Observers for multivariable systems - IEEE Xplore

APRIL,

1966

Observers for Multivariable Systems D. G. LUENBERGER, Abshacf-Often in controldesign it is necessary to construct estimates of state variables which are not available by direct meavector can be approximately surement. If a system is linear, its state reconstructed by buildingan observer which is itself a linear system driven by the available outputs and inputs of the original system. The statevector of an nth order system withm independent outputs can be reconstructed with an observer of order n-m. In this paper it is shown that the design of an observer for a system with M outputs can be reduced to the designof m separate observers for single-output subsystems. This result is a consequence of aspecialcanonical form developed in the paperfor multiple output systems. In the special case of reconstruction of a single linear functional of the unknown state vector, it is shown that a great reduction in observer complexity is often possible. Finally, the application of observers to control design is investigated. It is shown that an observer’s estimate of the system state vector can be used in place of the actual state vector in linear or nonlinear feedback designs without loss of stability.

MEMBER, IEEE

Typicallythere will beassociatedwiththesystem (1) an output vector y of dimension rn < n of the form

y

=

H*x

(2)

u-here H* is an m X n output matrix.’ I t will be assumed that the outputs represented by the components of y are independent-or equivalently, that the matrix H* has rank wz. The problem discussed in this paper is that of reconstructing the state vector from the available outputs. The system 1%-hichperforms the reconstruction is called an observer. One possible method for obtaining the state vector is to build a model of the given system, drive themodel with the same inputs as the original system, and use the state vector of the model as an approximation to the unknou-n state vector. In this method the dynamicbeI. ISTRODUCTION havior of the observer is identical with thatof the sysA4KY COSTROL system designs are based on tem it observes. If initial conditions v-ere not set propstate vector feedback, where the input to the erly or if there n-ere slight disturbances, the model gensystem is a function only of the current state erally n-ould notrecoverfastenoughtoprovidean vector. For example, in the case of a continuous, linear, estimate suitable for control. time-invariant system of the form Another approach is to differentiate the available outputs a number of times and then combine these derivatives appropriately to obtain the state vector. In this case, the estimate responds instantaneously to disturn-here bances, b u t iis t severely degraded by a small quantity of x is an n X 1 state vector additive noise in the measurements. ZL is a n r X 1 input vector I t is important todesign an observer which is a comA is an n x n transition matrix promise bet\\-een these tn-o simple procedures. I t is deD is an n X r distribution matrix, sirable that the dynamic elements of the observer be such a design xould expressu ( t ) as a function of x and t ; faster than those of the system it observes, but not so u ( f ) = F [ x ( t ) , t]. The function F is determined by the fast as to possess the undesirable characteristics of difparticular design scheme employed. It may be the con- ferentiators (n-hich correspond to poles a t - x ) . trol function n-hich in some sense optimizes system perI t has been shou-n [ l ] that an observer for the system formance or i t may be determined by some other design (1) can be constructed which is itself a linear, time-intechniquypossibly trial and error. Such state vector variant system driven by the system i t observes. The feedbackdesigns offer many advantages with respect observer need only contain n -m poles and these may to both sl-stem performance and analysis. There holvis, be chosen arbitrarily. ever, one major disadvantage. In many control situaIn Section I1 of this paper the general theory of obtions the system state vector is not available for direct servers is reviewed with particular emphasis on singlemeasurement. In these situations, i t is not possible to output spstems. evaluatethecontrolfunction F[r(f), f ] , andhence, Section I11 containsthemaincontributionofthis eitherthecontrolschememustbeabandoned, or a paper. A nen- procedure is developed for designing obreasonablesubstituteforthestatevectormustbe servers for systems which have several outputs. Essenfound. tially,theproblemisreducedtoaseries of observer designs for single-output systems-these individual dehlanuscript receivedSeptember 27, 1965; revised Januark- 11, signs being relatively straightforu-ard.

L M

1966. This ~ ~work ..~ . - was suDDorted in D a r t bv the Office oi Kava1 Research Contract KOI%R-%(83). The author is with the Dept. of Electrical Engineering, Stanford Vniversity, Stanford, Calif.

* The ‘star” notation represents the transpose of a matrix. Thus the nzXn matrix H* is the transposeofan n X m matrix H. 190

191

LUENBERGER. OBSERVERS FOR MULTIVARIABLE SYSTEMS

In this section attention isfocused on the problem of T h e new procedure,basedonaspecialcanonical form for multiple-output systems developed in the Ap- constructingobserversforsingle-outputsystems.In Section 111 i t isshown that the multiple-output case pendix, not only leads to simpler observer designs but can be reduced to this form. also to stronger theoretical results than obtained previTwo possible constructions for an observer are conously. sidered here; both based on a well-known [3] canonical In Section 11,’ the problem of reconstructing a single formforanobservablesingle-outputsystem.Anaplinearfunctional of thestateratherthantheentire state vector is considered. I t is shown t h a t considerable proach more suitable for computation, which does not employ canonical forms, may be found in [l]. reduction in observer complexity is then possible. This Suppose an nth order system is governed by procedure is similar to one given by Bass and Gura [2] for the design of single-input, linear control systems. Finally, the stability implications of using the reconstructed state vector rather than the actual state vector is discussed. I t is shoxn that observers may be used and has a single output2 y=k’r. I t is assumed that the system is completely observable [SI, i.e., the vectors h, to realize both linear and nonlinear control laws without , (A*)”-lh arelinearlyindependent.Forsuch loss of stability. Thus, it is concluded t h a t observers A*h, can effectively surmount the difficulties associated with a system there exists a coordinate system in which the system is represented in the special canonical form [3] control design when the stateis not measurable.

-

I I. OBSERVERSWITH

ARBITRARY

DYSAMICS

The basic observer configuration is illustrated in Fig. 1. The system S I is assumed to be a linear time-invaria n t system. I t is assumed initially that the system is free, Le., unforced. The available outputsof this system drive a second systemS, which is the observer. Theorem 1 (provedin 111) shows thatundertheseconditions, the observer’s state vector will nearly always tend ton-ard a linear transformation of the first system’s state vector. Th.eorenz 1 (Observatiotz of a free system): Let S1 be a free system: 2 = A x which drives S,: i = Bz+ Cx. Suppose there is a transformation T satisfying T A - B T = C. If z ( 0 ) = Tx(O), then z ( t ) = T?;(t)for all t 2 0 . Or more generally, z(t) = T.r(t)

+ e B t [ z ( 0 )- Tr(O)].

(3)

The system in this form is represented schematically in Fig. 2. Here the vector x is represented in the new coordinatesystem so thatthe new state variables XI, x2, . . . , x , do not necessarilycorrespondto the original state variables-the original variables are suppressed from consideration for the present purposes. I t is readily verified that the characteristic polynomial of the matrix in (7) is sn+ansn--l +oln-lsn-2+ .

-

+Cub

Consider an nth order observer for this system govKotice that in Theorem 1 i t is not necessary for A and B to be the same size; they only have to be square. erned by If A and B have no common eigenvalues, there is always a unique T satisfying T A - B T = C [l]. Assume now that the system SI is governed by i = Ax

‘+ Dzt

(4)

where u is an inputvector. An observer driven by both the plant input and outputs governed by i = Bz

+ CX + TDzt

Fig. 1. A simpleobserver.

2

(5)

will behave according to (3). Therefore, observers designedfor a free system can be applied to the forced system if the input is suitably connected to the observer. The construction of a n observer rests on the solution of the matrix equation T A - B T = C. T h e soIution T must have rank great enough to guarantee the recovery Fig. 2. Canonical form of observable system. of the unmeasurable state variables. As illustrated below, observer design consists in choosingB and the part of C which is unspecified in such a way that T has that 2 As a matter of convention the transpose of a vector h is repproperty. resented by h‘ throughout this paper rather than by It*.

192

IEEE TRANSACTIONS APRIL CONTROL ON AUTOMATIC

-61 1 o - - . o

(P1

sumption is that of complete observability with respect to the outputs. A system

- ad

x = -4x (Pn

- a74

BZ + CS.

y = H*s

(12)

iscompletelyobservable [3] if the nx (nnz) matrix [ H , A *H, . . . , (A *)n-lH]has rank n. Often a rank of In this case i t is readily verified that T A - B T = C is n will be achieved with a smaller number of powers of satisfied b y T = I so that state variables of z are in direct A * times H. The observability index v of the system (12) correspondence 11-ith those of x. Since the characteristic polynomial of this system is sn+pnsn-'+ . . +Dl and is defined astheleastpositiveintegerforwhichthe matrix [ H , A *H, . . . , ( A *)I-'H] has rank n . The obthe coefficientspi are arbitrary, it isclear that the poles servability index plays a key role in the theory of obof a n observer of this form are arbitrary. servers for nlultivariable systems. Consider now the problemof constructing an( n- 1)th For some systems the extension to the multiple-outorder observer for (7). If the observer is taken as put case is elementary. For example, consider the fourthorder system shown in Fig.3. I t is assumed that the two O1 variables x1 and x3 are available for direct measurement. Thefourth-ordersystemmayberegarded as two coupledsecond-ordersubsystemsasindicatedbythe dashed-line boxes in the figure. The output of the first box is the measurable variable x1 and the input is the measurable variable x3. Therefore, since i t is possible t o measure the input and outputof this second-order subsystem, a first-order observer may be constructed for this subsystem. Similar considerations apply to thesecond box, so i t is seen that an observer for the total system can be built up from two separate observers, each T A - B T = C is satisfied by observing a single-output subsystem. T h e seemingly fortuitous situation in the above example is actually commonplace. In fact, all multiple-output observing problems can be reduced to the observation of single-output subsystems. This is a result of the fol1 -Bn-1 lowing theorem (proved in the Appendix) guaranteeing the existence of a special canonical form. Theorem 3 (Canonical Representation of X d t i p l e Output S y s t e m s ) : Supposethatthenthordersystem i = A x n-ith associated output vector y =H*x is complete1)- observable n-ith observability index v. Suppose further that y consists of m independent components. Then there is a nonsingularlinearcoordinatechange such that in terms of the neu- coordinates the system has the representation shown in Fig. 4. In this form the system consists of m component subsystems, each with one observable output u-hich is a linear combination of the components of y. The orders of the subsystems satisfy nl+nl+ . . . +n,=n, and the largest subsystem is of order v. The subsystems are coupled to each other 0nl>7 through their outputs. From the previous developments it is obvious that an observer can be constructed for each of the singleoutput subs)-stems, since the inputs to thesubsystems areavailableformeasurement.Furthermore,thekth observercanbedesignedwith nk- 1 arbitrar). poles. Thus, by employing Theorems 2 and 3, i t is easyto deduce Theorem 4: observerforthesystem(12)canbe constructedoforder n--nz. (.t istherank of H*.) =

(8)

-

1.

I

LUENBERGER: OBSERVERS FORSYSTEMS MULTIVARI.4BLE

1966

Furthermore, the n--m poles of the observer are arbitrary. This result is slightly stronger than the corresponding conclusion of [ l 1. Furthermore, in the form developed herethereisestablishedasimplealgorithmforcomputing the observer in terms of the single-output subsystems, whereas the method of [ l ] is not so straightfonvard.

Example: The sg-stern illustrated in Fig. 3 is already in canonicalformappropriate fordesign of asecondorder observer. The poles of the observer are arbitrary and will both be chosen to be -3. T h e design is carried out separately for each subsystem. System S1 is governed by

193

According to the results of Section 11, an observer with a pole a t - 3 driven by xl = ,1 0 , x will produce Tx where

or T = ,1 - 1, . The observer will be governed according to ( 5 ) which in this case is i = - 3z

+

x1

- sa.

(15)

The estimate .t2 of the variable x? is constructed from the measurement x1 and z according to

A similar procedure applied to the subsystem Sz leads to the observer

s2

r----------1

SI r---------- 1

Thecompleteobserverforthesystem Fig. 5. Fig. 3.

Fonrth-order sytern of example.

a h ORDER

I

r

ORDER

a

w

I

a

a

ORDER

Fig. 4.

Canonical form of multiple-output system.

Fig. 5.

Observer for fourth-order s>-stern.

is shown in

IV. OBSERVING A SINGLE LINEARFUKCTIONAL Sometimes it is only necessary to estimate a single (but prespecified)linearfunctional of a sytem's state vector. This is the situation, for example, in the design of linear,time-invariant,statefeedback for asingleinput system. In these instances, an observer of considerably reduced complexity can often be constructed \vhich will produce this single quantity. Observation of a single linear functional is similar in concept to a feedback design procedure developed by Bass and Gura [ 2 ] . The methodin [ 2 ] is not an observer method in the sense of this paper and does not enjoy the closed-loop stability properties developed in Section V . T h e conclusions in this section concerning the required dynamic order of such an observer, however, coincide \vith the corresponding conclusions of [2]. Imagine an observer constructed for a multiple-output system according to to the scheme of Section 111. T h e o u t p u tof the observer is an estimate of the system state vector x. In order to obtain an estimate of a linear functional of x, say u'x, the same linear functional of the observer output is taken. The result is shon-n in Fig. 6. The largest block in the observer has exactly Y- 1 poleswhich may be chosen arbitrarily. Suppose these poles are chosen first. Then the poles of each of the other blocks of the observer can be chosen to be a subset of the poles of this largest block. Xou;;, corresponding to each output y k there is a transfer function of the form &(s)/A(s)from yn throughtheobserverto u'2. T h e polynomial A(s) is the characteristic poll-nomial of the largest block in the observer, and &(s) is a polynomial of degree no greater than that of A(s).

IEEE TRANSACTIOKS CONTROL ON AUTOX4TIC

194

Thus, i t may be concluded that the observerof Fig. 6 is equivalent to the form shon-n in Fig. 7 when the individual blocks of the original observer have common poles. Xn observer of this form can be realized by a system of order v- 1; therefore the follon-ing theorem is established. Theorem 5: X single linear functional of the state of a linear system can be observed b>a system lvith v- 1 arbitrarl- poles. (v is the observability index of the s!.stern.) .As pointed out in Bass and Gura [2], v- 1 is often considerabll; less than n-m, the order of a complete observer.Infact, (%,:’HZ) -1 < v - 1 I n - m . t\vent!-fifth order s!-stem with five outputs, for example, may require as fen as four arbitrar>- poles to construct an estimate of a single linear functional of the state vector.

APRIL

I I I for this system can be used as a first stepin the design procedure. Using the results obtained in the previous example .??

+

2 4

=

+ 3x3 - z

XI

-

6 ~ .

(19)

By carrying out some straightforward manipulations, the observer shovx in Fig. 8 is obtained. In general, i t is possible to shortcut the two-step design procedure outlined in this section. One may boldly hJ-pothesize the required form of the observer (according to Theorem 5 ) and solve for the unknown numerator pol>-nomialsdirectll--withoutfirstconstructingthe canonical form. I t turns out that this shortcut procedure saves only a small percentage of the labor involved and does not offer much insight into the observer process. This shortcut procedure follon-s the spirit of Bass and Gura [2], however,and is certainll-recommended for actual computations of observers of this type.

Example: Consider again the fourth-order system in Fig. 5 . Suppose i t is desired to reconstruct the single linear Y. CLOSED-LOOP ST-ABILITY PROPERTIES functional x2+x4. &Accordingto Theorem 5 an observer lvith a single arbitrary pole is sufficient. If the pole is -4s stated in the Introduction, the investigationof obchosen to be a t - 3 , the observer constructed in Section servers is aimed a t circumventing the difficulty of real-

izing state vector feedback controlwhen the entire state vector is unavailable for measurement. Consider a system governed by 2 = Ax

+

Dld

(20)

u-ithoutputs y = H * x . Supposethata (possiblynonlinear) control lax of the form 21 = F ( s ) has been derived for this s!-steIn by some design scheme. -An appropriate observer for (20) is f

i

--__-___

=

Bz

+ Cs + T D u

\\-here T A - B T = C. Cx must be derivable from the outputvector y : hence, C=GH* forsomeappropriate matrix G. The estimated state is a linear combination of the system outputs and the state vector of the observer

-1

Fig. 6. Observing a single linear functional.

.t = L s

1

Fig. 7.

-

(21)

+ Kz

(22)

\\-here L + K T = I (the identit>-). The control law 21 = F(.v) can be approximated by a controllaw 12 = F ( 2 ) based on the estimated state vector. The completesystem is then governed by the equations

1

+ DF(.f) i Bz + CX + TDF(.t) 2 = Lx + k’z. f = Ax

Reduced observer for single linear functional.

=

A

A

x 2 +xq

U

1 Fig. 8. First-order observer for example.

(23)

I t is thepurpose of thissection to investigate the stability properties of the control system governed by (23). The system equations (23) can be rearranged so t h a t man>- of theirstabilitypropertiesbecomeclearlyapparent. Defining Z = Z - Tx,f = . t - x and then subtracting T times the first equation in (23) from the second leads to the equivalent system

1966

+ DF(i)

Proof: 4 s a first step in the proof a quadratic Liapunov function is constructed for the observer i= BZ by the .? = Bz standardprocedure for stable,linear,time-invariant j = KE. (2.2) s1-stems [SI, [ 6 ] .For this purpose define P as the unique solution to the matrix equation PB+B*P= - I . I t is P so defined is positive I f , initially, the estimated state vectoris equal to the well known thatthematrix actual state vector, i.e., ~ ( 0 ) = 0equality , n-ill be main- definite and that 8'P8=::211 2 p is a Liapunov function i= B i withderivative (d::'dt)II%;? p = tained for all future time. This important fact is due to forthesystem overallsystem (24) define Tl'(.v, 2) what might be described as the complete u~zcontrollabil- -112'12. Forthe bV(x, 8) is clearl?; positive definite. ,Mso ity of the observer from zt. I t implies t h a t if there is = l,'(s) initial equality between the state and its estimate the W ( x , E ) = G,T'(x)[dx DF(P)] - l ( E j ) 3 closed-loop system using the estimate behaves exactly = L,7(x) V,V(x)D[F(.i-) - F ( s ) ] like the closed-loop sl-stem using the actual state to obtain the control. 5 c ( x > c ~ ! ~ G , v ( . x ) ~ ~ - lizI(? (26) Generally the initial equality between the state and where thepositiveconstant c1 is determined by the its estimate will not obtain, and stability properties of Lipschitz condition. the complete s>-stem, including the observer, must be Using ~ = K from Z (24), the above inequality can be investigated. first result in this regard applies to linear converted to control laws. If F ( x ) = FN is linear, the closed-loop system using the actual state is i = (A +DF)x. I t has been Ti'(s, z) 5 C ( s ) GZl~VJ(H)(I .11Z(1 - ' y . ( 2 7 ) shon-n [ l ] , [4] that if an observer with transition matrix L'sing the function T V i t will non- be shon-n t h a t a n y B is used to supply an estimate of the state vector, the trajectory of the system (24) is bounded. Obviously closed-looppoles of the overallsystem (23) arethe is bounded on anytrajectory.Condition 4) on the eigenv-alues of A f D F and of B. In other words, the obLiapunov function l,r implies t h a t for sufficiently large server does not disturb the poles of the original system x and bounded 3 the function l P ( x , 2) is negative defib u t merely adds its on-n poles. nite. Therefore, since W(x, E ) + = as i t is imIn a similar fashion, i t is possible to investigate the possible t h a t l l x l l increase without bound. Thus there effect of an observerin realizing a nonlinear control law. is an K>O such t h a t for all t > O , I l x ( t ) , l O thereis a i = -4s D F ( s ) (25) finite time T such t h a t for t > T , L ~ ( . ~ ) + c ~ ~ ~ V ~ l ~ ~ ( x < O throughout the annulus E < ' ~ . L ' / ; T is negative defin'ite in thisannulusand is asymptoticallp stable in the large [SI. I t is assumed thefunction 5 E . Since E was arbihere that the asymptotic stability of (25) is established x must tend to\-\-ard the circle trary, x tends to 8. This establishes asymptotic stability bytheconstruction of a continuouslydifferentiable Liapunov function l'(x) for the system n-hichsatisfies in the large. the following conditions d = As

+!IZ'~~~.

+

+

ilZI12

+

.llq

+

I~.Y~~-+=

I(

+

.II&:I

1.~11

1) T'(x)

VI. COXCLVSIOSS

> 0 for N #e, V(8) = 0

The observer theory developed in this paper can be comparedwithothermethods of state estimation. In the case of noisymeasurementsandunknon-n nois>: disturbance inputs, an optimal least-mean-square estia linear estimator The first three assumptions aresufficient to guarantee mate of the state may be generated by If the estimator operates on the infinite past and as>-mptotic stabilityof (25), while the fourth is an addi- [i]. is a tional assumption xx-hich is often satisfied in p r a ~ t i c e . ~the noise statisticsarestationaq-,theestimator lineartime-invariantsystemdrivenbythemeasureTheorem 6 shows that under relative117 mildcondiments. Such an estimator will act as an observer in the tions the observer scheme outlined above leads to an sense of t h i s paper if the noise disturbances are suddenly asymptoticall>: stable system. Theorem 4 : Assume that there is available a Liapunov disconnected so that the systembecomes free and noisebe regarded as function for the system 3=.4x+DF(x) nhich satisfies less. Therefore, optimal estimators can observers with their pole locations determined by the theconditions 1)-4) listedabove. If F ( x ) satisfies a uniform Lipschitz condition and the observer is asymp- statisticalproperties of the noise. Inmanypractical totically stable in the large, i.e.,B has its eigenvalues in situations (namely those in xi-hich the noise level is sigthe left half plane. The complete system (24) is asymp- nificant), statistically optimal estimators offer excellent advantages over other estimation schemes. As the noise totically stable in the large. leveldecreases,however,theoptimal pole locations move toward - M and in the limiting case of perfect 3 For example. if is a pd quadratic form, there is a c>O such (noise-free) measurements, the statistically optimal that - V / Vz.VII>c!ls!l+m. 2) T.,'(X)-+= as l l x l : + ~ 3) ~ ( xE ) T(i,) =vz~'(x) [ A x + ~ F ( x )mthesis, etc.). Genefall>-, there seems linear dependence, all columns of the form (A*)"j to be little reason to choose observer poles much faster where k > i can be skipped, since they also must be than the other poles of the closed-loop s>.stem. So far, dependent on the previous columns. however, other than in the statistical case? thereis little -4s a result of this procedure, there is defined a n arra5theory devoted to the problemof choosing observer pole of 71 independent x-ectors locations. The procedure outlined in this paper decomposes con/?I, - 4 * h l , ' . . , (-4*)"-'h1 troldesign into state reconstruction and control of a I??, -4*h2, . . . , ( A * ) i - v / 2 completelymeasurablesystem.Theeffectivenessand practicality of the method lies in the fact that stable opserver poles do notaffect overall system stability. Theorem 2 of [ l ] and Theorem 6 of this paper are but two where for each k, v p s u . Furthermore, by construction resultsconcernedwiththestability of combinedobthere are coefficients a i j ( k ) such t h a t server and control sj-stems. Other questions concerning m r-1 speed of response,time-varyingsystems.andvarious (A*)"%: = azj(k)(A*)i/zj (28) types of stability under various assumptionsoffer fruitj = 1 i=o ful areas for research. One of the most important results described in this where a i j ( k )= 0 for i > v k and a i j ( k )= 0 for i = up if j 2 k . T h e desired canonical form of the system \\-ill have a paper is the canonical form given in the -Appendix. This structure similar to the structure of the above arraJ- in canonical form can be used to extend many n-ell-kno\m that the kth subs>-stem \vi11 be of order VI;. Hon-ever, the properties of single output or single input systems to state variables of the kth subs>-stem \vi11 be defined in multivariable s>-stems. terms of vectors from the complete array rather than APPENDIX just the kth ron-. Because of the complexity of notation Theorem 3 (Canonical Representatiou of AllultipleOut- due to the several indexes required, the explicit transformation defining the appropriate nen- state variables p u t S y s t e m s ) : Supposethatthenthordersystem n-ill be .?=Ax n-ith associated output vector y = H * s is conl- is notparticularlyilluminatingandtherefore defined pletely observable Fvith observabilitJ- index u. Suppose suppressed.The ne\\- coordinatesareinstead subfurther that y consists of m independent components. implicitlJ- in terms of a schematic diagram. The kth s>-stem takes the form shon-n in Fig. 9. Thenthere is anonsingularlinearcoordinatechange such that in terms of the ne\\- coordinates the system has the representation sho\\-n in Fig. 4. In this form the system consists of m. component subs)-stems, each with one observable output which is a linear combination of thecomponents of y. Theorders of thesubsystems satisfy nl+n%+ . . . +x,=% and the largest subs)-stem is of order v. The subsystems are coupled to each other only through their outputs.

Proof: The first step in the proof is the generation of a certainset of n linear independent vectors. The procedure usedhere is identicalwiththatemployedby Bass and Gura [2] for another purpose. Since the matrix [ H ,A*H, . . . , A*'-'H] has rank n , n independent vectors can be taken as a certain n columns of this matrix. Todefine these vectors precisel>-

Fig. 9. kth subs!-atetn of canonical form

The outputs of the nz subs!,stenls are each linear com-

binations of the original outputs and hence are themselves measurable quantities. Conversely, the new outputs are 1inearlJ- independent so the old outputs can be recovered from the ne\v. (The independence of the ne\\outputs follo\vs from the fact that the transformation 121, h2, . . . , h , of the matrix relating the Startwiththecolumns old and nen- outputs is triangular matrix H . \\-ith 1's along the diagonal.) hdjointo these thecolumns A*/zl, A * h 2 , . . , In order to establish that the proposed canonical form A*h, one b\* one, checking that each nen- column is in fact a linear coordinate change of the original sysis linearly independent of the previous ones. (I-se tem, it is only necessary to verify that a l l variables of the Gram-Schmidt orthogonalization procedure.) the form z = K ' s in the canonical form satisfJ-5 = ( A * k ) ' x . If any of the ne\\- columns is found to be depenThat this requirement is satisfied b\- the kth subsystem dent, omit itfrom the matrix andgo on to the next. shonm in Fig. 9, fo1lon.s directly from ( 2 8 ) .

[4] D. G. Luenberger, “Determining the state of a linear system with of observers of low dynamic order,” Ph.D. Dissertation, Dept. Elec.Engrg.,StanfordUniversity,Stanford, Calif., 1963. [SI J.,LaSalle and S.Lefshetz, Stability by Liapzmoa’s Direct :1fethod w t h Applications. Sew York: -Academic, 1961. [6] R. E. Kalman and J . E. Bertram, ”Control system analysis and design via the ‘second method’ of Lyapunov--1. continuous-time systems,” Tmns. A S M E , ser. D , J . BasicEzgrg., rol. 82,pp. 371-393, June 1960. 171 R. E. Kalman and R. S . Rucv. “New results i n linear filtering-and prediction theory,” TraiZs. A X M E , sei. D, J. Basic Engrg., PC. 95108, March 1961. [SI A. E. Bryson. Jr., and I>. E. Johansen. “Linear filtering for ritnevarying s>-stems using measurements containing colored noise,“ ZEEE TTans. on Automati< Control, vol. AC-10, pp. 9-10, January 1965.

ACKNOWLEDGME?JT

The author lyould like to ackno\\?ledge - the benefit of several discussions with P. Haley during the course of this research.

REFERESCES [ l ] D. G. Luenberger.“Observing the state of alinear system,” MIL-8,pp. 74-80, , ZEEE Trans. on :liilitary E l e r t ~ o n i c ~vol. -4pril 1964. [2] R. 11’. Bass and I. Gura, “High ordersystem design via statespace considerations,” 1965 Joint &UtOnlQtif ContioI Conf. (Puep ~ i n t pp. ~ , 311-318. [3] R. E. K@lman. “Mathematical description of linear dpamical systems, J . S l d J I , sei. - 4 , on Contiol, vol. 1, no. 2 , pp. 152-192, 1963.

> .

O n the a priori Informationin

Sequential

EstimationProblems

Abslracf-In this paper, the effectof errors in the a priori inforout on mation is studied when the sequential estimations are carried the states of linear systems disturbedby white noise.Four theorems are derived to describe the mutual relations among the three covariancematrices,namelytheoptimum,calculated,andactual covariance matrices, where the last two are based on the incorrect a priori information. By finding the upper bound for the variance of the actual estimate, performance of the Kalman filter is prescribed and the knowledge is utilized for design of the combined system of analog and digital filters. A phase-locked loop receiver is used as an example of analog filter and the considerable improvement on the estimation process is deduced by the theory and it is confirmed by the experimental simulation on the digital computer.

I. ISTRODUCTIOS

T

H E O P T I l I r l L FILTER n-hich has been introduced by Kalman’.? into the field of system theory J-ields the minimum variance estimate of states of linearsystems, n.hich arecontaminatedbywhite Gaussian noises, when a set of sequential observations are carried out. The basic featureof this filter is that the estimate of states is up-dated by a sequence of observations so as to minimize its variance, or equivalently, if

noise is Gaussian, to maximize the conditional probabilit,: density of the current states after having a set of observations. The original theor)- assumes an a priori information on the initial states and its variances. Hon-ever, there are casesn-henthisaprioriinformationmaycontain certain errors or some informationmay not be available a t t h e beginning of the estimating process. Soong3 examined the effect of errors in the a priori data on the a posteriori variance of estimates for the single-stagecase,i.e.,for thenonsequentialcase in ‘15-hich estimation is carried out after a sequence of observation has been completed, and derived the deviations of thecalculatedandactualvariancesfromthe true minimum variance. Thusno real-time estimation is performed in this case. Also the state being estimated in this model is a constant state vector so that the state transition matrix is an identity matrix and no s>-stem noise is assumed there. The same problem is studiedinthispaper for the multistagecase,namelyforthecase \I-hen sequential observations and estimations are performedon the state in real-time. Also a general transition form of states is assumed subject to excitation of the system noise. Then there is a n investigation of hon- the error in the a priori information propagates through the sequential estimation process and how i t affects the final result of estima-

irlanuscriptreceived June 17, 1965;revised Januar!. 18, 1966. This paper represents one phase of research carried out at the Jet Propulsion Laboratory, California Institute of Technology,under SAS.4 Contract S.AS 7-100. ‘The author is with theJet PropulsionT.aboratory,Califor~lin Institute of Technology. Pasadena. Calif. K. E. Kalman. “X new approach to linear filtering and prediction problems.” TYQTZS. A S M E , ser. D , J . Basic Engrg., vol. 82, pp. 35-45. March 1960. * R. E. Kalman and R. S. Bucy, “Yew results i n linear filtering T. T. Sooyg, “On a priori statistics in minimum variance estima and prediction theory,” Trans. 4S:lIE, ser. D , J . Basic Engrg., 1-01, tion problems, Trans..ASME, ser. D , J . BasicEngrg., vol. 87, 83, pp. 95-107, llarch 1961. pp. 109-112, hlarch 1965.

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