Kyung-Chul Shin Department of Mechanical Engineering

and Applied Mechanics.

Pierre T. Kabamba Department of Aerospace Engineering.

The University of Michigan, Ann Arbor, Mich. 48109-2140

Observation and Estimation in Linear Descriptor Systems With Application to Constrained Dynamical Systems This paper considers the problems of simultaneous observation or estimation of the positions, velocities, and contact forces in a constrained dynamical system. The equations of such systems are not ordinary differential equations, but descriptor equations, i.e., differential equations where the coefficient of the highest order derivative is singular. An asymptotic observer in descriptor form based on pole assignment techniques is used in the time-invariant case to reconstruct the positions, velocities, and contact forces. For time invariant constrained dynamical systems subject to random disturbances, an optimal estimator in descriptor form is designed based on Wiener-Hopf theory. Constrained dynamical systems yield descriptor systems that are uncontrollable and unobservable at infinity. As a consequence, the observer and estimator may not change the infinite eigenstructure of the system. Ex­amples are given to illustrate the use of our method.

1 Introduction Historically, the success of analytical methods in dynamics

[1-5] has two major reasons: for unconstrained systems, they yield ordinary differential equations of motion, which are easy to simulate, at the expense of routine algebraic manipulations; and they eliminate from the formalism all forces which do not work virtually. Common examples of such workless forces are: contact forces in rolling-without-slipping and sliding-without-friction, and the internal forces maintaining rigidity of a body. However, the study of constrained dynamical systems still poses significant fundamental issues. First, the equations of motion of such systems are no longer ordinary differential equations. In both holonomic and nonholonomic cases these equations can be written as a descriptor system, i.e., a differential system where the coefficient of the highest order derivative is singular, and the simulation of descriptor systems presents substantial challenges [6, 9]. Second, these equations of motion contain explicitly the constraint forces. Even though it is in general possible to eliminate the constraint forces from the equations [5], such a process may not be desirable because it is quite tedious, and because knowledge of the constraint forces is crucial in some applications. As a con­sequence, despite the simulation difficulty involved, the im­portance of modeling constrained dynamical systems as descriptor systems has been recognized in the literature [6-8].

The purpose of this work is the indirect estimation and sens­ing of the positions, velocities and constraint forces simultaneously in a constrained dynamical system. When such

Contributed by the Dynamic Systems and Control Division for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT AND CONTROL. Manuscript

received by the Dynamic Systems and Control Division June 1, 1987.

a system is to be controlled, it is necessary to know the state variables and constraint force. However, not all state variables may be measurable, and it may be impossible to sense the con­straint force directly. Such a situation motivates our results. The linearized equations of motion are written in descriptor form. An asymptotic observer in descriptor form is designed based on pole assignment techniques to reconstruct asymp­totically the positions, velocities and contact forces. For con­strained dynamical systems subject to random disturbances, an optimal estimator in descriptor form is designed based on Wiener-Hopf theory [20]. Examples are given to illustrate the practical use of these methods.

Constrained dynamical systems have been studied since the foundation of analytical dynamics [1-5] and we shall not at­tempt to cite all the literature pertaining thereto. The reader is referred to the excellent books [4, 5] for methods of deriving equations of motion for such systems. The advent of com­puters has unveiled the difficulty of simulating constrained dynamical systems. See [10] for efficient solutions to these questions, together with discussions on numerical stability of available software. The kinship of constrained dynamical systems and descriptor systems is well known [8, 10], and has been applied to compliant robotic motion [6, 7].

Linear descriptor systems have been studied extensively dur­ing the last decade [8-9]. Rosenbrock [11] and Verghese et al. [12] worked in the frequency domain and Luenberger [8], Campbell [9], Yip et al. [13] performed a time domain analysis. The Kronecker canonical decomposition [11, 25] is effectively applied to the fast and slow dynamic analysis of the descriptor system [11-17].

Controllability and observability of linear descriptor system

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have been discussed in [11-13, 15, 19]. Yip and Sincovec [13] defined observability as the observability of the finite frequen­cy behavior in the case of admissible initial conditions (see Section 3). However, Cobb [15] used a geometrical approach to establish that observability and controllability are dual in both finite and infinite modes. Feedback and pole assignment in impulse controllable (see Section 3) systems have been studied by Cobb [14, 29]. Extensions of Moore's method for eigenstructure assignment [21] have been applied to impulse controllable systems. Optimal regulation problems for im­pulse controllable descriptor system have been studied by [16, 19], where nondynamic modes are reduced by coordinate transformation. Several methods for solving the Riccati equa­tions are introduced in [25].

The original contributions of this paper can be summarized as follows:

• We demonstrate that when constrained dynamical systems are modeled as descriptor systems, observers and estimators—also in descriptor form—can be used to reconstruct asymptotically the descriptor vector.

• We extend Moore's eigenstructure assignment method to impulse unobservable descriptor systems. This extension is similar to [17, 18] in dual form, but this latter work was restricted to impulse controllable systems. Note that this extension is necessary because constrained dynamical systems are not impulse observable, and is nontrivial for it requires careful consideration of the eigenstructure at infinite of matrix pencils.

• We design an optimal asymptotic observer for impulse unobservable descriptor systems. Here also, our work is similar to that of [16, 19] in dual form, but does not re­quire observability at infinity.

• The standard McFarlane method for solving Algebraic Riccati Equations is extended to Descriptor Algebraic Riccati Equations (see Section 5). This aspect is similar to the work in [19] however, we require neither reducibility of the equations of motion, nor impulse observability.

It is important to emphasize that the originality of the present work is to consider descriptor systems which are not impulse observable. Such systems are practically significant because they include constrained dynamical systems where the constraint force is not measured directly. We design asymp­totic observers and estimators which are also in descriptor form. These observers exhibit impulses at time t = 0 (see Sec­tion 3) which cannot be eliminated by feedback because of im­pulse unobservability [29]. However, these impulses are not a drawback because we are interested in the asymptotic behavior of the descriptor variable, rather than its initial condition.

It is natural to ask the question: why use observers and estimators in descriptor form, rather than in reduced form, like in [19]? This would be equivalent to eliminating the con­straint and constraint force from the observer equation, and would be similar to eliminating unobservable states from a state space description. Among reasons for using estimators in descriptor form are the same as those for modeling constrain­ed dynamical systems as descriptor systems, namely: let the constraint force be directly available, and avoid a tedious reduction. But in the case of observers and estimators there is an additional reason to use descriptor systems: reduction im­plies differentiating the input of the observer, which will amplify high frequency noise.

Using estimators in descriptor form clearly requires on-line integration of singular differential equations. In our examples of Section 6, we have used Baumgarte's "constraint stabiliza­tion method" [30]. This method may exhibit poor transients but has an excellent steady-state behavior, which is quite con­sistent with our purpose of asymptotic observation.

The content of the paper is as follows. In Section 2 we brief­

ly formulate the problem and show how a constrained dynamical system can be modeled as a descriptor system. In Section 3 we review definitions and properties of descriptor systems for later use. The material in Section 3 is standard and can be found in more detail in [13-19], Sections 4 and 5 present our results on observation and estimation, respec­tively. Section 6 presents some examples and we conclude with Section 7.

2 Problem Formulation

The most common model for dynamical systems with con­straints is that of Lagrange's equations with multipliers. For a finite-dimensional scleronomic system they have the form [3]

M{q)q + H(q,q)=(-~j \+T, (2.1)

<p(q,q)=0, (2.2)

(2.3) r = y{q,q),

where superscript T represents matrix transpose; q(t), q(t), q(t) € (R" are the generalized coordinates, velocities, and ac­celerations, respectively; M(q) € <R"X" is the symmetric positive definite inertia matrix; H(q, q) € (R" represents a force vector, assumed differentiable with respect to its arguments; <p: <R" x <R" — (Rp represents a set of p differen­tiable constraint functions; X € (Rp is the Lagrange multiplier vector; T€ <R" represents input forces acting as controls; and r e (Rm is an output vector. The constraint equation (2.2) may be holonomic or nonholonomic. Note that contrary to the common assumption, <p(q, q) needs not be linear in the velocities [4]. Also note that if the original constraints are in­dependent of the velocities, (2.2) requires differentiation with respect to time.

The system of equations (2.1), (2.2) can be rewritten as:

/„ 0 (T

0 M 0

0 0 0

d

It

' Q '

Q

X

q

-H(q,q)+(d<p/dq)T\+T

<p(q,<i)

(2.4)

where /„ is the unit matrix of order n. Equation (2.4) is not an ordinary differentiable equation, but a singular differential equation or descriptor equation because the coefficient of the derivative in the left-hand side is singular.

In this paper, we consider linear descriptor systems obtained by linearizing (2.3), (2.4). Assume we have a nominal trajec­tory q" (t), q° (t), q° {t), X° ( 0 , T (r) for (2.1)-(2.4). Define N(t) = M(q°(t)), F(t) = d/dq H(q, q) \q°, q°, G{t) = d/dqH(q,q)\q°,q°,D(t) = d/dq <p(q, q)\q\ q\ S(t) = d/dq<p(q,q)\q°,q°,Cp = d/dq *(q, q)\q°, q°, Cv = d/dq

*(<7,<7)l<7° X - X°, w = tions are:

\ 0 0

q°. Also definexy = q — q°,x2 = q — q°,x3 = •• T — T°, y = r — r°. Then the linearized equa-

0 N(t) 0

0 0 0

0 /„ 0

-F(t) -G(t) DT(t)

S(t) D(t) 0

" xx

x2

L*3 .

+

" 0 "

Im

0

(2.5)

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y=[Cp,Cv,0] (2.6)

Our purpose is to observe or estimate Xj, x2, x3 in (2.5) given the output y, despite disturbances corrupting (2.5) and (2.6). This problem resembles well known observation and estima­tion problems [22, 23] but the original feature is that (2.5) is a descriptor system.

3 Linear Time-Invariant Descriptor Systems

This tutorial section reviews elementary properties of linear time-invariant descriptor systems for later use. No proof is given, but the reader is referred to [13-19] for further details. Consider the following differential system which is a time-invariant version of (2.5), (2.6):

Ex(t)^Ax(t)+Bu(t)+T w(t), t>0, (3.1)

y(t)=Cx(t) + v(t) (3.2)

x(0) = x0> (3.3)

where x £ (R" is the descriptor vector, u € 6{p is the input vec­tor, y e (Rm is the output vector, w 6 (R? and v € (Rm are distur­bance vectors, A, B, C, T and E are real matrices of ap­propriate dimensions and Eis allowed to be singular. It should be emphasized that (3.1)—(3.3) are time-invariant, which is a limitation affecting all the results of this paper. We define a solution as a vector function x(t) satisfying (3.3), differen­t i a t e for t > 0, and satisfying (3.1). Notice that by our defini­tion, a solution may not be differentiable at t = 0. Also, it will be seen that existence of a solution requires certain com­ponents of Bu(t) + T w(t) to be sufficiently differentiable (See (3.13)), and is therefore input-dependent. The problem treated in this paper is to estimate the descriptor variable x given the measurement y and despite the disturbances v and w.

Associated with system (3.1) is the System Matrix Pencil S(s) = sE — A. The descriptor system (3.1) is called regular if its characteristic polynomial p(s) = det (S(s)) is not iden­tically nil. Throughout the paper we will assume that (3.1) is regular. The number of degrees of freedom r is the degree of the characteristic polynomial r, whose roots are the finite eigenvalues. The set of finite eigenvalues, with multiplicities counted, is called the finite spectrum. Clearly r < n if E is singular. The regular descriptor system (3.1) is said to possess n — r infinite eigenvalues. The algebraic multiplicity of a finite eigenvalue is its multiplicity as a root of the characteristic polynomial, and s = oo is considered an eigen­value of algebraic multiplicity n — r.lf s = \is a finite eigen­value, a right eigenvector (left eigenvector) associated with X is a nonzero vector £ 6 Q" satisfying S(X) £ = 0 (S(X)T £ = 0). An infinite right eigenvector (infinite left eigenvector) is a nonzero vector £ € C" satisfying E £ = 0 (ET £ = 0). If s = X is a finite eigenvalue, the geometric multiplicity of X is defined as n-rank (S(X)); similarly the geometric multiplicity of s = oo is defined as n-rank (E). For both finite and infinite eigen­values, the geometric multiplicity represents the number of linearly independent right (left) eigenvectors associated with the eigenvalue, and is at most equal to the algebraic multiplici­ty. If s = X is a finite eigenvalue with algebraic multiplicity a and geometric multiplicity y, the defect of X is defined as the non-negative integer a — 7; similarly the defect of s = 0° is the non-negative integer rank (E) — r. A finite (infinite) eigenvalue is called defective it has a positive defect, otherwise it is nondefective.

If 5 = X is a defective eigenvalue, a chain of generalized

eigenvectors of length k associated with X is a sequence of vec­tors £,, £2, . . . , £ t of Q" satisfying

( \ E - - 4 ) £ , = 0,

(\E-A)Z2 = ££ , ,

(\E-A)£3 = ££2 ,

•• (3.4)

(\E-A)Zk = Eik_x,

E^kiR(\E~A),

where R (•) represents range space. Similarly, if 5 = 00 is defective, a chain of generalized infinite eigenvectors of length k is a sequence of vectors £,, £2, . . . , %k of G" satisfying

£ £ , = 0 ,

E^=A^2, (3-5)

AZkiR(E),

Chains of finite and infinite left generalized eigenvectors are similarly defined by replacing E and A by ET and A T, respec­tively, in (3.4), (3.5). Chains of generalized eigenvectors associated with distinct eigenvalues are linearly independent. Chains of generalized eigenvectors associated with the same eigenvalue, but starting from linearly independent eigenvec­tors are also linearly independent. The eigenstructure of a (finite or infinite) eigenvalue of geometric multiplicity 7 is specified by giving the lengths of 7 chains of generalized eigenvectors starting with linearly independent eigenvectors, and is independent of how the eigenvectors are chosen and how the chains are constructed.

When E is the unit matrix, S(s) can be transformed by similarity into block diagonal form in several ways, e.g., Jor­dan canonical form or rational canonical form. Similarly, when E is singular but S(s) is regular with r degrees of freedom, there exist real nonsingular matrices M € (Rnxn,Ne (R"xn which transform S(s) into Kronecker canonical form, i.e., such that

M 0

0 /

sE-A B T

0 0

N 0 0 1 =

0 / 0

0 0 /

" sI-Al

0

. c,

0

I-sEl

C2

Bt

B2

0

r, " r2

0

(3.6)

where A, € <Rr E, € (R («-<•)>< <«-»•), Bx € <Rrxp, B2 e (R("- ' , x p , C, € (Rmxr, C2 6 (R'"x(«-'-)> i1! € <Rrx", T2 6 (R<n-r)x?i Ea = 0 a n d Ea-i .£ Q T h e i n t e g e r 3 is c a u e c i

degree of nilpotency of the pencil S(s). Clearly, rank (E) = r + rank (E^, so that rank (£,) is the defect of the infinite eigenvalue. Therefore iii = 0 iff the infinite eigenvalue is nondefective. Since Ex is square, of order n — r and is nilpo-tent with nilpotency degree 3, we also have d < n — r. In fact, the degree of nilpotency is the length of the longest chain of generalized infinite eigenvectors.

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In (3.6), partition N as N = [Nlt N2], iV, € (R"xr, JV2 € a l «x( / l - r ) i a n d d e f i n e v g (ft,, b y

x = Nv, (3.7)

and partition rj accordingly. Then (3.1), (3.2), (3.3) and (3.6) imply

•iliV) =A1rll(t) +Bxu(t) +T1w(t), (3.8)

ElTl2U)=V2U)+B2u(t)+T2w{t), (3.9)

j ' = C, i j1 (0+C2 i ,2 (0 + «(0 . (3-10)

Vw

Vio = N~lx0. (3.11)

Equations (3.8), (3.9) represent the slow dynamics and fast dynamics, while xl 4 N, rj ls x2 4 iV2 rj2 are called the s/ovc variable and /ar t variable, respectively.

A formal solution of (3.8)—(3.11) obtained using Laplace transforms is:

(3.12)

i j1(O=exp041fhio+]oexpG41(f-T))(fl ,K(T)

+ riW,(T))rfT,

moved arbitrarily by choice of G under the obvious constraint that the spectrum remain self-conjugate. The finite unobser-vable modes are invariant under feedback. System (3.1), (3.2) is detectable if all its unobservable finite eigenvalues have negative real parts.

Systems (3.1), (3.2) is infinite-observable (or impulse ob­servable) if the impulsive behavior of x(t) at x - 0" can be uniquely determined from y(t), t > 0 in the absence of input u(t) and disturbances w(t), v(t). If system is observable at infinity, its impulsive behavior can be removed by feedback. In other words (3.1), (3.2) is impulse observable iff there exists G 6 (R"xm such that s = oo is a nondefective eigenvalue of the pencil [sE — A — GC], while this pencil has the same number of finite mode as rank E.

The singular value decomposition (SVD) coordinate transformation of (3.1), (3.2) without disturbance gives a more transparent criterion of observability and controllability of impulse modes. In (3.1), decompose E as E = ULV, where U £ (R" x \ V € (R"x" are orthogonal matrices and E is a diagonal matrix having same rank as E, Define

" Z\ 1

_Z2 _

- K / l

. v2T _

X, -uf ^ 2 T _

H][F,K2] = -•4 u ^ 1 2

-421 ^ 2 2

V2(t)=- E 8 ( ' _ 1 ) ( O £ l ' l j 2 0 - E - E l ' ( 5 2 « ( " < 0

x{t)=Nl7]l(t)+N2rl2(t),

(3.13)

(3.14)

where 5(/) is the Dirac delta function and superscript (/) denotes /th derivative. As indicated by (3.12), the finite eigen­values of the pencil S(s) are the eigenvalues of the slow system, i.e., those of the matrix At. A descriptor system is asymptotically stable iff all its finite eigenvalues have negative real parts. Equation (3.13) shows the two types of pathologies which may plague linear descriptor systems where s = °° is defective. First, if ij20 j± 0, (3.13) shows that t)2{t) may con­tain impulses and derivatives thereof at t = 0 due to initial conditions. The initial condition (3.3) is called admissible if (3.11) implies ij20 = 0. Second, (3.13) shows that i)2(0 may contain impulses and derivatives thereof due to inputs if u(t) or w(t) are not 3 - 1 times differentiable. We will henceforth assume that T2 = 0, i.e., the fast subsystem is unperturbed. Notice that for a descriptor system where s = oo is nondefec­tive, (3.13) implies i)2(t) = 0, which represents a purely algebraic relationship between the components of the descrip­tor vector x{t). It should be emphasized that even though the Kronecker canonical form is useful for understanding the behavior of descriptor systems, its effectiveness is limited in practice for numerical ill-posedness reasons [24], similar to those limiting the use of Jordan's form.

Also associated with system (3.1), (3.2) are the observability pencil O(s) = [sET — AT: CT]T and the controllability pencil C(s) = [sE - A: B]. System (3.1), (3.2) is called finite-observable if we can uniquely determine the descriptor vector x(t), t > 0 through knowledge of the output y(r), r € (0, t] in the absence of input u(t) and disturbances w(t), v(t).

System (3.1), (3.2) is finite-observable iff rank O (5) = n for all finite s £ &. The finite eigenvalues where rank O(s) < n are called unobservable, and form the finite unobservable modes while the other finite eigenvalues form the finite ob­servable modes.

System (3.1) is unobservable iff there exists an nonzero eigenvector u which satisfied 0(X)« = 0 for some finite eigen­value X 6 C The finite observable modes are arbitrarily assignable by feedback, i.e., the finite observable eigenvalues of the pencil [sE - A - GC], where G € (R"xm, can be

- U?-

^ 2 T _

B = B{ -

„ 5 2 _ ,C[VlV2] = [ClC2

where [U2] [V2] contain columnwise bases of the null spaces of ET undE, respectively. Then (3.1, 3.2) imply

L 0

0 0

Z\

Zl

Au Ax ' zx '

_ z2 _ +

B,-

_B2_

y=[Clc2] Zx

z2

(3.15)

(3.16)

System (3.1) has no impulsive mode iff ^422 is nonsingular. In the SVD coordinates impulse observability can be more

systematically checked: system (3.3), (3.2) is impluse obser­vable iff [^422

r C2T]T has full column rank.

Controllability at finite modes, stabilizability and impulse controllability are dual to the observability at finite modes, detectability and impulse observability.

4 Observer Design for Deterministic Descriptor Systems

In this section, we consider the design of observers for linear time invariant descriptor systems (3.1), (3.2) in the deter­ministic case, i.e., w = 0, v = 0. We consider the system (3.1), (3.2) to be impulse unobservable, since we do not measure the constraint forces (see (2.5), (2.6)).

The observer of system (3.1), (3.2), which will reconstruct the descriptor variable X(T) from y(a) where a < T, is de­scribed by

Ex(t)=Ax{t)+Bu(t) + G(y(t)-Cx(t)), (4.1)

where x € (Rn denotes the reconstructed variable and G € (R"x"' is the observer gain. We call (4.1) the descriptor system observer.

Define the descriptor variable observation error

x{t)=x(t)~x(t) (4.2)

then we have error system

Ex(t) = (A + GC)x(t), t>t0. (4.3)

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The first question one could ask about system (4.3) is whether it is regular. This question has been treated in [18] where it is shown that (4.3) is regular iff it has n linearly in­dependent eigenvectors and generalized eigenvectors. For­tunately, the method we propose for choosing G in (4.3) in­volves an explict choice of the eigenvectors and generalized eigenvectors of (4.3), thus choosing them linearly independent will imply regularity.

Since we assume that (3.1), (3.2) is impulse unobservable, it is impossible to find a matrix G in (4.3) to remove the im­pulsive behavior of the error equation. Our purpose is therefore to find G in (4.3) to assign the finite spectrum of (4.3) while preserving its eigenstructure at infinity. In Appen­dix A, we show that for any choice of self conjugate finite spectrum, a feedback gain G in (4.3) assigns the finite spec­trum to the desired value while preserving the infinite eigenstructure iff it has the form

G = [vuv2, [w,,w2, • .w„J (4.4)

where [u,r, w,. r] r , 1 </ '<« are determined by A, C, E, and the desired closed loop spectrum through (A.3)-(A-5).

Choosing the desired eigenvalues [ ̂ . . . . X,} in the left-half plane will yield an asymptotically stable error system (4.3), implying that the reconstructed descriptor vector x(t) will tend asymptotically toward the true descriptor vector x(t).

5 Design of Optimal Estimators for Stochastic Descriptor Systems

In the previous section it has been assumed that the linear descriptor system is unperturbed. However, it is often the case that system and measurement perturbations are not negligible. In this section we consider the problem of optimally estimating the descriptor variables of a stochastic descriptor system. We assume that the fast dynamics is noise-free, or equivalently for a constrained dynamical system, the con­straint equation is noise-free. Without loss of generality, we can then replace r by E in (3.1) after properly redefining w. We also assume that all output variables are corrupted by white noise processes. The Weiner-Hopf equation [20] will be extended to descriptor systems. Consider the system (3.1), (3.2), where we neglect deterministic input terms for simplici­ty. We will obtain the optimal filter equation, gain matrix and descriptor Riccati equation.

We assume that the system and output disturbances w(t) and v(t) are zero-mean Gaussian white noise processes, in­dependent of the initial conditions, and satisfying:

Z[w(T)wT{o)]=Rw(r)b(T-o) (5.1a)

&[V(T)VT(<T)]=RV(T)8(T-O) (5.1b)

&[w(T)vT(a)] = 0 (5.1c)

for all T, a > t0, where RW(T), RV(T) are continuous and satisfy Rw (r) > 0, R„ (T) > 0 for all T > t0 and S denotes ex­pected value.

Assume that the optimal estimate can be written as:

f(o=[' m J'o

t,T)y(T)dr (5.2)

(5.3)

Define the estimation error

x(t)=x(t)-x(t)

Define the error covariance matrix

P(t)=&[x(t)xT(t)] (5.4)

then, we define the optimal estimate as that which minimizes the error covariance, i.e., the trace of P(t).

Proposition 5.1. An estimate x(t) is optimal iff the weighting pattern matrix H(t, T) satisfies the Weiner-Hopf equation:

&[x(T)yT(a)]-\' H(t,T)&\y(T)yT(a)]dT = 0 (5.5)

Proof: The proof is identical to that of [20, Theorem 8.1].

Theorem 5.1.

(a) The continuous optimal estimator equation for the descriptor system is given by

Ex(t)=Ax(t)+EK(y(t)-Cx(t)) (5.6)

for t > tQ.

Proof: See Appendix B.

(b) The optimal gain matrix for the estimator (5.6) is given by

(5.7) G = EK=EPCTRUX, for t>tQ

where G 6 <Jt"x'\

Proof: See Appendix C.

(c) Consider the error equation of the system (3.1), (3.2)

Ex(t) = (A -EKC)x(t) +Ew(t) -EKv(t) (5.8)

Then the matrix EP of the optimal gain matrix (5.7) is obtain­ed from the solution of the Descriptor Riccati equation:

EP(t)ET = AP(t)ET + EP(t)AT

-EP(t)CTR„~l{t)CP(t)ET + ERtv(t)Er (5.9)

fort > t0, whereP(t0) = P0. |||

Proof: See Appendix D.

A descriptor Riccati equation can be written as a nonlinear descriptor system. Define the vector form of a matrix A €

as (5.10)

(5.11)

\ec(A) = [aua21 • • • • anlan • • •

Then it is well known that

vec(XYZ) = [ZT®X\vtc{ Y)

where (g) denotes Kronecker product [26]. Then (5.9) has a vector form:

[E®E]vtc{P) = [E® (A -EKC)]\ec(P) +

[(A-EKC)®ET]vec(P)

+ [E®E]vec(R„) + [E®E\wtc(KRvKT) (5.12)

where the coefficient matrix [E (x) E] e (Rn x" is singular. Consider the following Generalized Hamiltonian Equation

(GHE):

ET 0

0 E

together with the equation:

\ = PETz

d

~di ' z

X

" -AT

_ ERWET

CTR/C~

A

' z

X (5.13)

(5.14)

Then, by differentiating (5.14), it is easily verified that (5.14) and (5.13) imply (5.9).

Proposition 5.2. Let the GHE (5.13) of the time-invariant system have finite eigenvalues (5, sr), then ( - $ ! , . . . . , — sr) are also finite eigenvalues of the equation (5.13). HI

Proof. See Appendix E.

If the system is completely observable and the GHE (5.13)

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has no defective eigenvalue, then the steady state error covariance matrix, P can be obtained from the eigenvectors of the Hamiltonian equation. This extends the MacFarlane-Potter-Fath method [25] to the GHE which is obtained from a descriptor system.

Theorem 5.2.

If the descriptor system (3.1), (3.2) is completely observable and infinite controllable:

(/) the steady state error covariance P € (R"xm satisfies the descriptor algebraic Riccati equation (DARE):

APET + EPA T - EPCTR „"' CPET + ERwET = 0 (5.15)

(«) Equation (5.13) can be reduced, after descriptor variable feedback, to a regular Hamiltonian equation such that the steady state error covariance P e 0|(n-m)x(n-»i) satisfies an algebraic Riccati equation. |||

Proof:

(/) Equation (5.15) is obtained from (5.9). This result was derived for reduced state space system by Kwakernaak [22]. Then the solution PET of the DARE can be com­puted from equation (5.13).

(») Equation (5.13) can be transformed, using the SVD coordinate system of Section 3 as follows:

E 0 0 0"

0 E 0 0

0 0 0 0

0 0 0 0

" i\ '

7 i

z2

_ 72 .

-AT, Ru -AT, R,

Qu An Ql2 Au

'Al\ -"21 — ^ 2 2 -"22

V21 A2l G22 ^ 2 2

7 l

Zl

72

(5.16)

where

UTAV= An A12

UTERWETU= Qu Qn

Q21 Q22

VCTRJCV= R\\ R\

R-,, R-) , IFZ =

' Zi

I z2 _ ,V*\ =

' 7 i "

L 72 J

Remark 5.1. The generalized eigenvalues with negative real part of the GHE are equal to the finite characteristic values of the descriptor error system (5.8). Moreover, the closed loop eigenvalues and generalized eigenvectors are subvectors of the eigenvalues and generalized eigenvectors of the GHE. Therefore, once the latter are computed, the eigenstructure assignment procedure of Section 4 is used to compute the op­timal gain.

Corollary 5.1.

(a) If the system (3.1), (3.2) is not detectable, the optimal estimator does not exist.

(b) If the system (3.1), (3.2) is finite observable and infinite unobservable, we can estimate the slow dynamics despite the unobservability at infinity. 111

Proof: See Appendix F.

Remark 5.2. To illustrate Corollary 5.1.a and the difficulty of finding an optimal estimator, consider the case:

E = " 1 0 ]

. 0 0 ,A =

" 0 1 "

L 1 0 J , C = [ 0 1], Rv=\,

Rw —

1 0

0 0 (5.17)

It is easily seen that this system has one infinite nondefective eigenvalue and one defective infinite eigenvalue. As a conse­quence, it does not have any detectable finite eigenvalue, and an optimal estimator does not exist. In fact the covariance computed from (5.15) is

P = 0 a

where a = 1 ± 2, and /3 is free, implying that P is not positive semidefinite.

Remark 5.3. If a descriptor system is finite observable, but infinite unobservable and the GHE has an infinite defective eigenvalue the steady state P obtained from the GHE (5.13), (5.14) will not always satisfy the DARE (5.15). This situation is similar to that in regular systems—see [27, 28]—for a discus­sion of Hamiltonian systems with defective eigenvalues.

To reduce equation (5.16), the matrix A22 R22

Q22 A2

should

be nonsingular. In general Qi2 = 0 because E is singular

Moreover, the matrix - ^ 2 2 -^22

0 A7

is invertible iff A-,

nonsingular. The idea is to use feedback to ensure that A22 is nonsingular. Indeed, if system (3.1) is impulse controllable, again K e (R"xm exists such that degree det (Es - (A + BK)) = rank E. Define A' = (A + BK), where

VA'V An Axi

A 21 A22

Then, nonexistence of impulsive

modes is equivalent toA22' being nonsingular (see Section 3). Therefore, equation (5.16) can be reduced to a regular Hamiltonian equation and P, '(R <"-"" x("-m) satisfies an algebraic Riccati equation. Q.E.D.

6 Examples

Consider the following constrained dynamical system:

1 0

0 1 J

01

. » 2 J +

1 1 "

L 1 1 J

01

L ^2 J +

2 0

0 1

[1 1]

1

^ 1

= 0

x+

y = 9i + v

(6.1)

(6.2)

(6.3)

where 0lt 62 € (R are generalized coordinates, X £ (R is the Lagrange multiplier, (w,, w2) and v are disturbance and measurement noise, respectively, and y is the measurement. Note that in the absence of the constraint, (6.1) defines an asymptotically stable system, whereas with the constraint, the

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— I 1 — 0.6 0.9

TIME Fig. 1(a)

Fig. 1 Response of descriptor vector observer

system eigenvalues are at ± j 1.225 implying marginal stability.

System (6.1)-(6.3) can be rewritten as the descriptor system

Ex=Ax + Ew (6.4)

y = Cx+v (6.5)

where

x=(01,02,61A,\)T (6.6)

w = (0,0,w,,w2,0)7

E =

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0

0

0

0

1

0

0 "

0

0

0

0

;A =

0

0

- 2

0

1

0

0

0

- 1

1

1

0

- 1

- 1

0

0 0

1 0

- 1 1

- 1 1

0 0

C= [ 1 0 0 0 0 ]

(6.7)

(6.8)

It is easily checked that the (6.4), (6.5) is not impulse ob­servable because (6.3) does not contain X. Therefore, observa­tion and estimation in (6.4), (6.5) must be performed without changing its infinite eigenstructure. Moreover, (6.4) has three

infinite eigenvalues and only one infinite eigenvector. Hence s = oo is a defective eigenvalue of (6.4) and of any observer or estimator of (6.4).

Example 1: Descriptor Observer. Assume w, = w2 = v = 0 in (6.1), (6.3). We use an observer of the form (4.1). Suppose we want the finite closed loop eigenvalues (i.e., those of (4.3)) to be - 5. and - 6. Using the construction of Appendix A, we obtain the observer gain

G=[11.2, -38 .9 ,31 .1 , -86 .8 , - 2 . 1 ] 3 (6.10)

Figure l(a-d) shows computer simulations of the error system (4.3) corresponding to initial conditions x(0) = (1,0, 0, 0, 0) r , (0, 1, 0, 0, 0) r , (0, 0, 1, 0, 0)7-, (0, 0, 0, 1, 0)T, respectively. Notice the large and fast transient which reflects the occur­rence of an impulse at t = 0.

Example 2: Optimal Estimator. Here, we assume that in (6.4), ,w and v are zero-mean stationary white noise processes, mutually independent, uncorrelated with the initial condi­tions, and with covariances

0 0 0 0 0

Rw =

0 0 0 0 0

0 0 1 0 0

0 0 0 1 0

0 0 0 0 0

; * „ = [!]• (6.11)

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o.o

— 2 .0 —|

- 4 . 0

— 6 .0

— B.O

-1 O.O

- 1 2 . 0 —

-1 +.0

- 1 6 . 0

-1 8.0

- 2 0 . 0

o.

x 2 X3

~r

T I M E

Fig. 2(a)

o.o —

- 2 . 0

- 4 . 0

- 6 . 0

- B . O

- 1 0 . 0

- 1 2 . 0

-1 4 .0

-1 6.0

- 1 8 . 0

- 2 0 . 0 T 1 4. i

T I M E

Fig. 2(b)

TIME

Fig. 2(d)

Fig. 2 Response of descriptor vector optimal estimator

The optimal steady-state estimator in the sense of (5.4) is a descriptor observer with gain

G=[0.563, -0.563,0.158, -0.158, 0.] r (6.12) and the eigenvalues of the error system are -0.28 ± j 1.26. Figure 2{a-d) is similar to Figure 1 and shows computer simulations of the error system (5.8) corresponding to initial conditions JC(0) = (1,0, 0, 0, 0)T, (0, 1, 0, 0, 0)7", (0, 0, 1, 0, 0) r, (0, 0, 0, 1, 0) r, respectively. Here also we notice the large and fast transient due to an impulse at t = 0. However, both the observer and the optimal estimator have quite satisfactory steady-state behaviors.

7 Conclusions

Deterministic observers and stochastic estimators have been designed for linear time-invariant continuous time descriptor systems. Such systems can be obtained from constrained dynamical system. In practice, the constrained dynamical systems are always infinite unobservable and infinite uncon­trollable. A design method is presented which preserves the in­finite eigenvalues and assigns the finite eigenvalues for observer asymptotic stability. Moore's pole assignment method has been extended to the design of descriptor system observer with multiple finite and infinite eigenvalues. An op­timal estimation algorithm in descriptor systems has been developed using the Weiner-Hopf approach. Also a computa­

tion algorithm has been developed to solve the descriptor Ric-cati equation in the steady state case. In both problems, no coordinate transformation is necessary. This paper is restricted to time-invariant descriptor systems. The extension to time-varying systems will be the topic of future work.

Acknowledgment

The authors are very grateful to the anonymous reviewers for their constructive comments.

References

1 Meirovitch, L., Method of Analytical Dynamics, McGraw-Hill, New York, 1970.

2 Goldstein, H., Classical Mechanics, Addison-Wesley, 1980. 3 Greenwood, D. T., Principles of Dynamics, Prentice Hall, Englewood

Cliffs, 1965. 4 Neimark, Y. I., and Fufaev, N. A., Dynamics of Non Holonomic

Systems, American Mathematics Society, 1972. 5 Kane, T. R., and Levinson, D. A., Dynamics: Theory and Applications,

McGraw-Hill, New York, 1985. 6 McClamroch, N. H., "Singular Systems of Differential Equations as

Dynamics Models for Constrained Robot Systems," Center for Robotics and Integrated Manufacturing, Report RSD-TR-14-84, Ann Arbor, 1984.

7 McClamroch, N. H., and Huang, H. P., "Dynamics of a Closed Chain Manipulator," Proc. ACC, Boston, 1985, pp. 50-54.

8 Luenberger, D. G., "Dynamic Equations in Descriptor Forms," IEEE Trans. Automat. Control, Vol. AC-19, 1974, pp. 201-208.

9 Campbell, S. L., Singular Systems of Differential Equations, Pitman, London, 1980.

262 / Vol. 110, SEPTEMBER 1988 Transactions of the AS ME Downloaded From: https://dynamicsystems.asmedigitalcollection.asme.org on 06/29/2019 Terms of Use: http://www.asme.org/about-asme/terms-of-use

10 Sincovec, R. F., Erisman, A. M., Yip, E. L., and Epton, M. A., "Analysis of Descriptor Systems using Numerical Algorithms," IEEE Trans. Automat. Control, Vol. AC-26, Feb. 1981, pp. 139-147.

11 Rosenbrock, H. H., "Structural Properties of Linear Dynamical Systems," Int. J. Control, Vol. 20, No. 2, 1974, pp. 191-202.

12 Verghese, G. C , Levy, B. C , and Kailath, T., "A Generalized Statespace for Singular Systems," IEEE Trans. Automat. Control, Vol. AC-26, 1981, pp. 811-831.

13 Yip, E. L., and Sincovec, R. F., "Solvability, Controllability, and Ob­servability of Continuous Descriptor Systems," IEEE Trans. Automat. Con­trol, Vol. AC-26, 1981, pp. 702-707.

14 Cobb, D., "Feedback and Pole-placement in Descriptor Variable Systems," Int. J. Control, Vol. 33, 1981, pp. 1135-1146.

15 Cobb, D., "Controllability, Observability and Duality in Singular Systems," IEEE Trans. Automat. Control, Vol. AC-29, 1984, pp. 1076-1082.

16 Cobb, D., "Descriptor Variable Systems and Optimal State Regulation," IEEE Trans. Automat. Control, Vol. AC-28, 1983, pp. 601-611.

17 Lewis, F. L., and Oscaldiran, K., "On the Eigenstructure Assignment of Singular Systems," Proc. CDC, Ft. Lauderdale, FL, 1985, pp. 179-182.

18 Fletcher, L. R., Kautsky, J., and Nichols, N. K., "Eigenstructure Assign­ment in Descriptor Systems," to be published.

19 Bender, D. J., and Laub, A. J., "The Linear-Quadratic Optimal Regulator for Descriptor Systems," Proc. CDC, Ft. Lauderdale, FL, 1985, pp. 951-962.

20 Meditch, J. S., Stochastic Optimal Linear Estimation and Control, McGraw-Hill, New York, 1969.

21 Moore, B. C , "On the Flexibility Offered by State Feedback in Multivariate Systems beyond Closed Loop Eigenvalue Assignment," IEEE Trans. Automat. Control, Oct. 1976, pp. 589-692.

22 Kwakernaak, FL, and Sivan, R., Linear Optimal Control Systems,, Wiley-Interscience, New York, 1972.

23 Sage, P. A., and White, C. C , III, Optimal Systems Control, Prentice Hall, Englewood Cliffs, 1977.

24 Van Dooren, P., "The Computation of Kronecker's Canonical Form of a Singluar Pencil, 103-140.

25 Kailath, T. 26 Graham, A.

Ellis Horwood, 1' 27 Kucera, V.,

Linear Algebra and Its Applications, Vol. 27, 1979, pp.

Linear Systems, Prentice-Hall, Englewood Cliffs, 1980. , Kronecker Products and Matrix Calculus with Applications, 181. "On Nonnegative Definite Solutions to Matrix Quadratic

Equations," Automatica, Vol. 8, 1972, pp. 413-423. 28 Martensson, K., "On the Matrix Ricatti Equation," Information

Sciences, Vol. 3, 1971, pp. 17-49. 29 Cobb, D., "Observability and Impulse Observers in Descriptor-Variable

Systems," Proc. 20th Allerton Conf. on Communications, Control, and Com­puting, 1982.

30 Baumgarte, J., "Stabilization of Constraints and Integrals of Motion in Dynamical Systems," Computer Methods in Applied Mechanics and Engineer­ing, Vol. 1, 1972, pp. 1-16.

A P P E N D I X A

Derivation of equation (4.4)

Assume system (4.3) is regular. For an arbitrarily given eigenvalue X,- 6 C, let us define a left eigenvector vt € Q" such that

E^v^iA + GC^Vi

and equivalently

(AT-ET\i)vi + CTGTvi = 0

by defining w,- = GT vh (A.2) becomes

v. [AT-ET\ CT] = 0

(A.l)

(A.2)

(A.3)

We can obtain a total of n independent vectors [v,7', vf, r] r by proceeding as follows:

Case (1). For each nondefective finite eigenvalue: Choose vectors [v:

T, wf]7 in the null space of [AT — .E^X,-: CT] in number equal to the multiplicity of the eigenvalue.

Case (2). For each defective finite eigenvalue: From (3.4), choose chains of generalized eigenvectors

satisfying

AT-\iET 0

-ET AT-\,ET

CT 0

0 CT

ET AT~\fiT: 0 0 CT

V:dk'

-Q (A.4)

where dk' is the length of a chain of generalized eigenvector at X = X;.

Case (3). For infinite eigenvalues: Obtain «-Rank (£) vectors in the null space of [ET 0]. Case (4). For defective infinite eigenvalues: From (3.5), choose chains of generalized infinite eigenvec­

tors satisfying

ET 0 0

-AT ET 0

0 0 -AT ET

0 0 0

-Cr 0 0

-CT 0

v?

w?

= 0 (A.5)

where ?;' is the length of chain of generalized infinite eigenvectors.

Since the eigenvectors and generalized eigenvectors v,, 1 < i < n are linearly independent [18], the equations ve, = GT v„ 1 < / < n can be solved for G, yielding (4.4).

A P P E N D I X B

Proof of Theorem 5.1a

Premultiply the Wiener-Hopf equation (5.5) by E £ (R"x" and differentiate with respect to t. Then the first term is

E(cl[xU)yT(a)]) = &[Ex{t)yT(<r)]

= &[(Ax(t)+Ew(t))yT(a)]

= A£,[x(t)yT(a)] (B.l)

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The second term is

4r\' EH(t,T)Z\y(T)yT(o)]dT at J(0

r dEH(t,r)

<o dt &\y(T)yT(o)]dT + EH(t,t)&\y(t)yT(o)]

(B.2)

From equation (B.2)

&\y(t)yT(a)] = &[(Cx(t) + v(t))yT{a)]

= C&[x(t)yT(a)] + &[v(t)yT(a)] (B.3)

The last term of (B.3) vanishes for t0 < a < t. From (B.l), (B.2), and (B.3):

AZ[x{t)yT (a)}-EHUJ)C&[x(t)yT (a)}

• I ' J'n

dEH(t,T) £,\y(T)yT(o)]dT=0 (B.4)

h0 dt

Apply equation (5.5) for &[x(t)yT (a)], then we obtain

\AH(t,T) -EH(t,t)CH(t,r)

From equations (C.2) and (C.3), equation (C.l) becomes

&[x(t)xT(t)]CT- \ H(t,T)&[y(T)xT(t)]CTdT J'o

-mt,t)Rv{t)=0 (C.4)

As we defined # ( 0 = H(t, t):

K(t)Rv(t)=E[(x(t)-^ H{t,T)y{T)dT)xT{t)^CT

= E[x(t)xT(t)]CT (C.5)

However

&[xV)xT(t)] = &[x(t)(x(t)+x(t))T]

= &[x(t)xT(t)] + &[x(t)xT(t)] (C.6)

Since the last term vanishes, equation (C.6) becomes

&[x(t)xT(t)]=P(t) (C.7)

where P(t) is the error covariance matrix. Then the optimal gain matrix can be obtained from equation (C.5):

EK(t)=EP(t)CTRv~1(t)

for tst0. Q.E.D.

dEH(t,r)

dt ]z[y(T)yT{a)]dT = 0

for t0 < a < t. A sufficient condition for equation (B.5) is

dEH(t,T) AH(t,T) -EH{t,t)CH(t,T) - = 0

at

(B.5)

(B.6)

for t0 < T < t. By differentiating equation (5.2) and premultiplying E, we obtain

Ex(t)=\' dEH(t,T)

y{T)dT + EH(t,t)y(t) (B.7) J'o dt

From equations (B.6) and (B.7)

Ex(t) = [ [AH(t,r) -EH(t,t)CH(t,T)]y(T)dT J'o

+ EH(t,t)y(t)

A^' H(t,T)y(t)dT + EH(t,t)[y(t)-C^ H(t,T)y(r)d^

(B.8)

(D.3)

J 'o L •> ' o

= Ax(t)+EH(t,t)\y(t)-Cx{t)]

By defining K(t) = H(t, t), we obtain

Ex(t) =Ax(t) + EK(t)\y(t)-Cx(t)]

for t0 < t.

(B.9)

A P P E N D I X C

Proof of Theorem 5.1b Consider the Weiner-Hopf equation (5.5) for a = t. Then

&l(x(t)yTU)]-\' H(t,T)&\y(T)yT(t)]dT = 0 (C.l) J'o

The first term becomes

&[x(t)yT {t)} = &[x(t)xT (t)}CT (C.2)

The second term becomes

&\y(r)yT(t)] = &ly(T)xT(t)CT+y(T)vT(t)]

^&\y(T)xT(t)]CT + Rv{t)b(t-T)

264/Vol. 110, SEPTEMBER 1988

(C.3)

A P P E N D I X D

Proof of Theorem 5.1c

Assume the solution of equation (5.8) has the form

x(t)=<p(t-t0)x(t0)+\ <p(t-T)(w(T)-Kv(T))dT (D.l) J'o

where

<p(t) =£~1[sE~ (A-EKC)]-{E, <p(fl) = I (D.2)

and the £ ~' is the inverse Laplace operator. Then, from equa­tion (D.l)

P(t)=&[x(t)x(t)T]

= <p(t-t0mx(t0)xT(t0)]<pT(t-t0)

+ S [ ( S / V>U-r)[w(T)-Kv(T)]dT^

[\t lwT(a) ~vT(a)KT]<pT{t-a)ba\)

where cross terms involving x (/0) and [w(t), t > t0] vanish. By applying equation (5.1)

P(t)=<p(t-t0)P(t0)<pT(t-tQ)+ \ <p(t-T)[Rw(T) J'o

+ KRv(T)KT]<fiT(t-T)dT (D.4)

for t > t0. Then

P(t)=<p(t-t0)P(t0)<pTU-t0) + <pV-t0)P(t0)<pr(t-t0)

+ Rw(t)+KRv(t)KT+\ <p(t~T)[Rw(T)

J 'o + KRv(T)KT]<pT(t-T)dT

+ ( ' <p(t-T)[Rw(T)+KRv(T)KT]pT(t-T)dT J'o

Equation (D.2) yields E<p(t-T) = (A-EKC)<p(t-T)

(D.5)

(D.6)

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In equation (D.5) pre and postmultiplying by E and ET, respectively, yields

EP(t)ET = AP(t)ET + EP(t)AT

-EP(t)CTRv~l (t)CP(t)ET+ ERw(t)ET (D.7)

w h e r e i n ) = P0- Q.E.D.

A P P E N D I X E

Proof of Proposition 5.2 Define the following matrices

H(S):

i oi r / o , T* =

PET l \ I -EP I

Also define

sET+AT -CTRv~lC

-ER„ET sE-A

Then

det(T*H(s) T) = det(r*)det (H(s) )det (T)

= det(H(s))

We see that

T*H(s)T=

I 0

-EP I

ETs+AT -CTRv-lC

-ER„ET Es-A

(E.l)

(E.2)

(E.3)

/ 0

PET I

ETs+AT-CTKTET -CTRv~lC

0 Es-A+EKC

From (E.3) and (E.4)

det (H(s)) = det(£s r + AT- CTKTET)det(Es -A+ EKC)

If we define

a(s)=dtt{Es-A+EKC)

(E.4)

) (E.5)

(E.6)

Then

a{ -s) =de t (£ r ( -s) +AT-CTKTET)

= -det(Es-A+EKC)T

= - f l (s ) (E.7)

Therefore, if s1,- € C are eigenvalues of the equation (5.11) - S/ € C are also eigenvalues of that equation. Q.E.D.

A P P E N D I X F

Proof of Corollary 5.1 The Kalman decomposition of the descriptor system was

derived by Verghese et al. [12]. One can decompose the Kronecker canonical form of the descriptor system into finite observable part, finite unobservable part, infinite observable part and infinite unobservable part. Then the observability pencil in decomposed form is:

si-A, 0 0

0 sI-A,0 0

0

0

C,

0

0

0

sE i0-I

0

Cm

*

sE w-1

0

(F.l)

of

Now we can compute the optimal estimates of finite modes and infinite modes separately.

(a) The proof is the same as for state space systems (see [22]).

(b) All finite eigenvalues are observable and we do not feed back the fast variables. Then (F.l) can be reformulated

sI-Ai

0 sii,

c, 7 c. (F.2)

The Kronecker canonical form of the error system can then be defined as

Is-Ai-GfCf

-GfC,

0

E,s-I (F.3)

Here we can assign the eigenvalues of the slow system by feed­back of the slow variables. Then the impulsive modes will re­main fixed.

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