Input design for identification of aircraft stability and control derivatives

APPENDIX A INPUT DESIGN IN TIME DOMAIN A.I PROBLEM STATEMENT Consider a continuous time varying system • x = F(t, 9)x + G ( t , 9)u (A.I) x(0) =0 0 <_ t <_ T where x is a n x 1 state vector u is a q x 1 control vector, and F(t, 6) and G ( t , 9) are n x n and n x q matrices, which depend on m unknown parameter 9.

There are noisy measurements of some linear combinations of state y = H ( t , 9)x +v A^ z + v (A.2) y is a p x 1 output vector and v is p x 1 white noise vector with zero mean and known covariance R ( t ) .

The problem is to choose u from a class of second order processes inde- pendent of v to obtain good estimates of parameters from the measurements of u and y. The class of inputs is such that T u (t) u(t) = 1 (A. 3) A . 2 INFORMATION MATRIX The information matrix for parameters 9 is If " (A.4) - / (It (A. 1) can be solved for state vector x,

r

x(t) = / <f>(t, T) G ( T , 9) U(T) dr ( A > 5 ) o where <p(t, T) is the transition matrix and follows the differential equation T) > = F(t, 9) c|,(t, T) (A. 6) <|>(T, T) = I Vector z can be written as z(t) = / H(t, 9) <j)(t, T) G(T, 9) u(x)dT = / H(t, o

q

r

= E / H ( t , ( ,9) <|>(t, T) G ( T , 9) u (T)dT ± ± (A 1=1 J 1=± -'o Therefore, t •gz (H(t, 9) <()(t T) G (T, 9)} u (T)dT f d o 1 1 / O

fc

1 f

A E / A . ( t , T) u.(r)dT ' (A.8) ~ 1=1 J Using (A.7) in (A.4), q / / T _1 L M = / E / A . ( t , T) u.(T)dT R - ( ) t J x 1 Jo i=l o

q

C

E / A (t, s) u.(s)ds dt J j=l 7 J o T T T

/" /" q q f / _i I

T T R s = / / E E / ^i^* ) ^^ A - ^ ' dtju (T)U.(S) dT ds O O SUp(T, S) where sup(t, s) is the higher of T and s. This can be written as T T r q / s / E M^CT. > u (T)u.(s)dT ds (A.9) ± / o o where M ,(T, s) = 1/2 { ( t , T) R C t ) A . ( t , s) ± A J J SUp(T, S) + A ( t , T ) R C t ) A (t, s)}dt ( A . 1 0 ) i It is clear that M .(T, s) = M (T, s) = T.(s, T) (A. 11) ±: M The next sections show how one can work with the three criteria of optimal- ity outlined in Chapter in .

A . 3 MAXIMIZATION OF A LINEAR FUNCTION OF THE INFORMATION MATRIX From (A. 9), a linear functional <£ (see Section 3.3 for definition of ,2?) of the information matrix is

r r -

. A^(M) = / / E (T, s)) U,(T) u.(s) dr ds x j J J i,j=l

•T /r

T,

u

I f ' (T) P ( T , s) u(s) dT ds (A.12)

J J

• J

i _ 0 0 where P (T, s) =^( _.( ) (A. 13) ± j M i T j s) P(t, s) is a symmetric positive definite matrix. To maximize a linear function of the information matrix for the given class of inputs, it is necessary to maximize (A. 12) under the constraint (A. 3) . It is straightforward to show that this is achieved by solving the following eigenvalue problem: P(T, s) u(s) ds = XU(T) (A. 14) 0 < T < T The maximum eigenvalue X is the maximum value of the trace of the information matrix and the corresponding eigenvector is the desired optimal input. Mehra shows this eigenvalue problem can be recast into a linear two-point boundary value problem. The solution technique to this problem is given in Chapter III.

A . 4 DETERMINANT OF THE DISPERSION MATRIX In this and the next sections, we prove some important theorems. These lead to a computation algorithm presented in Section 3.4.3.

*^ ;ta Theorem A . 1: A necessary condition that an input u (t) minimizes |D^ is that * * (i) u is an eigenvector of P with eigenvalue m.

k * (ii) Any other eigenvalue, X, with eigenvector u , of P follows the inequality 1 k £ m + 2 TrUM*" M* ) } (A. 15) where P*..(T, s) = TrCM*' (T, s)) (A. 16) M i j T /-T k (A 17) / ? M (T. s) u*(T) u (s) d T d s ' x / J i.J-1 ^ 3 n o o and /•T P*(T, s) u^s) ds = X, I^T) (A. 18) Proof: (a) From (A. 11) M* = / / Z M . . ( T , s) ufCi) u*(s) di ds 1J 1 3 J J i,j=l >J o o T /• T q / *-1 / Z Tr(M M (T, s)) U*(T) u*(s) dx ds / J 1 =1 1J J o o ?J

T

/" r *T

m = / / u (T) F^T, s) uvs) dT ds (A.19) • o o Therefore, X >m (A. 20) max — * * Equality holds if u (t) is an eigenvector of P corresponding to its maximum eigenvalue m.

(b) Consider an input u (t) which minimizes |D| . Since logarithm is a monotonic function, u (t) must also minimize log |D| . Let A- be * k an eigenvalue of P and u (t) the corresponding eigenvector.

Consider an input u, k u(t) = a u*(t) + B u (t) (A.21) The energy constraint (A. 3) requires 2 2 a + B + 2aBY = 1 (A. 22) where T T k ( A 2 3 ) Y = f u* (t)u (t)dt ' o giving C A 8B ~ ~ (a ^ The information matrix for input u(t) is 2 * 2 k * k K M = oM + 3 M + 2agM (A.25) where T /-T k M M T s U , t_, i^ ' > £(T) u*(s) dT ds (A.26)

•/

o "o

3. log ID I k

= -Tr{M* -"-(-2YM* + 2M* ) } fi n p=0 . - Z Y ( m - A ) k ( A * Since u minimizes |D| and y can be positive or negative, for all * eigenvalues of P A = m or Y = 0 (A. 28) k If Y ^ 0 for one particular eigenvector of P (except possibly u ) , * it cannot be zero for any other eigenvector of P , since the eigen- vectors of a positive self adjoint kernel are orthogonal. Then all * eigenvalues of P must equal m. In the light of Equation (A. 19), * * this implies that u is also an eigenvector of P with eigenvalue m. Then y = 0 for all other eigenvectors .

* If y = 0, then u is an eigenvector and the corresponding eigen- * * value is m. Thus, we show that u is an eigenvector of P with eigenvalue m.

% Since u minimizes |D|

3 log ID I

1 k 1 k k = -Tr{-M* (M* M*~ M* + 2M* - 2M ) ,2 3=0 36' _ , =4Tr{(M ^ ) } + 2m - 2X, (A. 29) > 0 or 1 k A < m + 2 Tr{ (M* M* ) } (A. 30) K.

Theorem A .2: A sufficient condition that an input u (t) is a stationary point for i i * * |D| is that u be an eigenvector of P with eigenvalue m.

m, Proof: Assume u is an eigenvector of P . Consider any input u (t) satisfying the energy constraint and define m (A.31) u ( t ) = au*(t) + 3u (t) Since the eigenvectors of P form a complete set and are orthogonal, m u ( t ) = Z (A. 32) 1=1 Z a. = 1 (A. 33) 1=1 Using (A.27),

, *_i * * *,•

x x

9 log |D|

= -Tr{M (-2a M + Z a M ) ± * * = 2a m - 2a m = 0 (A. 34) * * m * where a is the component of u in u . Since u is a stationary point of log |D| and logarithm is a monotonic function, it must be a stationary point for |D| .

A . 5 LINEAR FUNCTIONAL OF THE DISPERSION MATRIX * Theorem A. 3: A necessary condition that an input u (t) minimizes ^(D) is that * * * (i) u (t) is an eigenvector of Q with eigenvalue <2?(D ).

* (ii) Any other eigenvalue A, of Q follows the inequality ( A 3 5 l *-l *k 2 , ' > and . T , s - , s .

Q.*(T, s ) - t f O ! - ! ! , s ) ) (A 36) Proof: It is similar to proof of Theorem A . I .

* Theorem A. 4: A sufficient condition that an input u (t) is a stationary point for * * * <S?(D) is that u be an eigenvector of Q with eigenvalue «2?(D ) .

Proof: Again the proof follows along the same lines as in Theorem A. 2.

A . 6 EXTENSIONS We state one more theorem without proof.

* Theorem A .5: A necessary and sufficient condition than an input u (t) is a g * * stationary point for ,2?(D ) is that u (t) be an eigenvector of Q with p *$.

eigenvalue ^(D ) , where <T. 8)) This is an important theorem because by choosing the linear operator as trace and a high value of 2, the maximum eigenvalue of D can be minimized.

In other words, the maximum error in any direction in parameter space is minimum.

APPENDIX B INPUT DESIGN IN FREQUENCY DOMAIN B . 1 PROBLEM STATEMENT Consider the state space representation of a discrete time system x(k+l) = $x(k) + Gu(k) k = l , 2 , = . , N (B.I) and the noisy measurements ( B 2 y(k) = Hx(k) + v(k) ' > where x(-) is n x 1 state vector u(*) is q x 1 control vector y(O and v(-) are p x 1 measurement and noise vectors, respectively.

cp, G, and H are matrices of appropriate dimensions and contain m unknown para- meters 6. It is assumed that: (a) The system is stable and 6 identifiable.

(b) v is a second order process with known covariance.

Mehra developed a technique for determining the spectrum of input u which minimizes the trace or determinant of the dispersion matrix of parameter esti- mates . A summary is presented here .

103; B . 2 FREQUENCY DOMAIN APPROACH The approach consists of transforming the frequency domain representation of the system into a regression problem. The results of Kiefer and Wolfowitz *• ^ [14] and Fedorov, in the theory of optimal experiments, are subsequently applied to develop schemes for designing optimal inputs.

Fourier transform (1) and (2) to get: and (B 4) y(n) = Hx(n) + v(n) ' where _ N _ N 1 - 1 n = + 1 0) = 2TT/N and ~ over a variable denotes its finite Fourier transform.

From (B.3) and ( B . 4 ) jW n 1 y(n) = H(e~ o - 40' GS(n) + v(n) - T(n,0) u(nV-t- v(n) ( .

B 5) I .. ^m *> ) Z - T(n,0 ) + ~- (0-0 ) + 0(A9 ) u(n) + v(n) ( O ' do O I where 6 is the a priori estimate of 6 and u is a scalar. Thus, Ay(n) - y(n) - T(n,6 ) 5(n) o , ( B - 6 ) T = u(n) |i (9- 6 ) + v(n) O v O From (B.6) we can estimate A0 using generalized least-squares -1 N/2-1 * N/2 1 - * K6 Z, -=-r- S (nJ -r-r S (nJ — Re L •*-=• L •i Re Z -.«• S ^(n) u*(n) A y ( n ) ( B . 7 ) W N d -N/2 ' * -1 and cov(A6) = M where M is the information matrix given by N/2-1 * S T* ^T' 3T* M = Re Z VQ S (n) -r^- S (n) (B 8) uu -N/2 ^^ ^°-°^ S (n) and S (n) are the spectra of u and v and * denotes transpose and com- plex conjugate. " ' The average information matrix per sample is N/2-1 * ^ 1 1 ' C i T 1 /^T L J n 1 T> V " e~" -/',,'\ 2-=- <Z (n} CR Q^> — Re L TTT^ b (.n) r\ 3...A / ^o. ?^ a N As the number of sample points increases, the information matrix approaches infinity. However, the average information matrix per sample reaches a constant value .

We now take limits of equation (B .9) as N -»• °°. It is straightforward to show that Lt M -*• M , where where F is the spectral distribution function of u. In the input design procedure the total power in u will be constrained and, without loss of generality, we may fix it at unity. Such an input will be called normalized input and the correspond- ing information matrix is the normalized information matrix. We will only con- sider normalized inputs in the following development.

The normalized information matrix has many important properties. Some of them are given in Theorem 1 (see [2] for proof) .

Theorem 1: (a) M is a symmetric positive semi-definite matrix.

(b) If the spectrum of u contains fewer than -~— frequencies, the matrix £p M is singular and not all parameters can be identified.

(c) For any normalized input with mixed spectrum f,(co) , it is always possible to find another normalized input foC ) which has the same information matrix and has discrete spectrum with no more than m —- + 1 frequencies. In addition, if f(co) is an optimizing nor- malized input, the number of frequencies cannot exceed = .

This theorem has important implications . Thus, the optimal input is a sum of a finite number of sinusoidal waves . If k is the number of independent fre- quencies in the optimal design, k ^ 1 — ~2— (B.ll) There are two other theorems which are important for input design.

Theorem 2: The following are equivalent: (a) The normalized design f* maximizes the determinant of the information matrix (or minimizes the determinant of the dispersion matrix) .

(b) The normalized design f* minimizes max ij;(a),f) ( ) max iKu.f*) = m (B.13) c where „,<„,, f ) = Re Tr ( ^(«o) f M' ff (B - 14) S Theorem 3: The following are equivalent: (a) The normalized design f minimizes «5?(D(f*) (b) The normalized design f* minimizes max *Ko),f) wef-TT.ir] max .

(B where * ( f ) - D ( f ) S ( ) - D ( f ) (B - 16) - D(f)] U f U and SS is a linear operator (see Equation (3.9) for definition of All input spectra satisfying (a)-(c) of theorem 2 and their linear combin- ations give the same information matrix. The same holds of Theorem 3.

The above theorems are very important because they convert complex non- linear problems into simpler ones. Instead of minimizing |D| or ^(D), it is only necessary to minimize the max \|;(to,f) . This leads to a very powerful cos [-7t.il] computation technique given in Section 3.5.

Example 1: Consider a second order system in which we wish to estimate the frequency 0 0 1' X + (B. 17) -6 -1 y = x + v (B. 18) Discrete time equivalent to this system is (for small A) '0'

" i " A"

x(k) + u(k) (B.19) _-A6 -A.

_ A _ z(k) = x ( k ) + v(k) (B.20) We assume that v(k) is white with known power spectral density S .

vv re-^-1 0 - '°~ T(w,8) = [1 0] Jlu A0 ~ +Aj A e (B.21) V3T (B.22) 89 ja) 2 : e -l+A) -f A Q} w (B.23) jW 2 2 -l)(e -l+A) A e} {( + In this case, the maximum number of frequencies is one and can be determined by maximizing M(co) for coe[-Tt,Tt] . It can be shown that for small A, M is maximized at

.e >

e 0 f i if (B.24) 6 is not smaller than jfrom the assumption of a stable system. For high damping, it is best to use a constant input. With low damping, an oscillatory input at the damped natural frequency is the best.

Page Intentionally Left Blank

APPENDIX C COMPUTATION OF TIME DOMAIN INPUTS USING EIGENVALUE-EIGENVECTOR DECOMPOSITION In the time domain input design problem, a weighted trace of information matrix has to be maximized in each iteration, whichever criteria of Section 3.3 we may be using. This leads to a two point boundary value problem of the type (see Equation (3 .10)) : F V

e e

d — dt T -1 H R H _ A

_ e e e

A

(C.I) x (0) = 0 X(T) = 0 Q The smallest value of \i to give a nontrivial solution to the above equation is to [28] be determined. It has been shown by Bryson and Hall that the Hamiltonian matrix <3$fis symplectic. Therefore, the eigenvalues of 3£ occur in pairs +X, and -A.. Let SP, and ~SP, be the positive and negative sets of eigenvalues of 3£. Then the corresponding eigenvector matrix can be written in the partitioned form as X, X T - S = (C.2) A+ A- The eigenvectors are normalized such that (C.3) It can be shown that (see Bryson arid Hall) (C.4) -1 Premultiply both sides of Equation (C.I) by S (C.5) T T T T or 1 1 A x A x - X * a fl — a — — u _d_ (C.6) dt T T T x x X A +

+

Using the boundary conditions and simplifying, - - J T . T e + A A(0)' = 0 (C.7) -4 T T x (T) Q For a nohtrivial solution, -s.T.T e . + A is singular is singular (C.8) i.e. , has at least one eigenvalue equal to one.

Suppose for a certain \i and T , U has eigenvalues X close to one. Let ° ° ( 1 ) C D the corresponding normalized left and right eigenvectors of U be yy and Yp • Then

. "riV" 4" (C.9)

Differentiating with respect to T, y (1) M _(1) , 9 R 9X (1) TT = y y 9T R 3T 9T R J Premultiply by y and simph'fy to get 3T \L 3T R Similarly , (1) 3A (1) 3U (D y L

1= X

o T -1 -«? T -1 -s T + -1 -s T S+ x x e + A A S T S

- - -e + ^ • x_ V T

+ + + _ + — dy T Since ff> are the first half eigenvalues of <#?, + T \ 8H A r

°

/.T T \ A = ( -i --i 0 / \A.

T T G G A

-i e e i

+ T T or (C.13) It can also be shown that V X , A X X - + (G.14) = A A- + where < j < n9, 1 < i < n9 = G G

± " -j e e

G G A n9 < 2n9

- V Jl '

ne> e e i

+ < j < n6, n9 < i < 2n6 _ . _ / ( A . - A.)

( n Q ) x. G GQ A ,. N / ( X . - X.)

QN Q 0 (j - n8) 9 9 -(x - n6) j i' (C.15) n9 < j _< 2n9, n9 < i <_ 2n6

_

Equation (C , 14) can be used to compute - ^ - and — ^ - .

To find the change in [i for a small change AT in T, we use the relation y = 0 (C A If the desired eigenvalue is close to one, but not exactly equal to one, the approximate change in (j. required to bring this eigenvalue closer to one is given by For a. given \i, there may be several values of T for which the matrix U has at least one eigenvalue equal to unity. We are interested in the smallest T for which this is true. If we start with a correct \i ,T pair, this iteration tech- nique will always give the correct H,T pair for small changes, AT. Thus, if the desired T is "far" from T , u is updated in steps, each step involving a small ° (1) change in T . In each step (C. 17) is used to bring X closer to one. Once the optimum [i is found, ^,(0) can be determined. Then the state and control time histories and information matrix are evaluated.

This technique gives an excellent insight into the structure of the optimal inputs. By studying the eigenvalues and eigenvectors, we can determine if they consist of basically damped or undamped oscillatory functions of rising and falling exponentials . The time constants of these oscillatory functions and exponentials can usually be correlated with model time constants . This approach is excellent for obtaining good approximations to optimal inputs and building input generators . Instead of specifying the input as a function of time, it is possible to give its functional representation. The sensitivity of the nature of optimal inputs to the length of the experiment can also be studied.

Example Consider the following first order system with an unknown parameter 9, with a priori value one.

(C. 18) X = -X + 0U x(0) = 0 There is a continuous measurement of the scalar state x: y = x + v (C. 19) Let E(v(t)) =0, E(v(t)v(T)) = 6(t - T)

-i o

o -i

G

e

H fi = [0 1] (C.20) The controllability matrix of (Fo, GO) is

-i

The system is uncontrollable. The controllable part can be represented by F = [-1] (C.21) .G0 = [1] H = [1] e The Hamilton! an is -y (C.22)

.ye =

with eigenvalues + ^ \i - 1 j and the corresponding normalized eigenvector matrix -1 +/y - 1 j --5 +-=

'y - i

S = .(C.23)

v - i

/y - 1 (C.24) (2 (C.25)

- i

.(C.26) -1 tan One eigenvalue of U is one if or tan \ju - T T _ i — L V P - 1 (C.27) H - 1 T example, if = 2, 2.30.

T M

APPENDIX D PRACTICAL TECHNIQUES FOR SENSITIVITY FUNCTIONS REDUCTION IN LINEAR TIME-INVARIANT SYSTEMS D . 1 INTRODU CTION The problem of computing state sensitivities using reduced order models has become very important in parameter estimation involving high order models and many unknown parameters . These techniques allow a considerable saving in computation time which makes the determination of optimal inputs feasible for [29-31] practical systems. Most efforts to date have concentrated on finding bounds on the order of the model which can generate state sensitivities for all system parameters. Very little attention has been given to the formulation of practical techniques leading to these lowest order models . Formulations by Wilkie and Perkins, '• Denery, ^ "' and Neuman and Sood'" ^ lead to fairly complicated transformations and are not capable of exploiting the characteristics of the system in most cases .

A practical method for obtaining lowest order models for sensitivity func- tions computations is developed. The technique makes full use of special system characteristics and has general application to high order systems with a large number of unknown parameters .

D . 2 PROBLEM STATEMENT Consider a system • x = Fx + Gu x(0) = x (D.I) o where x is a nxl state vector, u is a qxl control vector. F and G are matrices of appropriate dimensions and are functions of m parameters 6.

A heretofore uncited property of systems , which depends on the para- meters 6, is important in sensitivity computation.

Definition 1 - Structural Controllability: A system is said to be structurally controllable if it is controllable for almost all values of parameters. The system may be uncontrollable if certain relations hold among the para- meters .

Definition 2 - Structural Linear Dependence: A set of vectors has structural linear dependence if a linear combination of these vectors is zero for almost all values of parameters. The particular linear combination may depend on the values of the parameters.

Example 1: Consider the system x = x + hi (D.2) The controllability matrix is 'l '8 + 9, The system is controllable unless 6, + 0_ = 6, + 9.. Thus, if 9, = &. = -1 and 0_ = 0_ = -5, the system is uncontrollable in the classical sense but structurally L* J controllable.

Initially, the following simplifications can be made: a. The system is made structurally controllable by dropping uncon- trollable states. Since the initial condition is zero, the system never moves into the uncontrollable subspace. This reduces the order of the system . Note that the states which are uncontrollable only for the given values of the parameters but which are structurally controllable should not be dropped.

b .

All structurally linearly dependent columns of G matrix are lumped with other columns. This reduces the number of effective controls.

[29] The state sensitivities for all parameters 0 can be written as x = F x + G u

e e e e

(D.3) x (0) = 0 Q FQ, Gp,and x~ are denned in Chapter III. If (F , G~) is uncontrollable, the fl corresponding controllability matrix is of rank less than (m+l)n, say r. Let Q, be the set of r independent columns in the controllability matrix. If Q ?

is such that Q, and (X form a set of n(m+l) linearly independent vectors, then

' • v" (D.4)

follows the differential equation \ • t (D.5) x 22/ V Since the initial condition in (D.5) is zero, the last (m+l)n~r uncontrollable states rei states remain zero throughout. The remaining r states, x , follow the differential equation X = F x + G U c c c C (D.6) x (0) = 0 c where ' t F A F =0 c ~~ 11 1 (D.7) Also, since other states in xL are zero. Note that (X is a pseudo-inverse of Q, depending on Q . The transformation from F^.G^ to F ,G and from x to & b . b c c c XQ does not involve Q~ explicitly . Therefore , Ol can be chosen to be any pseudo-inverse of Q, , for example,

CD 9)

'

It is assumed here that the inputs are linearly independent. If this is not so, the number of inputs can be reduced until they are linearly independent.

This will usually result in a reduction in the controllable subspace of (FQ,GQ) as shown in Section D .4. Under the assumption of linear independence of inputs, it is necessary and sufficient to solve a system of r linear equations (D.6) to determine the state and its sensitivities at all times. The next sections investi- gate the nature and dimension of the controllable subspace of (F ,G ) and f i f t explore efficient methods for finding Q, .

D . 3 SINGLE INPUT SYSTEMS In single input systems, F is a (m+ln x (m+l)n matrix and G is a fi fl (m+l)n vector . The following theorem holds .

Theorem D . 1: For a single input system, the rank of the controllability matrix of (F ,G ) is less than 2n.

Q Q o w Proof: The controllability matrix of (F ,G ) is f l f l m + (D.10) r " v r ' j?( G F c F

e: o 0: • •

e It is easy to show that P F G - 30, P F G A D ( F G ) A G + D (D.ll)

e e =

m If a (D.12) o 1 n-1 the (n+k)th column of C is Q ^G) - V {D*, a (D.13) i=0 The second term is a linear combination of n preceding columns of C . Thus, f l n 1 n-1 . ~ i • rank C = rank{D(G) ; ; D(F G) • E D*(a I)F G!

e 1=0 . } CD.14) 1=0 1=0 The (2n+k)th column of the right hand side matrix is n— 1 I'n i n— 1 n— 1 Z D ^ D F G - Z D*(a I) Z =0 1=0 j=0 >J _ ci. L D*(a. i)tf "' G (D. 15) J i j«0 i=0 which i& a linear combination of n previous columns for k > 0. Therefore, F 2n lG ] 2n k C = rank^F^: 6 ~ 6 - CD . 16) rail Thus, the order of the system required to compute all state sensitivities for a single input system cannot exceed 2n. In many practical cases, it is smaller as shown in Example 1.

Corollary 1: If the structurally controllable subspace of (F,G) is of the order p, the maximal order of the controllable subspace of CFrj.Gg) is 2p.

Since F^G is a linear combination of G, FG, . . . , F^ G, the corollary follows immediately.

Example 2: Consider the following system (D. 17) u x(0) \0 -] The state vector and its sensitivities for 6, and 8_ form a set of six differential equations . Since the number of states is two and the number of controls is one, only the first four columns can be independent in the controllability matrix of (F~,G ) . These columns are f l 0 8 " " 6 8 6 f /0\ / 2 \ f l 2 (D.18) l rank(C ) - rank[ (J). D^J), D( [ Q D p - Q Q O O ,| ft 6 0 9 9 6 + 6 0,0o ~ 9o 2 12 2 1 2 " 1 2 2

1 -1 1 -1

0 0 9 29 9 9 2 1 2 - 2 (D.19)' rank 0 0 0 0

9 9 + 1

0 1

1- 1

•l-

0 0 0 0 The first three columns are independent for 6 - ^ 0 . If 0., is zero, only the first two columns are independent and the required model is of the second order.

D . 4 MULTI-INPUT SYSTEMS We state and prove the following theorem.

Theorem D .2: The rank of the controllability matrix of (FQ,GQ) cannot exceed (q+l)n for a q inputs system.

In multi-input systems, G» is a matrix with q columns. The controllability matrix of (F~ , GQ ) has (m+l)nq columns which can be written thus: n ( m + 1 ) 1 ...:D(G ,FG ,...,F - G )] (D.20) q q q where G. is the ith column of G. By using Theorem 1, it is easily seen that the last (m-l)n columns involving any of vectors G.'s are linearly dependent on the first 2n columns for that vector. From (D.14) and (D . 15) rank(C ) - p = ran , F G .....F .

> (D.21) where all summations are from 0 to n-1. Let the structurally independent columns in the controllability matrix of (F,G) be This set of linearly independent vectors in the controllability matrix spans the complete n-dimensional space . So any vector can be represented as a linear combination of these vectors . In particular, l J - R f -4- 4- R 17^1 V ft i7 S,~ -r — 3 G + + 1 G q G - 1 1 ^ i ' " " V q 1 5 k £ q (D.24) Therefore, n-1 n-1 . n-1 1+:i 1 1 1+1 E D*(a.I)F G. = B E D*(a.I)F G. + 6 E D*(a.I)F G, x 1 1 2 1 1=0 - i=0 * 1=0 . . . . , + B E DM. ,_ ...

n z q (D.25) i=Q This is a linear combination of n vectors in the right hand side matrix of (D.24) for all j and k (the values of p. depend on j and k) . Thus, 11 n 1 p = ranktDCG, ,FG. ,. . . .F '^) :D(G_,FG , . . . .F^G.) I . . . . I D ( G ,FG , . . . ,F ~ G ) J _ ' _ L J - « ^ £ Z « » C J C J t j (D.26) 1 (q+Dn Thus , the procedure for finding independent columns of Cg consists of finding structurally independent columns of the controllability matrix of (F,G) choosing (q+l)n appropriate columns from CQ and checking to see if there is any further linear dependence.

Another simplification is possible in large order systems in which each input controls only a small number of states . If k! is the dimension of the con- trollable subspace of the ith input, no more than 2k.' columns involving G. can be linearly independent in the right hand side matrix of (D.26) as shown in Corollary 1 to Theorem 1.

Corollary 2: If for any single input u. the system is completely controllable, P = rank[D(G . . . , F ~ G ) ,D(G , . . . , F ~ G ) .... ,D(G , . . . . F " ) , 1 > ; L 2 2 1 n+1 1 .... ,ZD*(a I)F G , . . . ,ZD*(aI)F " G ] (D .27) J J The proof is obvious since G., . . . ,F G. form a set of n linearly inde- pendent columns.

D. 5 CONCLUSIONS A systematic method for finding the controllable subspace of the augmented system, in which the state vector is the system state and its sensitivities, is presented. It is necessary to start with no more than (q+l)n columns of the controllability matrix of the augmented system. These columns can be selected quickly by inspection of the controllability matrix of the initial system. If r is the dimension of the controllable subspace of the augmented system, it is necessary to solve r linear equations to evaluate the state vector and its sensitivities This method of sensitivity functions reduction fully exploits the character- istics of the system and the cases in which the sensitivity to all parameters in the system is not required. In other words, it leads to the minimal order model under the circumstances.

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