Robust nonlinear control of nonsquare multivarlable systems

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ROBUST NONLINEAR CONTROL OF NONSQUARE MULTIVARlABLE SYSTEMS Juan C. Cockburn· Srinivas Palanki **,1 Soumitri Kolavennu"·

• Dept. of Electrical Engineering,Florida State University, Tallahassee, FL 32310, USA ** Dept. of Chemical Engineering,Florida State University, Tallahassee, FL 32310, USA ... Honeywell Laboratories, 3660 Technology Drive, Minneapolis, MN 55418, USA

Abstract: In this paper, a nonlinear controller is designed in the Input/Output (I/O) linearization framework, for non-square multivariable nonlinear systems that are subject to parametric uncertainty. A non linear state feedback is synthesized that approximately linearizes the system in an I/O sense by solving a convex optimization problem online. A robust controller is designed for the linear uncertain subsystem using a multi-objective H2/ Hoc synthesis approach to ensure robust stability and performance of nonsquare multivariable, nonlinear systems. This methodology is illustrated via simulation of a regulation problem in a continuous stirred tank reactor. Keywords: Nonsquare Multivariable systems, Robust control, I/O linearization.

1. INTRODUCTION In the last two decades, there has been a significant effort in the development of the theoretical foundations of the differential geometric approach to design nonlinear controllers for multivariable nonlinear systems (Isidori, 1989). To make these nonlinear controller design procedures practical, it is necessary to consider the effect of uncertainty on the controller performance. The issue of robust controller design in the Input/Output (I/O) linearization framework for nonlinear systems has attracted attention recently. However results are available primarily for 8180 systems (see, for instance, Kravaris and Palanki (1988) ; Christofides et al. (1996) ; Kolavennu et al. (2000)) and square MIMO systems (see, for instance, Christofides et al. (1996) ; Kolavennu et al. (2001b)) ; the controller design issues for nonsquare multivariable systems (systems where the number of inputs is 1

not equal to the number of outputs) in the face of parametric uncertainty are not well understood. In this paper, we extend the multi-model approach of Kolavennu et al. (2001b) to nonsquare systems. Nonsquare systems occur frequently in the chemical process industry. However, for controller design purposes, they are often "squared" by adding or deleting the appropriate number of inputs or outputs from the system matrix (Reeves and Arkun, 1989). Once such systems are squared, then controller synthesis procedures developed for square systems (where the number of inputs and outputs is equal) are applied directly. However, it has been shown that there can be advantages in synthesizing a controller for the original nonsquare system. For instance, Morari et al. (1985) compared square and non-square structures of a reactor application study and concluded that for their system the non-square structure was less sensitive to modeling errors due to a smaller condition number. Despite such evidence, the literature

Correspondence: [email protected] .edu

77

full row rank, then there exists a diffeomorphism (1/,€) == T(x) given by

on controller design is sparse. Most of the available results in the literature of non-square systems are for linear systems (Reeves and Arkun, 1989). In the nonlinear systems area, Doyle et al. (1992) investigated the design ofl/O linearizing controllers for nonminimum phase nonlinear systems with two inputs and one output. The first input was used to achieve I/O linearization while the second input was used to stabilize the otherwise unstable zero dynamics. For non-square systems with more inputs than outputs and equal relative degree with respect to all inputs McLain et al. (1996) derived an analytical expression, where some inputs were used for I/O linearization and the remaining inputs were used to minimize input cost.

Z(j) 1

h;{x) ] Lfh;(X)

z(j) 2

z(j)==

==

z(j)

L,/-l hj (x)

rj

1/==[~][t:~~~~] 1/n:-r

In this paper, we propose a two step procedure that can be applied directly to nonsquare uncertain systems which I/O linearizable and mimimum phase. The first step involves the design of a state feedback that approximately linearizes the system in an I/O sense by solving a convex optimization problem online. The second step involves the design of a robust controller for the uncertain linear subsystem based on a multi-objective H2/ Hoo synthesis approach.

(3)

(4)


where
d
(5)

i=l ·(j) -

Z1

-

z(j) 2

(6) 2. PROBLEM FORMULATION Consider the following state-space model of a multi-input multi-output (MIMO) non linear system with parametric uncertainty

m

Z~;)

==

f(x, B)

+ Z)i(X, B)Ui i=l

y

==

j

==

1, .. . ,p

where D:j == L,/ hj(x) and ,Bji is the (j, i)th entry in the characteristic matrix.

(1)

h(x)

Equation (6) represents p subsystems, which form the linearizable part of (1).

where x E Rn is the vector of states, U E Rm is the vector of manipulated inputs, y E RP the vector of measured outputs, and B is a vector of uncertain parameters that takes values in a compact set 8 C RP. It is assumed that m ~ p since the case p > m (more outputs than inputs) leads to an uncontrollable situation. The cost of the inputs is represented in the following way

J

,Bji(€, 1/)Ui

i=l

m

:i;

== D:j(€, 1/) + L

Once the system has been transformed to the above normal form a feedback law can be designed to cancel the nonlinearities that appear in the equations

Zr

== D:(z,1/) + ,B(z,1/)u

(7)

where

== wtut + w~u~ + ... + w;,u;, == IIWull~ (2)

Z~~)] . (1)

where Wi is the cost of Ui and W is a diagonal matrix with Wi as its elements. For all B E 8 we assume that f and gi are sufficiently smooth vector fields on Rn, and hj are smooth real scalar valued functions. The objective is to design a controller utilizing all available inputs such that the total input cost is minimized and the closed loop system is stable and certain performance objectives, e.g., tracking, disturbance rejection, etc., are satisfied for all B E 8 .

zrl

[

.

. (p)

zrp

L'/h 1(X)] L?h 2 (x)

.

,

D:

==

[

. L,/hp(x)

j3 is the characteristic matrix and vector.

U

is the input

Remark 1: Consider an input-output linearizable, minimum phase MIMO nonlinear system, with relative degree T, that has a well defined normal form for all B E 8. There exists a diffeomorphism (1/,z) == T(x,B) which transforms system (1) into its normal form (eq (6) and (7)). This

If a system represented by (1) has well defined relative degree Tj for all outputs Yj with T == L Ti (T :::; n) and the characteristic matrix is

78

diffeomorphism results in a linear subsystem if the value of 9 is exactly known. However, since 9 is uncertain, the diffeomorphism has to be based on some nominal value 90 of 9. This results in inexact linearization which also leads to loss of decoupling. This could lead to performance degradation if a conventional I/O design is used.

Proof. For the system (8) with additive model for uncertainty, the nominal transformation (TJ, z) = T(x, ( 0 ) is given by

To overcome this loss of stability and/or performance the dynamics of the system obtained by using a transformation based on the nominal model are studied and a robust controller design methodology for this system is derived in the next section. The following assumptions are made: (1) Model uncertainty is assumed to be represented as uncertainty in the parameter vector 9. (2) The system represented by eq. (1) is an inputoutput linearizable, minimum phase MIMO nonlinear system, with finite relative degree r, that has a well defined normal form for all 9 E e. (3) The relative degree with respect to each output and stability properties of the original nonlinear system are not changed due to uncertainty. (4) The characteristic matrix f3 has full row rank.

In the new coordinates, system (1) can be written as

=
TJi

z~j)

= L~:lhj(x), 1 ~ i

z(j) t

= 9io(X) + 09i(X , 9)

= z(j) + ~(j)(TJ t.+1 t

z 9)

" ,

1 <_ i <_

r -

1 (13)

(j)

+ [f3oj (x) + D.f31 (TJ, x, 9)]u,

1~j ~P

where

= LofcPi ~I'/i (TJ, Z, 9)(1 xm) = LogcPi ~I'/i (TJ, z, 9)

A (j)( L..>.i TJ,

Z, 9)

(14) (15)

Li - 1 h

= LOf fo

(16)

j

~(j)(TJ z 9)=L'fU-1h , r " 0 fo j

~¥i(TJ,X,9)(1xm) = LogL/:lhj Ooj(TJ, z) f3oj(TJ,Z)(lxm)

(17) (18)

= LZhj

(19) r ' -1

= L gjoLIo

hj

(20)

f30j is the jth row of the characteristic matrix. The inner-loop controller is chosen to cancel the nominal nonlinearities as u(x)

= f3!(x)[-oo(x) + v]

(21)

This renders (13) equal to

= z(j) + ~ ", .+1

1< i < r - 1 -z(j)=D.(j)-D.(j)o +v , +(~(j))v r r f3 0 J f3'

z(j) •

(8) i = 1 ... m

(12) 1~i ~n - r

z~j) =ooj(x) +D.~j)(TJ , z,9)

Lemma 1. The system (1) with an additive model for uncertainty, i. e. of the form,

9i(X, 9)

~ rj, 1 ~ j ~ p (11)

+qi(TJ, z)u + Lil'/i (TJ, Z, 9)u,

In this section a methodology for the robust controller design is outlined. First it is shown how the system (1) is transformed by diffeomorphism based on nominal parameters using lemma 1. The uncertain transformed system is characterized in a convenient, approximate linear form using lemma 2. It is shown via Theorem 1 that this characterization provides a framework for robust controller design using multi-objective H2/ Hoo synthesis. Finally, it is shown via Theorem 2 that this controller stabilizes the original nonlinear system.

= fo(x) + of (x, 9)

(10)

rii = Pi(TJ, z) + ~I'/i (TJ , Z, 9)

3. ROBUST CONTROLLER DESIGN

f(x,9)

~n - r

where D.f3

(9)

= f3!~f31 '

(22) 1_
This is same as (10).

<>

1 _< J' <_ P

Remark 2: The calculation of the weighted right inverse, f3t (x) is equivalent to solving a pointwise convex optimization problem that minimizes the input cost represented by eq. (2) as shown in Kolavennu et aI, (2001a) . Essentially the m inputs represented by u are being utilized in an "optimal blend" to generate P new inputs represented by v.

,dj) = L of L fi -o1 hj (9)) A (j) h were L..>.i x, an d L..>.f3 = f3!L ogj LZ -1 hj (x , 9) . The subscript '0' refers to the the system at 9 = 90 , The vector v = [V1, ... , refers to external control inputs to the system after state transformation and f3t (x) is the weighted right inverse of the (p x m) matrix f30 at x .

We characterize the uncertainty in a suitable manner to design a outer loop controller. InputOutput linearization uses coordinate transformation and state feedback to reduce the nonlinear system to a linear one. However, in the presence of uncertainties, this method does not give a perfectly linear model. Perturbations appear in the canonical form, as nonlinear functions of z, due to

under the nominal transformation (TJ, z) = T(x, ( 0 ) and the nominal feedback law u(x) = f3!(x)[-oo(x)+ v] renders the subsystem:

z{j) = " z(j) rj

z(j)

t+1

+ ~(j)

= D.r J,(j) -

l'

1
~(j)o f3 0

-

-

J

'

+ vJ' + (~(j))v f3

1 <_ J' <_ p

vpV

79

the presence of uncertainties. A Jacobi linearization of these nonlinear perturbations around the steady states is used so that linear robust control techniques can be used. It may be noted that this is different from the Jacobi linearization of the original nonlinear system. Only the perturbations arising due to uncertainties are linearized but not the whole model.

where

biVi

-

(1) -

(2)

(p) T

-

+ ek

where k = I:~=1 rj, 6i = ir,c;, ir , is an ri x ri square matrix similar to the identity matrix but the ones appearing on the reverse diagonal and ek is the kth basis vector in Rr. The vector

+ Wdd

-

i=l

y=Cz

(30)

The vector, b; = [C1~{3; C2~{3i ... Cp~{3i 1

m

+L

:']J

Ci is a row vector of length r; whose first element is 1 and the rest all are zeros.

Lemma 2.The system of the form (10) can be characterized as i = A(9)z

c~ [1' -

-

-

T

~A = (8",1, .. . ,8"'r_m,8B1 ,···,8Bm )

(23)

The non-linear perturbations ~A are represented as external bounded disturbances. Let d; E £2[0,00), such that Ild;112 :::: 1, 1 :::: i :::: r . Stable linear time invariant weights W d , are chosen such that 116;112 :::: IIWd AI12, 1 5 i 5 r (31) Then the effects of ~A can be represented by Wdd, where Wd = diag(Wd1 , . ··, W dr ). The uncertainty in the input ~{3 is a function of the vector of uncertain parameters. This reduces (29) to (23) .

where IIdi l12 :::: 1 are the non-linear perturbations represented as external bounded disturbances and Wd is a linear time invariant stable, minimumphase weight.

Proof. Define the following vectors of perturbations: ~(1)

1

o

~Ct =

To complete the design we must find a robustly stabilizing controller for the uncertain system (23). This is a linear robust control problem that can be solved via multi-objective optimization techniques such as mixed H2/ Hoo synthesis with pole placement constraints. This technique can be used for robust design when the linear fractional representation of the plant is affine in 9. The multi-model H2/ Hoo state-feedback synthesis places the poles such that the system has good performance for all values of 9. This problem is represented in Figure 1. The term, w, contains all external disturbances, e.g. d, and Z2 and Zoo contain the relevant errors signals that we want to maintain small with respect to the 2-norm (average) and oo-norm (worst case), respectively. The generalized plant G(9) represents the plant model together with performance and normalization weights. The objective is to find a stabilizing controller K such that

(24) ~(p)

1

~(p)

rp-1

~B

=

[

~~~~ : ~g~ :0] (3

r

~~) +:~~)ao

~{3 =

[

(25)

0

.

~~1)]

~(2)

~

(26)

~(p) (3

Then, by formal Taylor series expansion we can write

~"'i

= 8",;(9)z + J"'i(1], z, 9),1

:::: i :::: r - p(27)

~Bi=8Bi(9)z+JBi(1],z,9)

is minimized, for all 9 E 8, where Tzoow and Tz. w are linear operators mapping w to Zoo and w to Z2 respectively and a, b are positive numbers representing the trade-off between the H2/ Hoo

(28)

where 8"'i, 1 :::: i 5 r - p are row vectors arising from the Taylor series expansion and J"'i and JB , contain the higher order terms. Then the system becomes:

objectives. For the problem to be tractable, G should be affine in 9. If the matrix G is not affine, it poses a non-convex, infinite dimensional optimization problem. For this reason, the uncertain state space model (23) is represented as a poly topic family of

p

i

= A(9)z + ~A + LbiVi

(29)

1

y=Cz

80

systems where the state space matrices are affine functions of the uncertain parameters i. e. of the form

Now, it is shown that the feedback controller found by multi-objective synthesis robustly stabilizes the original non linear system. Theorem 2. Consider a system of the form of equation (23) and assume that 1. (J is in a compact set 2. IIL\vlloo < 1 3. IIdi l1 2 :s 1 4. Wd LTI stable, minimum-phase weight .

where, q is number of uncertain parameters. Then the multi-objective problem (32) is solved by Linear Matrix Inequalities (LMI) using the following theorem.

If a controller K robustly stabilizes the system

Theorem 1. (Khargonekar and Rotea, 1991) Given a polytopic family of LTI systems, of the form

represented by (23) , then this controller also robustly stabilizes the nonlinear system represented by (1) .

p

i =Az +B1 d+ LbiVi

Zoo

(34)

i=l = C 1z + Dud + D 12 V

Proof. The proof is along the lines of the stability proof in Kolavennu et al. (2000) and is omitted here for the sake of brevity. 0

(35) (36)

= C 2z + D 22 v feedback v = Kz that Z2

The state robustly stabilizes the above system and minimizes the performance objective is given by K = Y X-1, where X and Y are obtained by solving the following LMI formulation of the multi-objective state feedback synthesis problem: Minimize a'·-? + b Trace(Q) for all k over Y, X, Q and satisfying

4. ILLUSTRATIVE EXAMPLE Consider the following process model of a reversible reaction A ~ B taking place in a constant volume CSTR:

,2

(

-

T+ Y TT) D12

A(X, Y) Blk XCI Bik -1 CIX + D12 Y Dll

(

Q

xci + yT D[2

dC

F

dCB

F

dT

F

TtA = V(CAi -

DE

<0

(37)

Tt = V (CBi

C2X +X D22Y ) > 0

(38)

dt = V(Ti

Trace(Q)

< "5

(39)

"I

< "15

(40)

<0

(41)

-,,? 1

fv(X , Y)

CA) - k 1(T)CA + k2(T)CB

- CB)

- T)

+

+ k1(T)CA

- k2(T)CB

(-L\H) pCp (k1(T)CA + k2(T)CB )

where ki (T) = Aiexp( ~~) . The objective is to control the concentration CB to a value of 0.4 by manipulating the feed temperature Ti and feed concentration C Ai . The uncertain parameter is A 2 , it has a nominal value of 5.0e3 and can vary between 3.0e3 and 7.0e3. The relative cost of both inputs is assumed to be the same and the robust controller has to use a blend of these inputs in an optimal way to regulate the output.

where A(X, Y) = AkX + XAI + BkY + yT BI , = [b 1 .. . b2 . ... .. bmJ, Ab B lk , Bk are coefficients in the poly topic representation (as shown in equation 33) of the parameter dependent state matrices A , B 1 , and B respectively,and and 110 are upper bounds on the Hoo and H2 norms respectively and iv(X, Y) specifies the pole placement constraints . 0

B

,0

Both inputs have a relative degree 2 with respect to the output. A conventional I/O linearizing controller for this system was developed by McLain et al. (1996) and is given by

The minimization problem posed by Theorem 1 can be solved using the standard software such as the LMI control toolbox in MATLAB (Gahinet et al., 1995).

v

= -O.0044z1 -

O.133z2

Using the LMI control toolbox from MATLAB the following robust control law is obtained

Remark 3 : If a linear controller K cannot be found by solving the optimization problem (32) in Theorem 1, this does not imply that a robustly stabilizing controller does not exist for the original uncertain non linear system. This situation can arise when a bound on J cannot be established or when the bound on J is so large that the performance level , cannot be satisfied for the uncertainty.

v

where

81

= -O.06 z 1 -

O.29z2

w v

Fig. 1. Multi-model H2I Hoo synthesis problem

0 · "'"

......

~ .~

Table 1. Values of the variables in Illustrative Example

o ."'"

o ,o;--=----=---"",,::--o-=--~-...J T_-.c ,

Symbol

Value

-t::.HI -t::.H2 -t::. H 3 Al A2 A3 El E2 E3 p Gp

4.5e5K J

5.0e5KJ 6.0e5KJ

··."".""... 0 · "'"

2.0e6min - 1 1.2e6min- 1 1.2e6Kmol- 1 min-l

·

o ."",

5.0e4KJ 6.5e4K J 5.7e4KJ lOOOKg / m 3 4.2KJ/ Kg .K

V F/V

O.Olm 3 O.lmin - I

To

295K lKmol / min

G Ao

Fig. 2. Output profile for Conventional I/O controller

--,

" .i;---':;--~-;;--:;_"=---:"'---= "'----'~~"""---,J,..

Fig. 3. Output profile for Robust controller

The performance of the nominal and the robust controllers are shown in Fig. 2 and Fig 3. It is seen that the conventional I/O controller performance degrades significantly in the presence of parametric uncertainty while the robust controller performs well in the face of uncertainty.

Isidori , A. (1989) . Nonlinear Control Systems. Springer-Verlag, Berlin. Khargonekar , P.P . and M.A . Rotea (1991) . Mixed h2 / hoo control: a convex optimation approach" . IEEE Trans . Automat. Contr. AC39, 824- 837. Kolavennu , S., S. P alanki and J .C . Cockburn (2000). Robust control of i/ o linearizable systems via multi-model h2lhoo synthesis. Ch em. Eng. Sci . 55(9) , 1583- 1589. Kolavennu, S., S. Palanki and J .C . Cockburn (2001a). Nonlinear control of non-square mu 1tivariable systems. Chem. Eng. Sci . p . in press. Kolavennu , S., S. Palanki and J .C . Cockburn (2001b) . Robust controller design for multivariable nonlinear systems via multi-model h2 / h= synthesis . Chem . Eng . Sci. p. in press . Kravaris , C. and S. Palanki (1988). Robust nonlinear state feedback under structured uncertainty. AIChE JoumaI34(7) , 1119-1127. ~&Lain. R.B ., M.J. Kurtz , M.A . Henson and F.J . Doyle (1996). Habituating control for nonsquare non linear processes . Ind . Eng. Ch em. Res. 35 , 4067-·W77 . Morari, M. , \V . Crimm , r-.U . Ogelsby and LD . Prosser (1985). Design of resilient pro cessing plants. viii : Design of energy measurement systems for unstable plants - new insights. Ch em . Eng. Sei. 40. 187. Reeves . D.E. and Y . Arkun (1989). Interaction measures for nonsquare decentralized control structures. AIChE J. 35 (4), 603 .

5. CONCLUSIONS A design procedure was developed for a class of uncertain MIMO nonlinear systems based on I/ O Iinearization and multi-objective H2 / Hoo synthesis. The procedure is applicable for minimum phase systems that are I/ O Iinearizable. The inner loop is based on nominal parameters of the model. The outer loop is designed to provide robustness to uncertainties in plant parameters and performs well despite the loss of decoupling. The controllers can be designed using off-the-shelf software and do not require restrictive matching conditions to be satisfied. This methodology was illustrated via simulation of a regulation problem in a CSTR. 6.

REFERE~CES

Christofides , P.D ., A.R . Teel and P . Daoutidis (1996). Robust semi-global output tracking for singularly perturbed non linear svstems . Int . 1. Contr. 65. 639--666. Doyle, F .J ., F . Allgower. S. Oliveria. E . Cilles and ;"1. r-.lorari (1992 ). On non linear systems with poorly behaved zero dynamics. Proceedings of the American Control Conference p . 25 71. Cahinet. P. , A. Nemirovski . A.J. Laub and :--1. Chilali (1995). LMI Control Toolbox. The r-.lathworks Inc . v

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