Geometric methods for nonlinear process control. 1. Background

I n d . Eng. C h e m . Res. 1990,29,2295-2310

2295

REVIEWS Geometric Methods for Nonlinear Process Control. 1. Background Costas Kravaris” Department of Chemical Engineering, T h e University of Michigan, A n n Arbor, Michigan 48109-2136

Jeffrey C. Kantor Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556

This is the first part of a review paper for geometric methods in nonlinear process control. We review here the mathematical and systems theory background, including linear results, tools from differential geometry, nonlinear inversion, and zero dynamics. The concept of feedback linearization of nonlinear systems is introduced at the end. 1. Introduction

Nonlinearities are the rule rather than the exception in chemical processes. Yet standard process control algorithms do not take into account the nonlinearities; they are based on a linear time-invariant approximate process model, valid in a neighborhood of an operating steady state. The performance of such a linear model-based controller can be unacceptably poor in the presence of severe process nonlinearities. The above considerations have motivated growing activity in the area of nonlinear chemical process control in the past 5 years (1-7,lO-12,14,18-21,23,24,2635,37-39, 42-53,56-67,6+71, 76, 79). This growing importance of nonlinear control in chemical processes poses an immediate question: Do we have the theoretical tools to be able to develop general synthesis techniques for nonlinear control systems? The theoretical literature of the 1950s and the 1960s concentrated on the stability analysis of nonlinear systems (25, 77) and on the optimal control of nonlinear systems (8, 15, 68). Although this early work provided valuable understanding of nonlinear stability issues and dynamic optimization problems, it did not address the controller synthesis problem. In the past decade, there have been major developments in the nonlinear systems theory literature using techniques from differential geometry (see Isidori (40)for a review of results obtained prior to 1985). These have been motivated by the geometric approach of linear systems theory (78); differential geometry provided the necessary mathematical tools that allowed “word by word” extension of most of the results in Wonham’s book (78) to nonlinear systems. Although these major theoretical developments did get noticed by the chemical engineering community, their impact was limited by two important factors: (i) There seems to be a lack of perspective in the systems theory literature in terms of the relevance of the results. Some of the results are extremely important in process control, whereas some others are merely nice theorems. (ii) The entire geometric nonlinear systems theory literature has adopted the so-called “modern view” of differential geometry, which uses the abstract language of *To whom all correspondence should be addressed. 08a8-5885/90/2629-2295$02.50/0

manifolds, bundles, distributions, forms, etc. Although no one can deny the clarity, rigor, and elegance of the “modern language”, the geometric picture is lost in the machinery and the abstraction and, more importantly, the results become unreadable to the majority of control engineers. These factors have motivated the authors to write a two-part review paper on geometric methods for nonlinear process control. Our intent is not to go through every theoretical result or every single application; instead, we will try to give a perspective on the field in a language that is understood by the majority of control engineers. We will review only those theoretical results that we believe are relevant to control applicatiofls and provide representative chemical engineering examples. Both parts of our paper will be written in a “classical” (advanced calculus) language; we will drop a lot of the “cosmetic mathematics” to be able to more effectively communicate the essence of the results. At the time of writing this paper, continuous-time SISO nonlinear systems are well understood from a systems theory point of view. Also, although there are still open problems in controller synthesis, all the main issues can be well-identified. For these reasons, we decided to deal exclusively with continuous-time SISO nonlinear systems of the form = f i ( x 1 , x2, 12

= fi(X1,

i n

= fn(x1, XZ,

x2,

*a*,

xn)

+ gl(x1, ~

2 ,

.*a,

..., x,) + g2(x1, x2, + gn(x1t ~

xn)

Y = h(xi, ~

e..,

2 ,

xn)u

x,)u

xn)u

..., x n )

2 ,

where u is the manipulated input, y is the output, and xl, ..., x , are the states. In a more compact notation, we will write

x2,

i = f(x)

+ g(x)u

Y = h(x)

(1)

where now x represents the vector of states and f ( x ) and g(x) are vector functions. In the special case where f ( x ) = Ax, g ( x ) = b, and h(x) = cx with A, b, and c being n X n, n X 1,and 1 X n matrices, respectively, (I) becomes the

0 1990 American Chemical Society

2296 Ind. Eng. Chem. Res., Vol. 29, No. 12, 1990

standard SISO linear system x = Ax + bu y = cx

Solution of

(11)

The linear system (IT) will be the starting point of our review. The next section will provide a collection of linear results for which direct nonlinear extensions are available and will be subsequently reviewed. Section 3 will provide the necessary background from differential geometry for the development of the nonlinear results. Section 4 will review key nonlinear systems theory concepts and results: relative order, zero dynamics, and inversion of nonlinear systems. Section 5 will introduce the problem of exact feedback linearization of nonlinear systems. This section will conclude the present first part of the paper. Part 2 will provide a review of controller synthesis results and indicate the open problems in the area. 2. Standard Results from Linear Systems Theory This section contains a collection of results from linear systems theory that will be extended "word by word" to nonlinear systems in the subsequent sections or in part 2. Its role is to facilitate the interpretation of the nonlinear results that will follow. None of the linear results are new or even recent; they can be found in classical books and monographs of linear systems theory like Kailath (41) and Chen (22). However, we will not present the results in their most popular form but instead in the form that is completely analogous to the subsequent generalizations. Consider a SISO nth-order linear system of the form (11). Its states x are, in a sense, the coordinates that we use to describe its evolution with time. Changing coordinates is a common operation that is necessary in the study of dynamic systems. In linear systems, a change of coordinates is called a similarity transformation, and it is of the form [ = Tx

(1)

where T is a nonsingular matrix. Under the transformation (11, the linear system (11) becomes 4 = At + 6u y = E[

Input

Input

(2)

where A = TAT-', 6 = Tb, F = cT-'. The systems (11) and (2) have, of course, the same transfer function because a change in coordinates cannot affect the input/output properties of a dynamic system. We say that (11) and ( 2 ) are two different state-space realizations of the same input/output system. In subsection 3.5, we will see how to transform nonlinear systems of the form (I) in different coordinates. 2.1. Relative Order and Inversion of Linear Systems. Consider the linear system (11). Its transfer function is given by cAdj(sI - A)b Y(S) - - - ~ ( S IA)-'b = (3) u(s) det(sI - A)

A series expansion of (3) gives cb cAb cA2b C(SI - A)-'b = - + - + - + ... s s2 S3 The quantities cb, cAb, cA2b, ...,cAk-'b, ..., are called the Markov parameters of the system, and by the above expansion, they completely characterize its transfer function. From (3) we see that the order of the denominator det(sI - A) is always n, but the order of the numerator cAdj(sI

Figure 1. State feedback control loop.

A)b can be anywhere between zero and n - 1. The relative order of the system is the difference between the order of the denominator det(sI - A) and the order of the numerator cAdj(sZ - A)b. Since the Markov parameters completely determine the system's transfer function, it is expected that they would determine the relative order as well. Proposition 2.1.1. The relative order of (ZI) is the smallest integer r for which cA"b # 0. From the above proposition, we see that a system will have relative order r = 1if cb # 0, r = 2 if cb = 0 and cAb # 0, r = 3 if cb = cAb = 0 and cA2b # 0, etc. Also observe that, from the definition of r as the difference of the degrees of denominator and numerator polynomials, it follows that 1 I r I n. An alternative characterization of relative order can be obtained in terms of the time derivatives of the output y, as indicated in the following proposition: Proposition 2.1.2. Consider a system of the form (II) whose relative order is r. Then,

-

dY _ - CAX dt

dP'y dt"

- cA'-~x

dry- - cA'x dt'

+ cA'-lbu

(4)

I n other words, r is the smallest order of derivatives of y that depends explicitly on u. In subsection 4.1, the concept of relative order will be

extended to nonlinear systems. The inversion problem for linear systems involves constructing a state-space realization of the transfer function of the process inverse: u(s)

--

y(s)

-

det(sI - A) cAdj(sI - A)b

(5)

A solution to the inversion problem can be easily obtained in terms of the relative order r and the system matrices A, b and c. Proposition 2.1.3. Consider a linear system of the form (ZI)whose relative order is r. Then, the dynamic system

1 u=---cA'-'b

d'Y dt'

cA' cA"b

z

is a state-space realization of its inverse ( 5 ) . It is important to observe that the order of (6) is n, whereas the order of the inverse (5) is n - r. Consequently, (6) is not a minimal order realization of the inverse. This can also be seen through a transfer function calculation for (6): there are r zero-pole cancellations at the origin. The result of proposition 2.1.3 will be extended to nonlinear systems in subsection 4.1.

Ind. Eng. Chem. Res., Vol. 29, No. 12, 1990 2297

2.2. A Normal Form for Systems of Relative Order r , In what follows, a normal form will be introduced, which will clarify the concept of relative order and will allow the construction of a minimal-order realization of the inverse of a linear system. The dynamic system

Zn-r+l = Zn-r+2 zn-l =

z,

zn

dry dtr

=

+ Yn

Equivalently, we can write the inverse as is a normal form for systems of relative order r, where A, r, dn, and Y,, are ( n - r ) X ( n - r ) , ( n r r ) X r, 1 X ( n - r), and 1 X r matrices, respectively, and b is a nonzero scalar. Calculating the Markov parameters of this system, one can see that the relative order is indeed r. Every linear system of the form (11) of relative order r can be transformed into the normal form (7). To see this, one must first observe that the row vectors c, cA, ..., cArl are linearly independent and b # 0. Consequently, one can rotate the indexes of the state variables xl, ..., x , so that (i) r - 1 linearly independent columns of the matrix

[:. ] cAra

are exactly in the ( n- r + l)th, ( n - r + 2)th, ..., ( n - 1)th places and (ii) b, # 0. Then, one can verify that the coordinate transformation bi

The system (9) is a minimal-order realization of the inverse system (5). The normal form (7) will be generalized to nonlinear systems in subsection 4.2. 2.3. Zeros and Zero Dynamics. The zeros of a linear system (11) are the roots of the numerator polynomial of its transfer function, Le., the roots of cAdj(sZ - A)b. Equivalently, one can view the zeros of (11) or (3) as the poles of its inverse (5). It is interesting to observe here that, since a minimalorder realization of (5) is given by (9),the zeros of (11)are exactly the roots of det(sZ - A ) = 0. In other words, the dynamics of the first n - r equations of the normal form (7) completely determines the system zeros. This can also be seen through a straightforward calculation of the transfer function of ( 7 ) :

-Y(S) - 4s) & & - r= Xn-?

&det(sZ- A )

bn-r

--

b, " s'det(s1-

&,-l

= cAr2x

[, = cAr-lx

(8)

is invertible and transforms (11) into (7). The normal form (7) provides a very clear interpretation of the concept of relative order (see also proposition 2.1.2); r is the number of integrations (state equations) that the input u has to go through in order to affect the output y -- x,,-,+~. Consequently, r is a measure of how direct the dynamic effect of the input u is on the output y. The normal form (7) is also useful in obtaining a minimal order realization of the inverse system (5). Since the choice of realization does not affect the input/output behavior, we can choose to work with ( 7 ) instead of (11) and apply proposition 2.1.3. This leads to

r

11

A ) - [y,det(sl - A ) + CnAdj(sZ- A)r

The extension of the concept of zeros to nonlinear systems will be discussed in subsection 4.3; it provides a key ingredient of nonlinear systems theory. As we will see, it is not possible to talk about nonlinear system zeros as a set of numbers; instead, we will be talking about the zero dynamics of a nonlinear system. The zero dynamics of a system is defined as the dynamics of its inverse. For a linear system (II), the dynamics of its inverse is governed by the state equations of (9), i.e.,

Depending on whether we wish to consider forced or unforced dynamics, we can accordingly define the zero dynamics:


Definition 2.3.1. Consider ( I I ) ,which is transformed to (7) through a n appropriate coordinate transformation. T h e n , the dynamic system

1

u=

F[V

- [ a , - B,,

an-l - Bn-l

...

CY' - 611x3

(16)

It is not difficult to generalize this relatively trivial pole placement result to an arbitrary controllable system i= Ax bu. All we need to do is to find an appropriate similarity transformation l = T x that will transform it in the form (15). Then the state feedback,

+

is called the forced zero dynamics of ( I n . The dynamic system (11)

is called the unforced zero dynamics of ( I I ) or simply the zero dynamics of ( I I ) . The unforced zero dynamics (11)completely determines the zeros of (II), and conversely, the zeros of (11)completely determine the zero dynamics (11) up to a coordinate transformation. For this reason, it is the unforced zero dynamics that can provide an alternative to the concept of zeros in the sense that (11) and the roots of the numerator polynomial of the transfer function contain exactly the same information. Consideration of the forced zero dynamics can arise in situations where the input/output characteristics of the inverse are important in addition to its internal dynamics. 2.4. Static State Feedback. Consider the system (11) being subject to the feedback law u = XU - kx (12) where v is a scalar external input, h is a scalar gain for external input precompensation, and k is a n row of feedback gains. The resulting closed-loop system is then described by i = ( A- bk)x

or

will make the closed-loop characteristic polynomial equal to Sn

Proposition 2.5.1. Consider the controllable system i = Ax bu (18) and denote by q a row vector that satisfies qb = 0

+

qAb = 0 qAn-2b = 0 qA"-'b # 0

( = T x = [:A:

(13)

(14)

These will be given in part 2, section 2. 2.5. Pole Placement by S t a t e Feedback. The dynamic system x, =

]

x

(20)

qA"-'

is invertible and transforms (18) into

The following are basic properties of linear static state feedback (12): (i) It preserves linearity of the system. (ii) It preserves the relative order of the system. (iii) It preserves the zeros (zero dynamics) of the system. (iv) It modifies the poles of the system. Similar properties hold for nonlinear systems of the form (I) under static state feedback of the form

u = P ( X ) + q(x)u

(19)

T h e n the transformation

+ (hb)u

y = cx

+ a1sn-1 + ... + an_1s + a,

XZ

E1

=

l2

i2

=

E3

En-1

=

En

in= q A n T 1 ( + qA"-lbu

(21)

Remark 2.5.1. The vector q = last row of [blAbl...I An-'b]-l satisfies (19) and in particular qAn-'b = 1.

Now that we know how to transform an arbitrary controllable system (18) to the canonical form, we can use formula (17) for pole placement. This leads to al]

x2 = xs

+ qAnT')Tx]

or xn-l

1, = - B n q -

= x,

6n-1X2

- ... - SIXn + 6u

(15)

where 6 is a nonzero scalar, is a canonical form for controllable linear systems. The characteristic polynomial of (15) is sR + + ... + & n - l ~ + B,. We would like to use state feedback to modify the characteristic polynomial to a polynomial of our choice, say s" + alsn-l + ... + CY,_~S + an.

The necessary state feedback is given by the simple formula

The result is summarized in the following proposition. Proposition 2.5.2 (Ackerman's Formula). Consider a n arbitrary controllable system (18)and denote by q a row vector that satisfies (19). T h e n , the state feedback u=-

1

qAn-'b

[V

-

(qAn + alqA"-l

+ ... + a,_,qA + a , q ) ~ ] (22)


places the closed-loop poles at the roots of the polynomial s"

+ Cqsn-1 + ... + (Y,-lS + an

The results of propositions 2.5.1 and 2.5.2 can be generalized to nonlinear systems. This will be done in part 2, section 3. 2.6. Input/Output Pole Placement for MinimumPhase Systems. Consider now a system of the form (II), whose relative order is r, and the problem of synthesizing a state feedback that provides the closed-loop system with a transfer function of our choice. Since static state feedback preserves the relative order of the system, the simplest possible closed-loop transfer function is an rth order lag: y-( s) 4 s ) s'

1

+ PlS?l + ... + ppls + p,

(23)

Proposition 2.6.1. Consider a system of the f o r m (II) whose relative order is r. T h e n , the necessary state feedback that produces the closed-loop transfer function (23),where P1, ..., p, are arbitrarily selected numbers, is given by [ u - (cA'

u=-

cAr-'b

the modern (manifold) perspective of differential geometry. 3.1. Vector Fields and Scalar Fields. Let Wn denote the usual n-dimensional Euclidian space. Its elements will be called "points" or "vectors" depending on the context. The inner product of two vectors x = (xl,..., x,) and y = (yl, ...,y,) is given by ( x , y ) = xlyl + xgv2 + ... + x , y n . The vectors el = ( 1 , 0 , ...,0), e2 = (0, 1, 0, ..., O), ..., e, = (0,0, ..., 0, 1) are a basis of W",and we can always write x = xlel + x2e2 ... x,e,. A scalar field on W"is simply a function with domain of definition an open subset U Wn and values in W. So, given an element x = (xl, x2, ...,x,) of U, a scalar field h ( x ) will associate to it the real number h ( x l , ..., x,). A scalar field h ( x ) will be called a C scalar field on W" if all partial derivatives dh/dxi exist and are continuous functions of (xl, ..., x,). h ( x ) will be called a C" scalar field on W" if all partial derivatives of arbitrary order exist and are continuous functions. A vector field on W" is simply a vector function with domain of definition an open subset U C R" and values in Rn. So, given an element x = (xl, x2, ..., x,) of U, a vector field g(x) will associate to it the vector

+ +

c

+ P,cA'-' + ... + P,-,cA + Prc)xl (24)

One way to verify the above proposition is a brute force calculation of the transfer function of the closed-loop system (13) for 1 A=cA'-'b and

In carrying out this calculation, one can see that the closed-loop characteristic polynomial is det(sZ - A + b k ) = cAdj(sI - A)b(s'

+ &s'-~ + ... + P r - l ~+ p,)

cA"b i.e., that the state feedback (24) places the closed-loop poles at the zeros of the process and at the roots of s' + &sP1 + ... + prw1s+ p,. Consequently, the closed-loop system will be internally stable (and therefore this control approach will "work") if and only if all zeros are in the left-half plane; Le., (11)is minimum phase. For minimum phase systems, the closed-loop response will be ISE-optimal in the limit as the roots of s' + filsr1+ ... + pr-ls + 0,tend to negative infinity (55). The result of proposition 2.6.1 as well as the results on internal stability will be generalized to nonlinear systems in part 2, section 4. 3. Several Basic Concepts from Differential Geometry The role of the present section is to provide the reader with the necessary background from differential geometry that will be used in the subsequent sections and in part 2. As pointed out in the Introduction, we will be using the classical (calculus) language throughout, sacrificing rigor and elegance for the sake of more effective communication of the concepts and results. The mathematically sophisticated reader is encouraged to refer to texts of differential geometry (13,16,54, 74, 75) for a broader picture of the subject, complete proofs of propositions and theorems, and

It is usually understood that the point of application of this vector is (xl, x,, ...,x,) instead of the origin. A vector field g(x) will be called a C1 vector field on W" if all the partial derivatives dgi/dxj exist and are continuous functions of (xl,..., x,). g ( x ) will be called a C" vector field on R"if all partial derivatives of arbitrary order exist and are continuous functions. A scalar field is completely characterized by its isoscalar surfaces. Given a scalar field h ( x ) and an arbitrary constant c, one can define a ( n - 1)-dimensional surface by the equation h(x)= c This will represent the set of points in W" for which the value of h is constant and equal to c; for this reason, it is called an isoscalar surface. An immediate consequence of the definition of isoscalar surfaces is that they can never cross each other. A scalar field is also completely characterized by its gradient, up to an arbitrary constant. The gradient dh(x) of a C scalar field h ( x ) is defined as the vector field whose components are the partial derivatives dhldx,, dh/dx2, ..., dhldx,. One can easily verify that the tangent plane to an isoscalar surface at the point no is described by the equation

(dh(x"), x - x o ) = 0 (25) Consequently, a vector applied at xo will be tangent to the isoscalar surface if and only if it is orthogonal to the gradient vector a t xo. Alternately stated, the gradient vector is always normal to the isoscalar surface that passes from that point. A vector field is completely characterized by its integral curves. These are defined as follows: Given a vector field g(x) and a point x, = (xlo,x,,, ..., x,,), let x i = @i(O; x,), i = 1, ..., n, be the solution of dxi/d0 = gi(x1, ..., x,)

i = 1, ..., n (26) For each 0 E W,(&(0; x,), &,(e; x,), ..., &(e; x,)) defines a point in R". Thus, x1 = @l(O; x,), xz = &(O; x,), ..., x , = xi(0) = xi,,


+,,(e; x,) defines a curve in Rn starting from the point x,; this curve is called the integral curve of the vector field g ( x ) passing from the point x,. An immediate consequence of the definition of the integral curves is that dx 1 gl(X1, .**, xn)

-

dx2

gz(x1, **., xn)

-- ... --

dxn gn(x1, ..., xn)

(27)

along the integral curves. This shows that the vector g(xl, ..., x,) is tangent to the integral curve at each point. Vector fields, integral curves, and isoscalar surfaces are key geometric notions in nonlinear analysis. A vector field generalizes the notion of a constant vector by allowing it to vary in magnitude and orientation as a function of x. The integral curves associated with a vector field generalize the notion of straight lines, which are associated with the orientation of a constant vector. The isoscalar surface h ( x ) = c generalizes the notion of a plane, which is defined by setting a linear function of x equal to a constant. 3.2. Linear Independence. Definition 3.2.1. T h e vector fields g l ( x ) ,...,g,(x) are said t o be linearly independent if for every x t h e vectors g l ( x ) ,...,g,(x) are linearly independent elements o f W". Definition 3.2.2. The C scalar fields t l ( x ) ,t p ( x ) ,..., t,(x) are said t o be linearly independent if their gradients d t l ( x ) ,d t 2 ( x ) ,..., dt,(x) are linearly independent vector fields. The first definition provides a natural extension of the concept of linear independence of constant vectors. The second definition needs a little further justification. Suppose that the scalar fields t l ( x ) ,..., t,(x) are related by (28) Q(t,b), t,(x)) = 0

(b) At each point, the vector g ( x ) is tangent to the isoscalar surface of w ( x ) passing from that point. (c) At each point, the integral curve of g ( x ) passing from that point lies entirely on an isoscalar surface of w ( x ) . A basic theorem in the theory of partial differential equations states that it is always possible to find n - 1 linearly independent solutions w = t l ( x ) , w = t z ( x ) ,..., w = tnTl(x)to the partial differential equation (30). In geometric terms, this result implies that (a) the span of the gradients d t l ( x ) ,dtg(x),..., dtn-,(x) will be the ( n - 1)-dimensional plane in Wn which is orthogonal to g(x); (b) there are n - 1independent families of surfaces that are tangent to g ( x ) at each point; and (c) there are n - 1 independent families of surfaces that are "composed of" integral curves of g ( x ) . From a practical viewpoint, the question is how to compute n - 1 independent solutions to the partial differential equation (30). The following proposition provides a standard algorithm for this purpose. Proposition 3.3.1. Given a vector field gl(X1,

g(X1,

Xn-11 x n )

*..)

=

.*.I

[

with gn(xl, ...,

x,) ~1

xn)

..., xn-l, x,)

gn-l(xl,

gn(X1,

Xn-1,

*../:

xn-1, Xn)

]

# 0, let

= k ( x n ; 41,

xn-1 = +n-l(Xn;

tn-1)

El,

***)

En-1)

(32)

be t h e solution of

a*.,

dx1 - gI(x1, _

where 9 is a nonconstant function. This would automatically imply that

a9 -(tl(X), m

*a*,

at

gl(X1, X Z ~ ...t xn)

g(X1, XZ,

..., x n )

=

gn(x1,

xn-1,

xn)

xn-1, xn)

5

x1(0) = t 1

tm(x)) dtm(x) = 0 (29)

i.e., that d t l ( x ) , ..., dt,(x) are linearly dependent vector fields. The above argument shows that, if the scalar fields t l ( x ) , .,., t,(x) are known to be linearly independent, they will be unrelated in the sense that no equation of the form (28) can hold. 3.3. Geometric Interpretation of the Linear Homogeneous First-OrderPartial Differential Equation. Given a vector field

[

dxn

.**I

g2(x1,

"'9

'''3

&(Xl,

xz,

..., x , )

xn)]

consider the problem of finding a scalar field w ( x l , ...,x , ) that satisfies the partial differential equation

T h e n , t h e equations in (32) are solvable f o r El, ..., En-1: E1

= tl(x1, *.*,

En-1

= tn-I(x1,

xn)

xn-1, xn)

Furthermore, the scalar fields t l ( x ) ,..., tn-l(x) are linearly independent and satisfy the partial differential equation (30).

Remark 3.3.1. The assumption gn(xl, ...,xn-l, x,) # 0 can always be met by appropriate rotation of indexes. Remark 3.3.2. Comparing (33) and (26), we see that XI

xn-1

Equation 30 can be written in a more compact way as (31) (dw(x),g ( x ) ) = 0 and has the following geometric interpretations: (a) At each point, the gradient of w ( x ) is orthogonal to gb).

xn-1,

= +l(xn;5 1 ,

= +n-l(xn;

..., tn-1)

El,

.*., En-1)

represent the integral curves of g ( x ) originating from the point (E1, ..., tn+0) of Wn. This shows that the solutions to the partial differential equation are indeed "composed of" integral curves of g ( x ) . 3.4. The Problem of Frobenius. As a direct generalization of the problem of the previous section, consider now a system of m linear homogeneous first-order partial differential equations. In particular, given m linearly in-

Ind. Eng. Chem. Res., Vol. 29, No. 12, 1990 2301 dependent vector fields ( m < n),

a=,

ax,

-(x) ax2

az, ... -b)

-(x) az2

-a:, (x) ax2

... axz axil

az,

-(x)

az

-ax (x)

=

ax,

ann

az,,

I

-(x)

(38)

...

-(x)

ax,

Lgmn(x1, ~ 2 *.*) ,

ax,

ax2

is invertible. In view of the definition of linearly independent scalar fields (definition 3.2.2), this translates into the following proposition. Proposition 3.5.1. The transformation (36)is invertible if and only if E l ( x ) , Zz(x), ..., Z n ( x ) are linearly independent. An invertible transformation x E W" [ = Z(x) E W" defines a curvilinear coordinate system. The new coordinates are F,, t2,..., .$, and the new basis vectors are given by

-

x.1)

it is desired to find a scalar field w ( x l , ...,x,) that satisfies

el = first column of

( $ ) I

E , = second column of

E , = nth column of

(E),

(!E)]

(39)

Transformation of a scalar field h ( x ) into a new coordinate system involves a simple substitution of the inverse transformation x = Z-'(t):

&E) or, in more compact notation, ( d u b ) , g j ( x ) ) = 0, j = 1, ..., m

(35)

In the previous section, we saw that, in the special case of m = 1, it is always possible to find n - 1 linearly independent solutions. In the general case, one would try to find n - m linearly independent solutions to (34) or (35). This is referred to as t h e problem of Frobenius. For m # 1, the problem of Frobenius is not always solvable (Le., there may be fewer than n - m linearly independent solutions). A necessary and sufficient condition for solvability will be given in subsection 3.10 using the concept of involutive vector fields. 3.5. Invertible Transformations and Curvilinear Coordinate Systems. In this section, we consider transformations from IR"to W" of the form F1

= Zl(x1, 3 ~ 2 ,

xn)

E2

=

Z 2 ( ~ 1 r~ 2 * *,* 7

xn)

En

=

Zn(X1t ~ 2 ,

*a*,

*..,x n )

(36)

where Zl(x), E&), ..., Z,(x) are C scalar fields on W". In a more compact notation, we write [ = Z(x)

(37)

The inverse function theorem states that such a transformation will be invertible if and only if the Jacobian matrix

=

[h(X)l,=,l([)

(40)

Transformation of a vector field g ( x ) into a new coordinate system is more tricky; its new components El((), g2(5), ...,g,@) will be along the new basis vectors e,, C2, ..., e,, instead of e,, e2, ..., e,. They will be given by

E l ( [ ) = [(dZl(x),g ( X ) ) l x = - l ( [ ) E2([) = [(dZz(x), g ( X ) ) l X = r l ( [ )

En(E)

= [(dEn(x),g(X))]x=z-l([)

(41)

In the subsequent sections and in part 2, we will be applying coordinate transformations on the states of SISO nonlinear dynamic systems. In particular, given a nonlinear system of the form (I), where f ( x ) and g ( x ) are vector fields on W"and h ( x ) is a scalar field on W",we will want to change coordinates to 5 = Z(x). Under such a transformation, (I) will become


Comparing (43) with (40) and (41), we see that transforming a dynamic system (I) in new coordinates means transforming the vector fields j ( x ) and g ( x ) and the scalar field h ( x ) in new coordinates. 3.6. "Straightening Out" a Vector Field. A classical problem of differential geometry is the following: Given a vector field g ( x ) , find a curvilinear coordinate system ( = E ( x ) so that the vector g ( x ) is aligned with e,, the nth basis vector of the curvilinear coordinate system:

&(5) gn-l(t)

Repeated use of the Lie derivative operator is possible. For example, one may wish to differentiate a scalar field h ( x )first in the direction of the vector field f ( x ) and then in the direction of the vector field g ( x ) :

c

. I

=0

Alternately stated, the composition of the Lie derivative operators L , and L f is a second-order linear partial differential operator given by

=0

g n ( 5 ) tf 0

(44)

If such a coordinate system can be found, the integral curves of g ( x ) in 5 coordinates will be straight lines; it is in this sense that g is "straightened out". In view of (41), conditions (44) translate into (Gi(x), g(x)) = 0

In the case of repeated use of the Lie derivative operator with respect to the same vector argument, the following notation is standard:

Lf = LfLf (dXn-l(x),g ( x ) ) = 0 (dZn(x), g ( x ) ) # 0

L? = LfLj (45)

Consequently, Xl(x), ..., En-l(x) must be n - 1 linearly independent solutions of the linear homogeneous partial differential equation (30) and & ( x ) must be any function that does not satisfy the partial differential equation (30). This means that the problem of straightening out a vector field reduces to the problem of finding n - 1 linearly independent solutions of (30). The latter problem is already solved in subsection 3.3. A natural generalization of the problem of straightening out one vector field is the problem of straightening out m vector fields at the same time ( m < n). Using a similar argument, it is possible to show that this reduces to the problem of Frobenius. 3.7. Lie Derivatives. Given a C scalar field h ( x )and a vector field j ( x ) , the Lie derivative of h ( x )with respect to j ( x ) is defined by (46)

La = L f L p

3.8. Lie Brackets. One interesting question is whether the composition of two different Lie derivative operators is a commutative operation or not. In other words, whether LfLg is the same operator as L,LP A straightforward calculation gives

This shows that in general L& # L&,, so the composition operation is not commutative. This is analogous to the matrix algebra result that matrix multiplication does not commute. What is remarkable is that the difference LfL, - L$,, is not a second-order operator as might be expected but is instead a first-order operator. In fact, if we define a new vector field [ j , g] by

In other words, the Lie derivative is simply the directional derivative of the function h(x)in the direction of the vector f(x).

The Lie derivative operator is a linear first-order partial differential operator defined by (47)

Some properties of the Lie derivative operator are listed here. Let f ( x ) and g ( x ) be vector fields and h(x)and t ( x ) be C ' scalar fields. Then, L f ( h + t ) = L& + Lft (48)

L f ( h t )= hL,t

+ tLfh

Lf+&! = Lf + L,

then its components will be given by

(49)

(50)

Lhf = hL, (51) The first two rules represent properties of any linear differential operator; the second pair of rules states the linearity of L f with respect to its vector argument. These rules are relatively simple to use in control applications.

(53)

Thus, we will be able to rewrite (55) as

or

Ind. Eng. Chem. Res., Vol. 29, No. 12, 1990 2303 LfLg - LgLf = L[f,g]

(57)

The quantity [f,g ] defined by (55) is referred to as the commutator or the Poisson bracket or the Lie bracket of f and g. Note carefully the sign convention since the literature is not always consistent. The Lie bracket has a number of algebraic properties that result in its extreme importance in control applications. If f ( x ) , g ( x h fib),fib),gl(x),and gz(x) are C' vector fields and c1 and c2 are real numbers, the following properties hold: 1. [f,gl = -[g, f l (antisymmetry) (58) 2. clgl + Czgzl = cdf, g1l + c 2 [ f , g2l [fl

[ c J l + cd2, gl = cl[fl, gl 3.

[[ti,f 2 1 7 f31 + ' [ [ f 2 ,

+ c2[f2,gl

f 3 1 , f11

(linearity)

Le., the rows of the observability matrix of the linear system. Calculating the iterated brackets o f f and g (cf. (62)), we find

a d ] g ( x ) = -Ab ad:g(x) = A%

(59)

+ [ [ f 3 , f i l , f21

=0 (Jacobi's identity) (60)

Furthermore, if h(x) and t ( x ) are C' scalar fields, [hf, &I = ht[f, gl + h(Lft)g - t(L,h)f (61) In this paper we will be frequently using iterated Lie brackets. A standard recursive notation is as follows:

ad?&?= g

a d f g ( x ) = (-l)kAkb Thus, the vector fields ( g ( x ) , -ad:g(x), a d i g ( x ) , ..., ( - l ) n - ' a d ~ - ' g ( x ) become ) (b, Ab, A2b, ..., A"-'b), Le., the columns of the controllability matrix of the linear system. Finally, observe that L,h(x) = cb L,Lfh(x) = cAb

adtg = If, gl

[f, [f, gll a& = [f, [f, [f,glll a&

Thus, the gradients of

=

L&-'h(x)

adkg = [f, ad)-'g] (62) Finally, we should mention a very important consequence of (57) or of LgLf = LfLg - L[f,g] By applying the operator Lf from the right, we obtain L,L? = LfLgLf - L[f,g]Lf = LfGfLg - L[f,g])- L[f,g]Lf = LBL, - LfL[f,g]- L[f,g]Lf = L%L,- LfL,,,, - (LfL,,,, - L[f,[f,gll) (63) = L!L, - 2LfL[f,g,+ L[f,[f,g]] Repeating the same argument, we obtain by induction the higher order operator identity

= cAk-'b

Thus, the Lie derivatives of h(x),L&), ..., L!h(x), ..., with respect to g become the Markov parameters cb, cAb, ..., cAkb, ..., of the linear system. The conclusion that we draw from the above simple calculations is that Lie derivative and Lie bracket operations do arise in linear systems. We don't need to view them as such because matrix algebra is sufficient to derive and express all results. In nonlinear systems, Lie algebra will replace matrix algebra; Lie derivative and Lie bracket operations will play the same key role as matrix operations in linear systems. 3.10. Involutive Vector Fields and the Theorem of Frobenius. Consider the system of partial differential equations of subsection 3.4, and let us now write it in Lie derivative notation Lg,w = 0 Lgzw= 0 L,,w = 0

The above formula is sometimes referred to as the Leibnitz formula; it will be very useful in the subsequent sections and in part 2. 3.9. Significance of Lie Derivative and Lie Bracket Operations in Linear Systems. Consider the SISO linear dynamic system (II), which is a special case of (I) for f ( x ) = Ax, g ( x ) = b, and h(x) = cx. A straightforward calculation gives L&(x) = cAx

L!h(x) = cA2x Lgh(x) = cAkx

(66)

Applying the operator L,, on the first equation and the operator L,, on the second equation and subtracting by sides, we find (Lg1Lgz - LgzLgJw= 0 and therefore, from ( 5 7 ) , L[Pl&lW = 0 Repeating the same argument with all other pairs of equations, we conclude that the solution(s) w will have to satisfy (67)

(65)


in addition to (66). In other words, the isoscalar surfaces of w will have to be tangent not only to gl(x), ..., g,(x) but also to all combinations of Lie brackets [gi,g,](x). This suggests that the system of partial differential equations (66) may have fewer than n - m linearly independent solutions unless all the vectors [g,,g,](x)are linear combinations of gl(x),...,g,(x) for every x . The foregoing considerations motivate the concept of involutivity and the theorem of Frobenius. Definition 3.10.1. A set of linearly independent vector fields gl(x), ...,g,(x) is said to be involutive if there exist scalar functions CY,,~(X)such that

Using the Leibnitz formula (a), one can show by induction the following proposition. Proposition 4.1.1. For a system of t h e form ( I ) ,r is t h e smallest integer for which

L a d p g h ( x )Z 0 A characterization of relative order in terms of time derivatives of the output y can be obtained from the following proposition. Proposition 4.1.2. Consider a nonlinear system of the form ( I ) whose relative order is r. T h e n ,

m

[g,, g,l(x) =

Cayk(x)gdx)

for every i , j

k=l

(68)

Theorem 3.10.1 (Frobenius' Theorem). Given t h e linearly independent vector fields gl(x), ..., g,(x), t h e system of partial differential equations (66)admits n m linearly independent solutions if and only if g l ( x ) ,..., g,(x) f o r m a n inuolutiue set. The theorem of Frobenius is one of the most important theorems of differential geometry. Although the key idea is the one presented previously, a rigorous proof is more involved and the reader is referred to texts like refs 13,74, and 75. Remark 3.10.1. Involutivity is automatic in the case of m = 1. For this reason, when we have only one partial differential equation, it always has n - 1 linearly independent solutions (see subsection 3.3). Remark 3.10.2. Involutivity is also automatic when the vector fields gl(x),... g,(x) are constant since in this case all Lie brackets [g,,g ] ( x ) are identically zero. Remark 3.10.3. "he system of partial differential equations (66) can never have more than n - m linearly independent solutions. If it did, then their gradients and the m linearly independent vector fields g l ( x ) , ..., g,(x) would form more then n linearly independent vectors in IR". Remark 3.10.4. When g l ( x ) ,...,g,(x) form an involutive set of linearly independent vector fields, the n - m linearly independent solutions will be "composed OFintegral curves of gl(x),...,g , ( x ) . The calculation, however, is quite involved when m > 1. 4. Nonlinear Inversion. Zero Dynamics 4.1. Relative Order and Inversion of Nonlinear Systems. Consider the nonlinear system (I). It is assumed that f ( x ) and g ( x ) are C " vector fields on R"and h ( x ) is a C scalar field on R". Since such a system does not, in general, possess a transfer function, it is not possible to define a relative order for (I) in terms of the degrees of denominator and numerator polynomials like in linear systems. Hirschorn (36)suggested the following definition of relative order. Definition 4.1.1. T h e relative order of a system of the form ( I ) is t h e smallest integer r for which

L,L;-'h(x)

f

0

The above definition is consistent with the relative order concept for linear systems. As we saw in subsection 3.9, the quantities L&-'h(x), k = 1, 2, ..., become cAk-'b, k = 1, 2, ..., in the linear case. Thus, definition 4.1.1 can be viewed as a generalization of the statement of proposition 2.1.1. A nonlinear system (I) will have relative order r = 1 if L,h(x) # 0, r = 2 if L,h(x) = 0 but L L&) # 0 , r = 3 if L,h(x) = L,L&(x) = 0 but L,L:h(x? # 0, and so on.

dr-'y

- = LF-'h(x) dtr-l

+ (69) dt' In other words, r is the smallest order of derivatives of y that depends explicitly on u. The result of proposition 4.1.2 is a direct generalization of the one of proposition 2.1.2 for linear systems. The last important property of the relative order is that r 5 n. This was obvious in linear systems. For nonlinear systems, this is a consequence of the following proposition. Proposition 4.1.3. T h e scalar fields h ( x ) ,L,h(x), ..., L;-'h(x) are linearly independent. Since the above r scalar fields are linearly independent and R" cannot have more then n linearly independent elements, we will necessarily have r I n. Using the concept of the relative order, Hirschorn (36) was able to construct a state-space realization of the process inverse in a manner that is completely analogous to the linear case. His major result is summarized in the following theorem. Theorem 4.1.1 (Hirschorn Inversion). Consider a dynamic system o f the form ( I ) whose relative order is r. T h e n t h e inverse o f ( I ) can be calculated via _ - L $ h ( x ) L&lh(x)u dry

dry

- - LFh(2) u=

dt' L&;i-'h(Z)

Compared with the linear result (proposition 2.1.3), the analogy is evident. Consequently, one would expect that the realization (70) is not of minimal order. It will be seen in the next section that this is indeed the case. Example: Penicillin Fermentation. An important bioengineering process is the fermentation of penicillin. The dynamics of a continuous glucose-fed system can be modeled as (Bajpai and Reup (9)) dX/dt = p(S, X ) X - D X d P / d t = T ( S ) X- KP - DP dS/dt = -u(S, X ) X

+ D(SF - S )

where X is the cell mass concentration, P is the penicillin concentration, SFis the glucose feed concentration, and D is the dilution rate of glucose added to the fermentor. The parameters p(S, X ) , a(S), and u(S, X ) are the specific

Ind. Eng. Chem. Res., Vol. 29, No. 12, 1990 2305 growth rate, specifice product formation rate, and specific substrate utilization rates, respectively, and are given by the empirical expressions

P

C

c

Figure 2. Output-error feedback control loop. PPS

n(S) =

Kp + S(1+ S / K * )

1 a(S, X ) = - p ( S ,

X )

1 + -n(S) +m

yxs

YPS

The parameter K is the degradation constant of penicillin. The dilution rate D is the most convenient manipulated input for the system. Out of X , P, and S , none can be easily and accurately measured on-line. However, the cell mass concentration X can be very accurately estimated on-line through COz and O2 measurements and atom balances. So, X is a natural output for the system. With u = D as input and y = X as output, the input/ output relationship in a continuous penicillin fermentor is described by

we will outline an algorithm for constructing a coordinate transformation that transforms an arbitrary system of the form (I) with relative order r into (73). A t first, one must observe that, since the scalar fields h ( x ) , L f h ( x ) , ..., LF2h(x) are linearly independent (see proposition 4.1.3), the ( r - 1) X n matrix

aLp

ax,

... aL+

ax,

has rank r - 1. Therefore, we can always rotate the indexes of the state variables so that y

=x

(71)

Let's compute now the Hirschorn inverse of (71). Since L & X , S ) = -X # 0, it follows that the relative order of (71) is r = 1. Thus, the Hirschorn inverse can be calculated via

u=

dy/dt - L & ( Z ) Lgh(Z)

Substituting the particular f , g, and h, we obtain the following inverse of (71):

u =

dyldt - ~ ( G z 1 ) Z i

(72)

-21

rL

d e t r aLyh n;*'

ax,_,,

... .,.

aLyh # 0

(74)

ax,_,

Also, since g ( x ) # 0, we can rotate indexes to assure that gn(x) + 0 (75) Theorem 4.2.1 (Byrnes-Isidori), Assume that the indexes of x i in ( I )have been rotated so that (74) and (75) are satisfied and let t l ( x ) , ..., t,-,(x) be linearly independent solutions of the partial differential equation (dw(x),g) = 0 which have been calculated with t h e algorithm o f proposition 3.3.1. T h e n , the transformation

(76)

4.2. The Byrnes-Isidori Normal Form. The dynamic system 1, = F , ( x )

xn-l - x, in =Nx

) + G(x)u

(73) where F l ( x ) ,...,Fn-r(x),+(XI, and G ( x ) are all scalar fields on W",with G ( x ) # 0, is a normal form for systems of relative order r. In fact, the reader can easily verify that the relative order of (73) is indeed r. This normal form is due to Byrnes and Isidori (17). In the following lines, Y = Xn-r+1

The Byrnes-Isidori normal form clarifies the concept of relative order: r is the number of integrations (state equations) that the input has to go through in order to affect the output. (See also proposition 4.1.2.) The normal form is also useful in obtaining a minimal order realization of the inverse of (I). Since the input/ output behavior of a dynamic system is coordinate-independent, one can first transform the system in normal form (73) and then apply Theorem 4.1.1 to calculate the inverse.


which is in Byrnes-Isidori normal form. Note that (82) can be somewhat simplified if we further transform the first coordinate:

This leads to 2, = F l ( Z ) zn-r

= Fn-r(Z)

This leads to

Zn-r+l = Z n - r + ~

61'

zn-l= 2,

= -EI'P(SF 42

+ [1'[2,

t2)

= 52P(sF +

tl'E2, E 2 )

Y =

dry dt

2,= 7

- u(SF -

+ El'l2,

52)

[2u

(84)

E2

still in the Byrnes-Isidori normal form. Finally, one can use (80) to compute a minimal-order realization of the inverse:

Equivalently, we can write the inverse as 21,

...)z n - r , Y,

dy y , -,

Z1, ..., Z,-,

u =

-1

(Compare with the Hirschorn inverse, eq 72.) 4.3. Zero Dynamics. The zeros of a linear system are the poles of its inverse. In other words, the zeros of a linear system are completely determined by the dynamics of its inverse. A nonlinear system does not have a transfer function and therefore cannot have zeros as a set of numbers that are roots of the numerator polynomial. However, a nonlinear system does have an inverse, and the inverse does have dynamics! These considerations motivate defining the notion of zero dynamics of a nonlinear system as the dynamics of a minimal-order realization of its inverse. From (801, we see that these are governed by

dy dr-ly z, ..*, dtr-l

dt

d'-'y dtr-1

..., -

\ I

The system (80) is a minimal order realization of the inverse. As an illustration of the Byrnes-Isidori normal form, consider again the penicillin fermentor example of subsection 4.1. This system has relative order r = 1 and is not in normal form. To transform (71) in the ByrnesIsidori normal form, we need to change coordinates as suggested by Theorem 4.2.1. Since the state X is the output, we want to keep it in the new coordinate system. The other coordinate will be a solution of the partial differential equation

Following the algorithm of proposition 3.3.1, we find that

is a particular solution. Thus, the coordinate transformation SFX (1

=

E2 = x (81) will transform (71) into the Byrnes-Isidori normal form. Indeed, a straightforward calculation gives

Y =

t2

(82)

The output y and its derivatives up to order r - 1 are the inputs of the inverse, whereas Z1, ..., Zn-r are the states of the inverse. Definition 4.3.1. Consider a nonlinear system of the form ( I ) that has been transformed to the Byrnes-Isidori normal form (73) by an appropriate coordinate transformation. Then, the dynamic system 2,= Fl(Z1, ..., Z,-, U1, ..., Ur)

...) ur) is called the forced zero dynamics of ( I ) . zn-r

= Fn-r(Z1,

..., z n - r ,

u1,

(86)

The above definition considers the dynamics of the inverse as a system with inputs. As we saw in subsection 2.3 in the discussion of linear systems, this contains more information than the system zeros; it is the unforced inverse that contains exactly the same information as the system zeros. For this reason, it seems more justified to define the zero dynamics as the dynamics of the unforced inverse, although there are situations where the effect of its inputs is important. Definition 4.3.2 (Byrnes-Isidori). Consider a nonlinear system of the form (I)that has been transformed to the Byrnes-Isidori normal form (73). Assume that the origin is the reference steady state of (73). Then, the dynamic system


2, = Fl(Z1, ..., Z,-,

0, ..., 0)

Zn-? = Pn-#l, ..., z,-,, 0, ..., 0)

(87)

is called the unforced zero dynamics or simply the zero dynamics of ( I ) . The concept of zero dynamics allows a classification of nonlinear systems into minimum phase and nonminimum phase. Of course, the precise classification will depend on the concept of stability used for (86) or (87). Byrnes and Isidori suggested the following definition. Definition 4.3.3. A nonlinear system is called minimum phase if its zero dynamics (87) is asymptotically stable around the origin. Otherwise, it is called nonminimum-phase. Depending on the context, more or less stringent stability requirements may be necessary. 5. The Idea of Feedback Linearization 5.1. Approximate versus Exact Linearization. Consider the nonlinear system (I). The conventional concept of “linearization of a nonlinear system” involves approximating the nonlinear system by a linear one in the vicinity of a reference steady state. In particular, (I) can be approximated by d -(x - x,) =

Control is done by manipulating a valve located in the exit stream. A model for the liquid level is then k -dh-- Fin- -uh1/2 dt A where h is the liquid height, A is the cross-sectional area, and k is a valve constant. The control variable u corresponds to the valve stem position. A static control law can be constructed that linearizes the response of h to an external input u. That is, we can define a rule for computing u that brings (91) to the form

dh/dt = Fin- a h

+u

(92)

where l / a is a chosen time constant. Comparison of (91) and (92) yields the necessary control law (93)

5.3. Temperature Control of a Nonisothermal CSTR. Consider the problem of synthesizing a controller to regulate the temperature T of a nonisothermal CSTR where an exothermic irreversible first-order reaction takes place. The inlet stream’s temperature is assumed to be unregulated. Control is done by manipulating the heat u removed by a cooling jacket. A model for the nonisothermal CSTR is given by

dt

(94)

dt

in the vicinity of the steady state (us,x s , y e ) . In nonlinear control theory, “feedback linearization” has an entirely different meaning. It consists of finding a state feedback control law u = P(X) + q ( x ) u

(89)

where p ( x ) and q ( x ) are algebraic functions of the state variables and u is an external reference input, such that the closed-loop system dx/dt = [ f b+ ) p ( x ) g ( x ) l+ [ q ( x ) g ( x ) l u Y = h(x)

(90)

is exactly linear. Figure 1 provides a pictorial representation of the state feedback loop. In the next two subsections, we will give two examples of feedback linearization. The first example will be a liquid level system with a square-root nonlinearity; the second will be a CSTR with an Arrhenius-type nonlinearity. It will be seen in both cases that the linearizing feedback will contain the same type of nonlinearity as the process. This idea of building a nonlinear controller that contains a replica of the process nonlinearity is central in previous work on nonlinear control, which was based on intuition or experience; for example in Shinskey (72, 73) and in the recently developed “extensive variable control” approach of Georgakis (32,331. Note, however, that these empirical approaches do not, in general, provide closed-loop linearity in any sense. 5.2. A Simple Level Control Example. Consider the problem of synthesizing a controller that is to regulate the level of a fluid in a tank of constant cross-sectional area. The inlet flow to the tank is assumed to be unregulated.

= 9 ( T i n- T )

V

+ (-AH)koe-E/(Rnc- u

(95)

where c is the concentration of the reactant, koe-EJ(Rnis the rate constant, (-AH)is the heat of reaction, F is the flow rate of inlet and outlet streams, V is the reactor volume, and p and cp are the density and heat capacity of the fluid, respectively. A static control law can be computed that linearizes the response of the output T to an external input u. More precisely, we can request that T depends on u according to dT F = -Tin - a T + u (96) dt V Comparison of (95) and (96) yields the necessary control law:

5.4. Different Concepts of Exact Linearization. From a theoretical standpoint, there are two major concepts of feedback linearization: the Su-Hunt-Meyer linearization and the i n p u t l o u t p u t linearization. The former refers to making the closed-loop state equations linear; the latter refers to making the inputloutput behavior of t h e closed-loop system linear. In particular, for a nonlinear system of the form (I),the following linearization problems can be defined. A. Su-Hunt-Meyer Linearization Problem. Find a static state feedback control law u = P(X) + d x ) u

so that the closed-loop state equations = [fb) + g ( x ) p ( x ) l+ g ( x ) q ( x ) u are linear or can be transformed into linear equations with an appropriate coordinate change. Moreover, one would


want the control law to allow assignment of the closed-loop poles a t prespecified locations. This problem will be solved in part 2 , section 3. B. Input/Output Linearization Problem. Find a static state feedback control law u = p ( x )+ q ( x b

so that the closed-loop system

Y = h(x1 has linear input/ output behavior. Moreover, one would want the control law to allow assignment of a prespecified transfer function for the input/output linear closed-loop system. This problem will be solved in part 2 , section 4. There is a limited class of systems where one can obtain closed-loop linearity both in the Su-Hunt-Meyer sense and in the input/output sense. This is referred to as full or total linearization because both the state equations and the output map in the closed-loop system are linear. Conditions under which total linearization can be achieved will be given in part 2, section 5. Examining the previous examples, we see that in the liquid level system total linearity was obtained (one-dimensional system!), whereas in the CSTR only input/ output linearity was obtained (the closed-loop state equations were nonlinear). A natural question to ask is why should a system be feedback linearized. There are two possible answers. 1. Once the system becomes linear under the state feedback, we know how to control it. We can design a linear control loop around it. 2. The basic philosophy in synthesizing linear controllers is requesting desirable closed-loop dynamics. The same philosophy must be used in synthesizing nonlinear controllers for nonlinear systems. But it is only linear dynamics that we clearly understand and for which we can easily express preformance specifications. Thus, requesting closed-loop linearity is a natural first choice in an attempt to develop a nonlinear controller synthesis methodology. According to the first answer, the attempt is to make the nonlinear control problem reduce to a linear control problem. According to the second answer, the attempt is to extend linear controller synthesis theory into a nonlinear controller synthesis theory. The first interpretation of feedback linearization is meaningful only if input/output linearity is achieved; this was the case in both examples. In this work, both interpretations of feedback linearization will be addressed. In sections 3 and 4 of part 2, the solutions of the Su-HuntMeyer and input/output linearization problems will be interpreted as extensions of the linear results presented in subsections 2.5 and 2.6. In subsection 6.2 of part 2, linearization will be viewed as a means of reducing the nonlinear control problem to a linear one; an external control loop with integral action will be placed around the linear u - y system for offsetless control to set point. 5.5. A "Macroscopic"Linearization Problem. So far, we have taken the state-space perspective in terms of formulating the feedback linearization concept. An alternative would be the input/output perspective, in which both the process and the controller can be viewed as abstract nonlinear operators relating their input and their output. Such a point of view is macroscopic in the sense that the internal state-space structure is not considered at all.

Figure 2 depicts an output-error feedback control system, where P and C represent the process and controller operators, respectively. A formal manipulation of nonlinear operators shows that the closed-loop operator is PC(Z + PC)-' where I is the identity operator. Thus, one can pose a "macroscopic" linearization problem as follows. Given a nonlinear operator P, find a nonlinear operator C so that the closed-loop operator PC(Z + PC 1-l is linear. In subsection 6.1 of part 2, a solution to the above linearization problem will be provided through a nonlinear analogue of Q parametrization. This will automatically lead to the nonlinear IMC structure.

Acknowledgment We are grateful to the reviewers Frank Doyle, Manfred Morari, Coleman Brosilow, Jean-Paul Calvet, and Yaman Arkun for their helpful comments and suggestions, which have "shaped" the structure and content of the revised version. We are also indebted to Prodromos Daoutidis for numerous suggestions that resulted in major improvements in the technical content of the revised version.

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Received for review July 14, 1988 Accepted November 7, 1989

Geometric Methods for Nonlinear Process Control. 2. Controller Synthesis Costas K r a v a r i s * Department of Chemical Engineering, T h e University of Michigan, Ann Arbor, Michigan 48109-2136

J e f f r e y C. K a n t o r Department of Chemical Engineering, University of Notre Dame, Notre Dame, Indiana 46556

This is the second part of a review paper for geometric methods in nonlinear process control. I t focuses on exact linearization methods including Su-Hunt-Meyer, input/output, and full linearization. T h e internal model control (IMC) and globally linearizing control (GLC) structures are reviewed and interpreted in the context of input/output linearization. Further topics of current research interest are also identified. 1. Introduction

In the present second part of our review paper, we will discuss controller synthesis methods for SISO nonlinear systems of the form 1 = f(x) + g(x)u Y = h(x)

(1) where x is the vector of states, u is the manipulated input, y is the output, f ( x ) and g ( x ) are vector fields on W",and h(x) is a scalar field on W". In the first part, we have already provided a brief review of results for the linear case 1 = Ax bu y = cx

+

where A , b, and c are n X n, n X 1, and 1 X n matrices, respectively; these results will now be generalized to nonlinear systems of the form (I). In the first part, we have also set up the machinery of Lie derivatives, Lie brackets, and coordinate transformations, which will be instrumental in the derivation of control laws for nonlinear systems. Finally, the first part has introduced the concepts of relative order and zero dynamics of nonlinear systems; these will be essential in both deriving and interpreting controller synthesis methodologies for nonlinear systems. The main mission of the present part is to pose and solve the feedback linearization problems introduced in the first part and to show their application in controlling nonlinear processes. Research directions and open problems in the area of nonlinear process control will also be identified at the end. The next section will provide a review of basic properties of static state feedback. Sections 3 and 4 will present a theoretical overview of the Su-Hunt-Meyer and input/ output linearization problems and illustrate their solutions in a chemical reactor example. Section 5 will address the full linearization problem and its applications and discuss the advantages, disadvantages, and limitations of each

* To whom all correspondence should be addressed. 0888-5885/90/2629-2310$02.50/0

linearization approach. Section 6 will deal with control structures for minimum-phase nonlinear processes. The IMC structure will be reviewed first and will be interpreted as providing input/output linearization in a macroscopic sense. It will be followed by the GLC structure, which is directly based on the state feedback theory of the previous sections. Finally, section 7 will briefly address further topics in the area that correspond to either current research areas or major unsolved problems. 2. Basic Properties of Static State Feedback In this section, we will establish some basic properties of static state feedback, which are completely analogous to the properties of linear static state feedback for linear systems (see subsection 2.4 of part 1). Consider a nonlinear system of the form (I) subject to static state feedback u = P(X) + d X ) U

(1)

where u is an external reference input, p ( x ) and q(x) are scalar algebraic functions of the state vector, and q ( x ) # 0. The resulting closed-loop system is then described by =

[fb) + g(x)p(x)I + [g(x)q(x)lu Y = h(x)

(2)

In the present section, we discuss some fundamental properties of state feedback of the form (1). Comparing the open-loop and the closed-loop systems ((I) and (2), respectively), we immediately observe that they have the same structure: their right-hand sides are nonlinear in x but linear in the input. (In more precise mathematical terms, one should say that the right-hand side is affine in the input.) The reason that the structure of the equations is preserved is that the state feedback (I) is linear in the external input u. This seemingly trivial observation has considerable importance in the theoretical developments that will follow. For this reason, we state it as a proposition: Proposition 2.1. A static state feedback, which is linear in the external input, preserves the linearity of the 0 1990 American Chemical Society

Geometric methods for nonlinear process control. 1. Background

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