariant systems the control vector attains its (82) is not the time-optimal control for the system (79). maximum magnitude and is oppositely directed to the The reason is that the control (82) minimizes (81) o n l y state vector a t each instant of time; such a control is locally [because (83) is a function of x l ( t ) and x2(t)]. In simultaneouslytimeoptimaland fuel optimal.However, as pointedout,thefuel-optimalcontrol is not contradistinction, the control (12) leads to (19) which states unique, because there exist other controls, different from the time-optimal control, which require the same amount of minimum-fuel(althoughtheyrequire a longerre(84) sponse time). For minimum-energy control of norm-invariant systems it was shown that the control vector must be op- and since M is a constant, the controlu * , given by (12), positely directed to the state vector; the magnitude can minimizes the time rate of change of I(x(t)l(globally. It must be strongly emphasized that the optimal conbe found from the normof the initial state andfrom the trols obtained in this paper are independent of g [ x ( t ); t ] . value of the desired response time. As long as the homogeneous nonlinear system has the I t is worthwhile to note that the optimal controls structural property ( g [ x ( t ) ; t ] , x ( t ) ) = O , the state x (t) have been derivedwithoutthe use of theMaximum at any time t contains all the sufficient information to Principle [21]. T h e proofspresented werebasedon determine the optimal control. Thus, although the physicalreasoning andstandardmathematicaltechniques (such as the Schtvarz inequality). The reasoning actual values and the time dependence of the elements employedfor theminimum-timeproblemwasas fol- of the vector g [ x ( t ) ; t ] do influence the state x ( t ) , the lows: in order t o force any initial state to zero in the control is independent of the changes in g [ x ( t ); t ] . This shortest possible time it is necessary to reduce the norm independence is not found in other optimal control sysof bang(ix.,the Euclidean length) of the state vector zero to in tems. For example, the switching hypersurfaces bang systems depend very strongly upon the actual minimum time; a controlwhich will result in the fastest rate of change of the norm of the state vector,for every values of the eigenvalues. state in the state space, must be the time-optimal control, X. CONCLUSIOS which was found to be zmique for the systems treated. A natural question which may arise is the following: In this paper it has been demonstrated that fora class Given an arbitrary system and some constraints on the of nonlinear systems, called norm invariant, the time-, control(s),it ispossible todeterminethecontrol(s) fuel-, and energy-optimal controls can be determined as which will result in the fastest rate of change of the simple closed form functions of the state of the system. norm of the state vector and, so, derive a relationship T h e class of systems treated in this paper is a special between the control(s) and the state variables;will this one. However, i t is indeed reassuring and refreshing to
IEEE T R A X S A C T I O N S ON A U T O M A T I C C O N T R O L
202
find nonlinear sl-stems for which the optimal control can be found in a closed form a s a simple function of the state variables. L~CBSO\VLEDGMENT
R. E. Kuba and L. F. Kazda, “A Phase-Space Method for the Synthesis of Nonlinear Servomechanisms,” Trans. A I E E , vol. 1 5 , (Applications azd Indztstry), pp. 282-290; Kovember, 1956. 0. J. XI. Smith,“FeedbackControlSystems,“IIcGra\v-Hill Book Co., Inc., Ke\v York, h-.Y.; 1958. D. \V. Bushaw, “Optimal discontinuous forcing terms,!’ in “Contributions to the Theory of NonlinearOscillations,” vol. IV,
Princeton University Press. Princeton, N. J., pp. 29-52; 1958. hI. Athans, P. L.Falb.and R. T. Lacoss, “TimeOptimal \-elocit>- Control of a Spinning Space Bod?-,“ IEEE Paper KO. 6398. L. \Y. Seustadt,“Timeoptimalsystems u-ithposition and vol. 3, integrallimits.” J . Xatlz. A?zaIysis andApplications, pp. 406-427; December, 1961. X i . .Athanassiades, “Optimal Control for Linear Time Ifvariant REFERENCES Plants with Time, Fuel, and Energy Constraints, IEEE AKD IKDL-STRI., vol. 81, pp. 321-325; TnAss. os APPLICATIOXS [l] XI. .lthans, P. L. Falb, aFd R. T. Lacoss, “On Optimal Control January, 1963. of Self-.Adjoint Systems, to be presented at 1963 Joint -AutohI. .Athans, tXiinimum Fuel Feedback Control Systems: Second matic Control Conf. IIinneapolis, IIinn.; June 19-21, 1963. OrderCase, I E E E TRASS.os APPLICATIOSS AKD IKDUSTRT. [2] L. I. Rozonoer, “L. S. Pontr;-agin’s maximum principle in the x-01. 82, pp. 8-1i, IIarch, 1963. theory of optimumsystems, Axtontation and Renlote Control, 11..lthans. “1Iinimum Fuel Control of Second Order Svstems vol. 20, pp. 1288-1302, 1405-1421, 151i-1532; October, So\\-ithReal Poles,!’ to bepresented a t 1963 JointAutbmatic vember, December, 1959. Conf., hiinneapolis hlinn.; June 19-21, 1963. [3] R. Bellman, I. Glicksberg, and 0. Gross, “On thebangbang I. Flugge-Lotz and H. IIarbach,“TheOptimalControl of control problem,” QZ4Q.d. Appl. -lIaath., vol. 14, pp. 11-18; -April, Some .Attitude ControlSystems for Different Performance 1956. Criteria,” ASlIE Paper 62-J-ACC-6, presented a t 1962 Joint [A] C. A. Desoer, “The bang-bang servo problem treated by vari.Automatic Control Conf., Kew York, X. Y.; June 27-29, 1962. ationaltechniques,” InformationandControl, vol. 2, pp. 333J. S. Rleditch and L. \V. Seustadt, :;In .Application of Optimal 348; December, 1959. Control t o Xlidcourse Guidance, The Aerospace Corp., [j] E. E. Lee, “hIathematicalaspects of thesynthesis of linear Los .Angele.;, Calif., Rept, .I62-1732.2-5; July, 1962. minimum response time controllers,’ I R E TRASS. O S -3t-TOL. S. Poutryagin. et ~ l . “The , liathematical Theor>-of Optimal MATIC COSTKOL, 1-01. XC-5. pp. 283-289; Septem+, 1960. Processes,” Interscience Publishers, l-ew York, S . Y.; 1962. [6] J . P. LaSalle, ”Timeoptimalcontrol s!.ste~ns, Proc. .\-ut’l R. E. Kalman, “The Theory of Optimal Control and the Calcudcad. Sci., vol. 45, pp. 573-577; >April,1959. Ius of \.ariations,” RI.%S, Baltimore. hId., Rept. 61-3; 1961. [i]AI. .Athanasiades and 0. J. 11. Smith, “Theory and design of high-order bang-bang control systems,” I R E Tnass. O S - 1 ~ ~ 0 - B. Friedland, “The Design of Optimum Controllers for Linear Processes with Energy Limitations,“ .ISXIE Paper 62-JACC-8. MATIC COSTROL, x-01. .4C-6, pp. 125-134; XIay, 1961. .1. E.Pearson,“S>-nthesis of a RIinimum EnergyController [8] X. hI. Hopkin, “.I phase-planeapproach tothe design of at 1962 Subject to an .Average PowerConstraint.”presented saturating servomechanisms.” Trans. A I E E , vol. 70, (Coaznzzrx. J.ACC Sen- Yo&, S . Y.: June 27-29, 1962. and Electronics), pp. 6 3 1 4 3 9 ; 1951. S . S . Krasovskii, “On the theory of optimum control,” =1pp[. [9] D. C. hIcDonald,“Sonlineartechniques for improvingservo Jlath. and ,Ileclmnics, vol. 23, pp. 8?%919; Februar>-, 1959. performance,” Proc. L\-ut‘l. Electronics Cotzf., vol. 6. pp. 100H.Goldstein, “Classical Mechanics,-lddison-\\-esleyPublish421; 1950. ing Co., Inc., Reading, Mass.; 1959. [IO] I. Bogner, “-in Investigation of theSwitchingCriteriafor E. E. Lee,“Discussion of SatelliteXttitude Control,‘! d R S . , HigherOrderContractor Servomechanisms.’CookResearch pp. 981-982; June, 1962. Labs., Chicago, Ill., Rept. So. PR-16-9; 1953.
The authors wish to thank Dr. F. C. Schweppe and L. X. Gardner, Jr., for the stimulating discussions, and J. F. Nolan for the most critical reading of the manuscript.
’
An Expanded Matrix Representation
for
Multivariable Systems“ L. E. McBRIDE,
J R . ~m
K. S. XARENDRAt,
Summary-Thepurpose of this paper is todevelop a matrix representation for linear multivariable systems which describes bilateral propertieslike output interaction andcan be used to simplify the analysis of systems composed of interconnected multivariable elements. This representation is analogous to the circuit parameters used to describe two-port communications networks. Examples of the application of the proposed matrix to a physical system are given and certain properties are discussed which can be described in terms of either the system matrix or the principle of invariance.
* Received Januarl- 15, 1963; revised manuscript received April 3,
1963.
Division of Engineeringand verslty, Cambridge, hlass.
.Applied Physics,Han-ard
Yni-
~ M I ~ E R IEEE ,
INTRODUCTION H I L E A GREAT variety of useful techniques have been developed for the analysis of linear systems, the\- have generally been applied only to problems involving the effect of a single externally manipulated input variable on a single externall>; observedoutput.1Ian)-realcontrolsJ-stems,however, involve more than one input and output. Although such systems have been treated with some success by breakingthemdowninto a combination of single-variable components orblocks, such a decomposition is often completelyartificialanddifficult toarriveat. AIoreover,
