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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 3, MARCH 1995 441

Approximate Decoupling and Asymptotic Tracking for MIMO Systems

D. N. Godbole, Student Member, IEEE, and S . S . Sastry, Senior Member, IEEE

Abstract-This paper presents an algorithm for approximate input-output decoupling of nonlinear MIMO systems that are either numerically ill-posed or exhibit nearly singular behav- ior in the application of decoupling algorithms. Although the systems considered are regular, so that the exact decoupling algorithms are applicable in this case, they require inversion of an ill-conditioned matrix, and yield high gain feedback solutions that may result in actuator saturation and cancellation of high frequency zeros. The approximate algorithms of this paper are numerically robust, and provide solutions that do not cancel far off right-half plane zeros. This latter characteristic is especially valuable when some of the far off right-half plane zeros are unstable. The algorithms are inspired by and are generalizations of some examples in the flight control literature ill, i219 PI.

I. INTRODUCTION

HE nonlinear control toolbox has grown enormously T in the last decade. Central to this development is the theory of feedback linearization for nonlinear systems (see [4], [5]) and the solution to the multi-input multi-output (MIMO) decoupling problem. Several algorithms have been proposed in the literature for solving the problem of exact decoupling for nonlinear MIMO systems, see for example [4]-[9]. These algorithms all require the determination of the inverse of a so- called decoupling matrix. Practical implementation of these algorithms, however, is difficult when the decoupling matrix is ill conditioned or close to singularity, in which case the decoupling of such systems needs excessively large control effort. Further, the algorithm is not numerically robust since it requires the inverse of an ill conditioned matrix.

In this paper, we propose a numerically robust input-output decoupling algorithm for invertible nonlinear MIMO systems. Our efforts are motivated, in part, by the use in Hauser et al. [2], the work of Singh [l], [3] of such techniques to aircraft flight control problems and the work of Barbot et al. [lo] with applications to models of electric motors and the Belousov-Zhabotinsky reaction kinetics. In these examples, the intuition for approximation and the choice of parameters to be approximated is provided by the physics of the problem. This paper attempts to formalize the theory involved in

Manuscript received March 26, 1993. Recommended by Associate Editor, A. M. Bloch. This research was supported in part by Army Research Office under Grant DAAL-91-G-091 and NASA under Grant NAG 2-243.

The authors are with the Department of Electrical Engineering and Com- puter Sciences Intelligent Machines and Robotics Laboratory, University of Califomia, Berkeley, Berkeley, CA 94720 USA.

IEEE Log Number 9408267.

these examples and provide an algorithm for more general MIMO nonlinear systems, whose physical derivation may not be explicitly known by the designer. Another recent paper by Grizzle and Di Benedetto [ l l ] provides an approximate decoupling algorithm for systems that are not decouplable by the exact decoupling algorithms by reason of their not being regular. The approximate algorithm in this paper aims at those systems that are regular and so decouplable by the exact decoupling algorithms, but the numerics of the decoupling are poorly conditioned.

In addition to the numerical robustness of our approximate algorithm, it also serves another important purpose: The exact input-output decoupling algorithm is essentially a pole-zero cancelling control law. Thus, this law cancels zeros of the open-loop system regardless of whether they are in the left half plane or the right-half plane and regardless of their magnitude. In particular the input-output decoupling control law can only be applied to minimum phase nonlinear systems. For systems with far off zeros, cancellation may not be necessary, since the zeros do not play a large role in the system dynamics and indeed may cause instability when the zeros lie in the right-half plane. In either case, cancellation of far off zeros results in high gain controllers. Our approximate decoupling algorithm does not cancel the far off zeros of the open-loop system, thereby pro- viding reasonable gain, practically implementable solutions. The price to be paid is the replacement of asymptotically exact tracking control laws by approximate tracking control laws. This connection between regular perturbations of nonlinear systems and the far off zeros was first pointed out in [12] for the single-input single-output (SISO) case and in [13] for the MIMO case. Systems in which these far off zeros are in the right-half plane are called slightly nonminimum phase systems in [2]. The approximate decoupling controller, in this case, results in a stable closed-loop system.

It is possible to develop several different approximate decoupling algorithms starting from the different decoupling algorithms in the literature. The algorithm presented here is based on the Descusse and Moog dynamic decoupling algorithm (see [6], [14]). Section I1 reviews the Descusse-Moog algorithm. In Section 111, we state the approximate algorithm and Section IV compares its convergence properties with that of the Descusse-Moog algorithm. In many cases, input-output decoupling is a first step in designing a tracking controller. Section V examines the effect of the approximate decoupling on the performance of tracking controller.

0018-9286/95$04.00 0 1995 IEEE

442 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 3, MARCH 1995

11. DECOUPLING ALGORITHMS FOR NONLINEAR SYSTEMS

Consider the square (i.e., number of inputs is equal to the number of outputs) MIMO nonlinear control system described by

where z E !Xn, f(z), g l ( z ) , . . . , g m ( z ) are analytic vector fields on !Xn and hl (z) , . . . , h, (z) analytic functions on W. For convenience, these equations are written as

j. = f(z) + g(z)u Eo: { y = h(x).

Throughout the analysis, we will assume that zo is an equilibrium point of the autonomous system, that is f(z0) = 0. We will assume (without loss of generality) that h(s0) = 0. All the analysis in this paper will be local and will be valid in a given open neighborhood U of 20. We now review some algorithms for decoupling of MIMO nonlinear systems.

We assume in what follows that each output y j has a well defined relative degree yj, i.e., there exists an integer rj such that

L,,Lffhj(z) E 0 V l < rj - 1, V I 5 i 2 m, Vx E U.

Collecting these calculation, we have

A ( z ) is called the decoupling matrix. If A ( z ) is invertible at every point in U, then the static-state feedback given by

U = ( A ( x ) ) - ' [ - b ( ~ ) + V] (3)

will result in a closed-loop system that is decoupled from input 'U to output y. This decoupled and input-output linearized system is given by

(4)

If the matrix A ( % ) is singular, we cannot use a static state feedback to decouple the nonlinear system ([4]), and we have

1 I I 1

to look for a dynamic state feedback to achieve input-output decoupling.

A. Dynamic Decoupling

If decoupling of the system CO of (1) cannot be achieved by static state feedback, it may still be possible to find a dynamic compensator of the form

( 5 ) i = D ( z , z ) + E ( z , z)v U = F ( z , Z ) + G ( z , z)v

with z E %"c, v E Rm, such that the closed-loop system denoted by E, (for extended system)

x = f(.) + g ( z ) F ( z , z ) + g(z)G(z , z)v i = D ( z , z ) +E(& z)v Y = h ( X )

(6) ce: { is decoupled from v to y. The dynamic feedback that decouples the system CO of (1) is actually a static feedback to decouple the extended system E, of (6). There are a number of algorithms in the literature for dynamic decoupling. The approximate decoupling algorithm we will propose is based on the Descusse and Moog algorithm of ([6], [141). We will review the original algorithm to fix notation.

Descusse and Moog Dynamic Decoupling Algorithm: Define the extended system at the end of iteration IC - 1 to be Ck having ze as its state and equilibrium point xg. The algorithm will start at IC = 0 with the given system CO having state z E Rn and 20 as its equilibrium point. The outputs of the system are unchanged during the course of the algorithm.

Step i: Compute the relative degrees r,"(i E { 1, . . , m}) for the m outputs of zk. Define the decoupling matrix A k ( z e )

to have its ijth entry given by

Let T k be the normal rank of A k ( z e ) in an open neighborhood

Step ii: If T k < m, define a square and nonsingukar matrix @ k ( z e ) such that the (m - T k ) last Columns Of &(ze):= A k ( Z : " ) G k ( ze ) are identically zero. Moreover, this process can be carried out such that there exists T k rows of which the nonzero elements form anTk x TI, nonsingular diagonal matrix. It is shown in [6] that G k ( z e ) always exists and is an analytic function of xe.

There are T k columns of A k ( z e ) with nonzero elements, out of which q k columns have two or more nonzero elements. Design a permutation matrix P k such that the first q k columns of A k ( z e ) 4 have two or more elements nonzero, followed by the (Tk - q k ) columns having only one nonzero element and finally the last (m - T k ) columns having all zero elements. Denote G k ( z e ) = Gk(ze)Pk. Define an intermediate input C by

of Xg. If T k = m, Stop.

GODBOLE AND SASTRY: DECOUPLING AND TRACKING FOR MIMO SYSTEMS 443

Step iii: The system now is

X e =f(Ze) + g(Ze)Gk(Xe)G y =h(x").

E, has a well defined vector relative degree [yf, ... ,yk] at x& Let ye:= Ezlyyz". We can construct a local change of coordinates +(xe) = (6 , 77) with 6 = col(?), such that +(x;) = 0, by choosing

Add integrators in series with the first qk inputs. This creates 9 = col(h:(xe), Lf.hf(xe), * * . ,L$-lh:(xe)) the new input vector ii of the form

: = col (& E;, . f * , g:) (10) r t, i

J um

Thus the new system after adding these integrators is

1.:. I I .. 61 I

y = h(x") (9)

where j ( x e ) = g(ze)Gk(xe) . Call this system Ek+l. Step iv: Go to Step i and resume the procedure with k t

k+l, new state variables xe c {xe}U{~i}(i,l,...,qk) and new input U c 6. Let f, g , h still denote the extended f, g , h for notational simplicity. Let U still denote the open set of interest containing the equilibrium point of the extended system. 0

It has been shown by Descusse and Moog that if system (1) is right invertible and satisfies the accessibility rank condition (cf. [5, p. 861) at ZO, then the above algorithm converges in a finite number of steps, L, to an extended system CL which is decouplable by static state feedback.

Note: At the end of ith iteration, the above algorithm adds qi integrators (qi 5 ri) to the system Xi. Thus the dimension of xe increases by 4;.

The Dynamic Extension algorithm (cf. [4]) and Singh's algorithm ([7], [15]) are similar, except that Singh's algorithm involves nonlinear transformations of the output space instead of the input space as in the Descusse-Moog algorithm.

The computation of the rank of Ak(ze) will be greatly simplified if &(le) satisfies a regularity condition (see [16]), which guarantees that the normal rank of Ak(xe) is the same as the local rank for every xe E U . In this paper, we assume that CO is a regular system. If the given system is not regular, then one can use the procedure of [ 111 to get an approximate system that is regular and call this system Co.

B . Normal Form Let us assume that the Descusse-Moog dynamic decoupling

algorithm converges after L steps to a system of the form of (6). Let us denote this extended system by E,. Let (f", g e , he) be the triple characterizing E,, xe = (x, 2) E W+nc its state, ue its input, and ye its output. Let xg = (20, Z O ) be the equilibrium point of interest. This system

and remaining (n + nc - ye) complementary coordinates 77. In these coordinates, E, takes the normal form (see [4])

i ; - '

1 - G!

for i = 1, . . . , m, where

a;#, Q) = Lg;L$-lh:(+-l(<, 7 7 ) ) 1 5 2, j 5 m.

The static state feedback that decouples the system E, is given by

(12)

The decoupling state feedback renders the 77 dynamics unob- servable. The zero dynamics of system Ce are the dynamics of the 77 coordinates in the subspace < = 0 with the decoupling feedback law of (12) (with v = 0), i.e.,

(13)

For a detailed discussion of the zero dynamics and the transmission zeros of nonlinear systems we refer the reader to [ 171, [4, Chapter 61, [5, Chapter 111.

U e = (A"(<, 77) ) - l [ -be( t , 77) + 4.

li = d o , 77) - P(0, 77)[(Ae(0, 77))-lbe(0, 77)l.

111. APPROXIMATE DYNAMIC DECOUPLING ALGORITHM The difficulty in implementing the decoupling algorithm

comes from the ill-conditioning of the decoupling matrix or from a situation in which the decoupling matrix may be nonsingular but close to singularity. To be precise, this occurs if the smallest singular value of A e ( x e ) is smaller than a certain prespecified E > 0 for any x E U. In this case, the algorithm calls for the inverse of an ill-conditioned matrix. In addition to the fact that the inverse is not numerically robust, it may cause large feedback gains in the controller and also cause the cancellation of far-off zeros (see [13, Section 41). To alleviate these difficulties, we propose the following numerically robustified decoupling algorithm: while the algorithm appears to have numerical considerations in mind, it is, in fact, valuable for the reason that it does not cancel far off right-half plane zeros and may help control the magnitudes of the control inputs. To state the algorithm, recall a few basic facts and definitions about the numerical rank of a matrix.

-- I

444

I 8 ,

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 40, NO. 3, MARCH 1995

Definition I : A matrix A E ERnxn is said to have E

numerical rank r if

infrank{B: IIB - All < E } = T.

The norm in the above definition is the induced norm of the matrix induced by the Euclidean norm.

Thus, the numerical rank of a matrix is the lowest it can drop to in an E neighborhood of the given matrix. In particular, if the matrix has (n - r ) of its singular values less than E, then its numerical rank is r.

This algorithm starts at k = 0 with the given system CO having zo as its equilibrium point. We are given a threshold E > 0. Let the extended system at the end of iteration (k - 1) be denoted by xk having xe as its state and zg, the corresponding equilibrium point.

Step i: Compute the relative degrees of the outputs, namely,

be the normal rank of A k ( z e ) in U. If rk = m and if the smallest singular value of Ak(ze) is greater than the threshold t uniformly on U, stop.

Step ii: If all the nonzero singular values of Ak(xe ) are less than E uniformly on U, approximate Ak(ze) by a zero matrix. Go to Step i and recalculate the relative degrees -yt with this approximation.

If rk = m, go to Step iii, with &(ze) = Ak(xe ) and

Design a square, analytic and noFsingular matrix G k ( z e ) such that the last m - T k columns of Ak(ze ) : = Ak(xe)Gk(xe) are identically zero.

Step iii: If the smallest nonzero singular value of A k ( z e ) is greater than the threshold E, go to Step iv, with Ak(xe ) = A k ( X e ) , G k ( Z e ) = Imxm and W k = rk.

If the smallest nonzero singular value of A k ( x e ) is smaller than E, then theTe exists a positive integer wk(< r k ) such that the E rank of Ak(ze) is wk uniformly on U, i.e., ( r k - wk) nonzero singular values of A k ( z e ) are less than E uniformly on U.

Design a square analytic nonsingular matrix Gk(xe)l such that

Approximate Dynamic Decoupling Algorithm:

-yi k . , z = 1,. . . , m, and the decoupling matrix Ak(ze). k t rk

Gk(ze) = Imxm.

A&") = A k ( z e ) G k ( z e )

= [&Ee), . . . , a i k ( z e ) , Ea;k+l(ze), . . . , . ,a:, (8), 0,. * . , 01. (14)

Approximate the rk - wk columns, which are small in norm, by identically zero columns. Go to Step iv with

A#) = [ ,:( ,e) , . . . , & ( z e ) , 0,. . . ,O] .

Note: The E numerical rank of A, (ze) may not be constant over U. If the E numerical rank passes through singularity at x; , then the approximate algorithm will not work.' Throughout the analysis, we assume that the E numerical rank of Ak(xe) is constant in U.

' The proof of existence of such a matrix is given in the Appendix. *The problem may be solved by reducing the value of E . Since CO is a

regular system (cf. [16]), the singularity does not exist for E = 0. However, reducing the value of E defeats the purpose of the algorithm, by causing high gain solutions to the original decoupling problem.

Step iv: Out of wk nonzero columns of Ak(xe) , qk columns will have two or more nonzero elements. De_sign a permutation matrix Pk such that the first q k columns of A k ( z e ) q ( ~ ' ) have two or more nonzero elements, followed by the (wk - q k ) columns having only one nonzero element and finally the last (m - wk) identically zero columns. Denote Gk(ze) = Gk(ze)Gk(ze)Pk. Define an intermediate input by

ii = [Gk(ze) ] - l~ .

Step v: The system Ck now is

X e = f ( z e ) + g(ze)Gk(xe)ii y = h(z").

Add an integrator in series with the first qk inputs. This creates the new input vector G of the form

Thus the new system after adding these integrators is

y = h(xc") (16)

where i ( x e ) = g(ze)Gk(ze) . Call this system Ck+l. Step vi: Retum to Step i and resume the procedure with

k t k+l, new state variables x e t { z e } U { i i ~ } ~ ~ = ~ , . , , , ~ k ) and new input U t G. Let f , g , h still denote the extended f , g , h for notational simplicity. Let U still denote the open set containing the equilibrium point of the extended system. 0

Let us assume that the approximate dynamic decoupling algorithm converges after L steps to a system of the form of (6). Let us denote this extended system by E,. Let (fe, g e , L e ) be the triple characterizing ce, 2" = (z, z ) its state, Ge its input, and jje its output. Let 2; =-(zo, 20)

be the equilibrium point of interest. The system E, has a well-defined vector relative degree [;Y;, . . . , T;] at 2;. Thus we can construct a local diffeomorphism of the form of (10) such that E, will be transformed into the normal form (c, f i ) coordinates given by

= col ( L : ( z e ) , Lp&:( f" ) , . . . , L p z f ( 2 " ) ) -. -.

= col ((1, . . . , q: ) (17)

and remaining (n+fi, - "U;) complementary coordinates fi. In the course of the approximate dynamic decoupling algorithm, we have neglected order t terms at each iteration of

-1 I Tlr 1 1

GODBOLE AND SASTRY: DECOUPLING AND TRACKING FOR MIMO SYSTEMS 445

the algorithm. Thus th_e original or exact system in the normal form coordinates of e, will be

i " - p 1 - 2

m

m

j=1 m

j = 1

Note: If we substitute E-= 0 in (18), then we get the rep- resentation of the system E, in its normal form, coordinates ((, 6). The static state feedback that decouples the system

is given by

6" = ( A e ( ( , ij))-1[-P([, 6) + U ] . (19)

If we compare the approximate and exact decoupling algorithm, we see that

If the exact decoupling algorithm converges with a decoupling matrix A"(%") whose smallest singular value is greater than t uniformly on U, then the approximate decoupling algorithm yields the same result. If, on the other hand, A"(x") obtained by exact decoupling algorithm has its smallest singular value of the order of e uniformly on U, then the decoupling control law (12) will have terms of the order of 1 / ~ that will result in a high gain controller that may not be practically feasible if the actuators have saturation limits.

The approximate decoupling algorithm is of use if we can

1 ) If the given system (1) is right invertible and locally accessible (i.e., the exact decoupling algorithm converges in L steps), then does the approximate decoupling algorithm converge in a finite number of steps?

2) If the answer to the first question is yes, then how much error do we introduce in tracking the reference outputs by using the approximate decoupling and tracking law instead of the exact one?

These questions will be answered in the following sections.

answer the following two questions:

Iv. CONVERGENCE OF APPROXIMATE DECOUPLING ALGORITHM

The approximate decoupling algorithm is based on the Des- cusse and Moog dynamic decoupling algorithm. Consequently, the convergence properties of the approximate decoupling algorithm will be compared with that of the Descusse and Moog algorithm. Ideally the approximate decoupling algorithm should preserve the convergence properties of the Descusse and Moog algorithm.

The algorithm of Descusse and Moog adds dynamics to the given system Ea, to get an extended system CL which is decouplable by static state feedback. The following theorem [6], relates this notion to the nonsingularity of the decoupling matrix AL(x).

Theorem 1: System E of the form of (1) is decouplable by static state feedback, if, the decoupling matrix for is invertible.

The approximate decoupling algorithm converges to an extended system that is robustly decouplable by static state feedback. This notion is defined as follows.

Definition 2: An m x m matrix A(z) is E robustly invertible in an open set U with respect to a threshold E if the E numerical rank of A(x) is m uniformly on U.

of the form of (1) is E robustly decouplable by static state feedback with respect to a threshold E, if, is decouplable by static state feedback and the decoupling matrix A(z) is robustly invertible with respect to t.

The following lemma considers the effect of one iteration of the approximate decoupling algorithm on a system that is decouplable by using the Descusse and Moog algorithm. Lemma 1 : Suppose that the Descusse and Moog algorithm

converges for a system ck of the form of (1). Apply the approximate decoupling algorithm. After one iteration, we get the extended system ck+l. One of the following is true for

Definition 3: A system

L + 1 :

1) y$+l = 00 for some i. 2) At+l(x) is singular and the Descusse and Moog decou-

3) 7"' = n and Ak+l(x) is not robustly invertible.

4) Ak+1(x) is singular, but the Descusse and Moog algo-

5) Ak+l(x) is nonsingular, not robustly invertible and

6) Ak+1(x) is robustly invertible.

pling algorithm does not converge for

Ck+l

rithm converges for Ck+l.

yk+' < n c k + l '

= the dimension of

state space of Ck+,. Proof: This is a list of all the cases after application of

one step of the approximate decoupling algorithm. It is easy to check that the list exhausts all the possibilities. 0

We will analyze each of the above cases in detail to understand why in some cases the approximate decoupling algorithm may not converge.

While applying the Descusse and Moog algorithm, we differentiate each output until at least one input appears on the right hand side. Some of the inputs show up earlier than others making the decoupling matrix singular. Integrators are

C,+l where 7"' = C,"=,$" and n

I


added in front of these inputs to delay their appearance for at least one more step of differentiation.

In this process, at a particular step, some of the inputs might be weakly connected to the outputs, i.e., the functions multiplying them are smaller than the threshold E uniformly in U. These functions are approximated by zero in the approximate decoupling algorithm. This modification in the original Descusse and Moog algorithm might make the resulting systems noninvertible. The classification of various cases in the previous lemma helps detect such systems in the following manner:

Analysis of Lemma I : Recall that 7," are the relative degrees of the outputs of Ck. We have (""I ' k = b k ( X ) 4- A k ( X ) U

Y 2

rank of A k ( x ) = r k . At the end of Step ii, (m - rk) columns of &(.) are identically zero and there is at least one nonzero element in each row of A k ( x ) .

At the end of Step iv, some of the rows of & ( X ) G k ( l C )

might be identically zero because of the approximation of (Tk - W k ) columns by zero. The number of such rows is less than or equal to (Tk - W k ) . Let us denote the set of outputs corresponding to these rows by P k .

( W k - q k ) columns of A k ( i ) G k ( z ) have only one nonzero element. By construction, these nonzero elements will be in ( W k - q k ) different rows. Let us dcnote the ( W k - q k ) outputs corresponding to theze rows by Y k . The remaining outputs will be denoted by Y k .

The outputs of Ck do not change in the process. Thus for C k + l ? we have . k + l = k y . Ta , 2 s.t. yi E Y k $-+' = 7," + 1, vi s.t. yz E Y k

y$+l 2 7," + 1, v i s.t. yz E Pk. Case 1 : If the only nonzero entries in the ith row are in

the j column, then when the j th column of A k ( i ) G k ( x ) is approximated by zero, the ith row is made identically zero. yz is only affected by uJ , and after the approximation, ug never appears on the right hand side again. This makes y$+l = 00.

Case 2: The situation here is that Case 1 recurs, but not immediately at the end of the k - 1st step but later in the algorithm. Thus the system C k loses invertibility because of the approximation. Some rows of A k + l ( x ) will be dependent for all I > 0. This case can also be avoided by normalizing the jth input.

In the course of the approximate decoupling algorithm, Case 2 goes unnoticed until you reach Case 3. Thus the reason for Case 3 is in fact the occurrence of Case 2 during one of the previous iterations.

If the approximate decoupling algorithm converges to a system C L that can be decoupled by static state feedback, the outputs ya and their respective derivatives up to the order of (7," - 1) qualify as a partial change of coordinates. In the normal form notation these are the [ coordinates.

During each iteration of the approximate decoupling algorithm, the state space dimension of is extended by

Qk whereas at least (m - W k + q k ) new < coordinates are introduced. The difference between the state space dimension of Ck+l and the dimension of coordinates decreases by (m - W k ) during each iteration of the approximation decoupling algorithm.

Case 3: If we go through (k + 1)th iteration, dimension of [ coordinates will exceed the dimension of the state space of Ck+2.,Thus we cannot proceed further.

Cases 4, 5, and 6 lead towards the convergence of approx-

Definition 4: Suppose CO satisfies the hypothesis of Des- cusse and Moog algorithm. Apply the approximate decoupling algorithm, for given E > 0. Let k 2 0 be the smallest integer such that either case 1, 2, or 3 of the previous lemma is true for ck+l. Then the system CO is said to be E-unnormalized.

Theorem 2: Suppose CO satisfies the hypothesis of the Descusse and Moog dynamic decoupling algorithm. Apply the approximate decoupling algorithm for a given E > 0. Then one of the following is true.

The approximate decoupling algorithm converges in a

The system CO is E-unnormalized.

imate decoupling algorithm. 0

finite number of steps.

Proofi During each step of the approximate decoupling algorithm, the difference between the state space dimension and the dimension of coordinates decreases by (m - W k ) .

Thus, if the first three cases of the previous lemma are avoided during each iteration, the algorithm has to converge in a finite number of steps.

The discussion following the previous lemma shows that the first three cases correspond to the underlying system being unnormalized. 0

In general, it is not possible, a priori, to find out whether a given system is normalized or not. If the approximate decoupling algorithm does not converge in n steps, then the system is unnormdized, provided it was decouplable by using the exact Descusse and Moog dynamic decoupling algorithm.

The design parameter E is an input to the approximate decoupling algorithm. If the approximate algorithm does not converge for a given t, because of the system being E unnormalized, then the value of E can be reduced. This might result in convergence of the algorithm but the original problem of numerical robustness and high gain solutions still exists. In the limit that E - 0, one recovers the Descusse-Moog algorithm. For E = 0, the E unnormalized systems are noninvertible. Thus, given an invertible system that satisfies the accessibility rank condition, one can always find an E so that the approximate algorithm converges. In particular, the algorithm converges for the specific examples of 111, [21, [31.

A . Multiple Time Scale Zero Dynamics Since the zero dynamics of a system does not change by

addition of integrators to its input channels [4, p. 3891, or by input space transformations or by state feedback, the zero dynamics of CO is same as that of CL, where C L is the extended system at the end of Descusse and Moqg algorithm. The next lemma compares the zero dynamics of C L and C L , where E L is the extended system at the end of approximate

I I I 1 -- 1 -

I 1


It can be shown (e.g., see [2]) that if The reference trajectory and its derivatives are bounded

Zero dynamics of (23) (i.e., the equilibrium point i j o = 0

(24)

is _exponentially stable. q ( [ , f i ) +_P([, i j )Ge( [ , 6, w) is locally Lipschitz continuous in I , 71 then limt+oo e;((t) = 0 Vi . and the states I, 71 remain bounded.

The controller _of (20) is designed for the approximate extended system E,. If we apply this controller to the exact system, we get the system equations in the (e, 71) coordinates given by

and small enough.

of the system)

6 = 4 0 , i i) + P(0, ?)Ge(O, ?, 0)

6' = Ti"ez + c / ~ ( z ~ ) ~ ~ ( z ~ ) 1 I i 5 m Ir = 445 ii) + P ( t , ?Fe& 6, w) (25)

where

p"=

represents the dynamics that was neglected during the approximate decoupling algorithm. Each $ is 1 x m row vector of functions of z, the first W O elements of which are identically zero.

The tracking control law (20) was designed for a system of the form (25) with E = 0. The following theorem shows that it works for the approximate system with nonzero E as well. This theorem is motivated by and is similar to the one for slightly nonminimum phase systems as in [2].

Theorem 3: If Zero dynamics of system (25) (i.e., the equilibrium point 60 = 0 of (24)) is exponentially stable in a neighborhood U. The functions p" ( ze)Ge (ze) are locally Lipschitz continuous in a neighborhood U with @(zg)G"(zg) = OVi = 1, . . . , m. q ( ( , I s ) +-P(i, i j ) i ie(( , i j , w) is locally Lipschitz continuous in I , 6, U.

Then for c sufficiently small and for desired trajectories with derivatives small enough, the states of system (25) are bounded and the tracking errors satisfy

l l e ~ l l = II'$ - Yd, 1 1 5 Kc

for some K < CO.

Note: Since we know that f, g, h are smooth functions of z to start with, the functions p"(ze)Ge(ze) will be locally Lipschitz so long as the matrix G k ( X e ) is a smooth function of xe at each iteration of the approximation decoupling algorithm.

Proofi From (22) and the fact that the desired trajectory and its derivatives are bounded (by bd), we get

11C11 I I l e l l + bd. (26)

The transformation that transforms Fe into (i, 3) coordinates is a diffeomorphism, thus there exits I, > 0 such that

(27) 1141 5 ~z(11i11 + 1 1 ~ 1 1 > ~ As the functions @(ze)Ge(ze) are locally Lipschitz continuous with @(z;)G"(zg) = 0, there exists a positive constant Zp such that

112PP(.">Ge(z"> II I M1ze II (28)

where p ( x e ) is the block diagonal matrix with p2(ze) being its diagonal blocks. Since the zero dynamics

Ir = q(0, 6) + P(0, 71)GLe(0, ?, 0) (29)

is exponentially stable, by a converse Lyapunov theorem [22] there exists, V( i j ) and positive constants kl, k2, k3, k4 such that

klll?l12 I q i i> I ~211?112

Thus from the exponential stability of zero dynamics and the Lipschitz continuity of q + Puk, we get

1 1 171

GODBOLE AND SASTRY: DECOUPLING AND TRACKING FOR MIMO SYSTEMS

-1 A1

0 1

1 0

1 0 0

0 0

1

0 1

~

449

Let

7 j = k3

4 ( l p L + k41,(1 + lv))2'

Then, for all 11, 5 12 and all E 5 min (6, 1/4ZpZZ), we have

Thus V < 0 whenever llfjll or [le11 is large. This implies that 11ij11 and [le11 and also, I l i l l and llxll are bounded. The above analysis shows that if we choose the initial condition sufficiently close to x; and bd sufficiently small, we can guarantee that the states will remain in U. Using the boundedness of x and the continuity of P(x)ii"(x), we see that

ci(x) = 2 e i + Epi(x)iie(x> are m SISO exponentially stable linear systems driven by order E input. Thus we conclude that the tracking errors ei converge to a ball of order E. 0

VI. CONCLUSION

A numerically robust algorithm for input-output decoupling of nonlinear dynamical systems has been proposed. This algorithm provides low gain, practically implementable controllers that do not cancel far off zeros. The use of this algorithm for a slightly nonminimum phase system (i.e., one that has far off right-half plane zeros), results in an overall stable closed-loop system. It is shown that the tracking controllers constructed by using this approximate decoupling algorithm result in bounded tracking with stability. Controllers based on this theory already exist for a few specific examples in the literature and this paper can be thought of as an attempt to formalize the techniques used in those particular examples. Detailed calculations to establish one-to-one mapping between the neglected dynamics and the formal structure at infinity are still to be worked out. The ongoing work on a CAD package called APLIN ([23]) based on this approach will help automate the application of

this theory. Although this approximate algorithm is based on the Descusse-Moog dynamic decoupling algorithm, a similar algorithm based on the dynamic extension algorithm (see [4, Chapter 71) can be worked out in similar fashion.

APPENDIX EXISTENCE OF AN ANALYTIC MATRIX G k ( x )

The proof of existence of G k ( x ) is given by Descusse and Moog in [6 ] . Thus we have a matrix A,(,) whose E numerical rank is W k and the last m - Tk columns are identically zero.

From the definition of E numerical rank of A k ( x ) , it is possible to find a W k x W k minor, say, &(x) , such that all the singular values of & ( x ) are bigger than E in U . Without loss of generality, we assume that &(x) is the block formed

let 6(x) represent its determinant. By definition, any minor of & ( X ) having size bigger than

wk, will have at least one singular value smaller than E uniformly in U. Thus the determinant of this minor will be of the order of E x S(z). Thus we get

by the first wk rows and the first W k COlUmnS Of A k ( x ) , and

~,

det I

T k

6 ( x ) A i j ( x ) + C A l ( z ) A i l ( x ) = order E x S(x), i=l

where 6(x) is in the jth row and column. The first T k - 1 elements in the jth column of & ( z ) g j ( x ) will be zero. The rkh element will be the determinant given by (32), which is

1,


of the order of E . The rest of the elements in this column must be of the order of t, or else there will be a minor of A k ( : r ) g k ( x ) having all its singular values more than t and having more than W k columns and rows. This will contradict the definition of the E numerical rank of a matrix. Thus this particular procedure makes the elements of jth column of the order of t as compared with S(x). We can have T k - Wk

matrices of these form making one column of A k ( x ) g i ( x ) small at a time. It is clear that the matrix Gk(z) will thus be nonsingular, square,Aand an analytic function of z.

Thus Gk(z): = Gk(z)Gk(x)Pk(z) is a square invertible matrix of analytic functions of z.

ACKNOWLEDGMENT

The authors would like to acknowledge helpful discussions with M. Di Benedetto, J. Grizzle, J. Hauser, R. Kadiyala, and P. Kokotovii..

REFERENCES

S. N. Singh, “Control of nearly singular decoupling systems and nonlinear aircraft maneuver,” IEEE Trans. Aerosp. Electron. Syst., vol. 24, no. 6, pp. 775-784, 1988. J. Hauser, S. Sastry, and G. Meyer, “Nonlinear control design for slightly nonminimum phase systems-applications to V/STOL aircraft,” Automatica, vol. 28, no. 4, pp. 665-679, 1992. J. J. Romano and S. N. Singh, “1-0 map inversion, zero dynamics and flight control,” IEEE Trans. Aerosp. Electron. Syst., vol. 26, no. 6, pp. 1022-1029, 1990. A. Isidori, Nonlinear Control Systems. Springer-Verlag, 2nd ed., 1989. H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems. Springer-Verlag, 1990. J. Descusse and C. Moog, “Decoupling with dynamic compensation for strong invertible affine non-linear systems,” Int. J . Contr., vol. 42, no. 6, pp. 1387-1398, 1985. S. N. Singh, “Decoupling of invertible nonlinear systems with state feedback and precompensation,” IEEE Trans. Automat. Contr., vol. AC- 25, pp. 1237-1239, 1980. H. Nijmeijer and J. M. Schumacher, “The regular local non-interacting control problem for nonlinear control systems,’’ SIAM J. Contr. Optim., vol. 24, pp. 1232-1245, 1986. H. Nijmeijer and W. Respondek, “Decoupling via dynamic compensation for nonlinear control systems,” in IEEE Contr. Dec. Conf., pp. 192-197, 1986. J. P. Barbot, N. Pantalos, S. Monaco, and D. Normand-Cyrot, “On the control of singularly perturbed nonlinear systems,” in Proc. IFAC Works. Nod. Contr. Syst. Des., NOLCOS 11, 1992. J. W. Grizzle and M. D. D. Benedetto, “Approximation by regular input-output maps,” IEEE Trans. Automat. Conrr., vol. 37. no. 7, pp. .. 1052-1055, 1992. S. Sastrv, J. Hauser, and P. KokotoviC. “Zero dynamics of regularly - . perturbed systems may be singularly perturbed,” Syst. Contr. Lett., vol. 13, pp. 299-314, 1989. A. Isidori, S. Sastry, P. KokotoviC, and C. Bymes, “Singularly perturbed zero dynamics of nonlinear systems,” IEEE Trans. Automat. Contr., vol. AC-37, no. 10, pp. 1625-1631, 1992. J. Descusse and C. Moog, “Dynamic decoupling for right invertible nonlinear systems,” Syst. Contr. Lett., vol. 8, pp. 345-349, 1987. S. N. Singh, “A modified algorithm for invertibility in nonlinear systems,” IEEE Trans. Automat. Contr., vol. AC-26, pp. 595-598, 1981. M. D. Benedetto and J. Grizzle, “Intrinsic notions of regularity for local inversion, output nulling and dynamic extension of nonsquare systems,” Contr. Theory Advan. Tech., vol. 6, pp. 357-381, 1990.

[ 171 A. Isidori and C. Moog, “On the nonlinear equivalent of the notion of transmission zeros,” in Lecture Notes in Control and Information Sciences, C. Bymes and A. Kurzhanski, Eds. New York: Springer- Verlag, 1988, pp. 147-158.

[ 181 P. KokotoviC, H. Khalil, and J. O’Reilly, Singular Perturbation Methods in Control: Analysis and Design. Academic Press, 1978.

[19] J. Huang and W. J. Rugh, “On a nonlinear multivariable servomech- anism problem,” Automarica, vol. 26, no. 6, pp. 963-972, 1990.

[20] A. Isidori and C. I . Bymes, “Output regulation of nonlinear systems,” IEEE Trans. Automat. Contr., vol. 35, no. 2, pp. 131-140, 1990.

[21] J. Grizzle, M. D. D. Benedetto, and F. Lamnabhi-Lagarrigue, “Necessary conditions for asymptotic tracking in nonlinear systems,” University of Michigan, Ann Arbor, Tech. Rep., Control Group report no. GCR 91-4, 1991.

1221 M. Vidyasagar, Nonlinear Systems Analysis, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1992.

[23] R. R. Kadiyala, “AP-LIN: A CAD tool box for nonlinear control design,” Ph.D. dissertation, Dept. Elect. Eng., Univ. of Califomia, Berkeley, Califomia, 1992.

Datta N. Godbole (S’93) received the B.E. degree in Electrical Engineering from University of Pune, India, in 1987, and the M. Tech. degree in Systems and Control Engineering from the Indian Institute of Technology, Bombay, India, in 1989. He is currently a candidate for the Ph.D. degree in the Electncal Engineering and Computer Science Department at the University of Califomia, Berkeley.

His research interests include nonlinear control theory, automated highway systems, control of hy- bnd dynamical systems, applications to transporta-

Mr. Godbole is a recipient of the University Gold Medal in engineenng tion systems and vehicle dynamics.

from University of Pune, India

S. Shankar Sastry (S’79-SM’90) received the B.Tech. degree from the Indian Institute of Technology India, in 1977, the M.S. and Ph.D. degrees in electrical engineering in 1979 and 1981, respectively and the M.A. degree in mathematics in 1980, all from the University of Califomia, Berkeley.

He is a Professor in the Department of Electrical Engineering and Computer Sciences, University of Califomia, Berkeley, and was a Gordon McKay Professor of Electrical Engineering at Harvard

University in 1994. He has held visiting appointments at the Australian National University, Canberra, the University of Rome, the laboratory LAAS in Toulouse, and as a Vinton Hayes Visiting Professor at the Center for Intelligent Control Systems at MIT. His areas of research are nonlinear control, nonholonomic motion planning, robotic remote surgery, control and verification of hybrid systems and biological motor control.

Dr. Sastry is a coauthor (with M. Bodson) of Adaptive Control: Stability, Convergence and Robustness, (Prentice-Hall, 1989), and (with R. Murray and Z. Li) of A Mathematical Introduction to Robotic Manipulation, (CRC Press, 1994). His book, Nonlinear Control: Analysis, Stability and Control, is to be published by Addison Wesley in 1995. He is an Associate Editor of the IMA Journal of Control and Information, the International Journal of Adaptive Control and Signal Processing and the Journal of Biomimetic Systems and Materials.

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