Dual-mode adaptive control of nonlinear processes

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PSE '97-ESCAPE-7 Joint Conference

is invariant for the nonlinear system controlled with the local state-feedback from step one, and b) system (1) controlled with the local controller is asymptotically stable for initial conditions x(0) E 12. In the third step a simple nonlinear adaptive feedback based on the concept of adaptive high-gain stabilization of nonlinear systems (Allg6wer and Ilchmann, 1997) is constructed, that guarantees that the system state is driven to region 12 from any initial condition ~:(0) E ]Rn with x(0) ~ 12. The linear controller from step one will be employed as long as the system state stays within region f~. As soon as a large disturbance causes the system to leave region 12 the dual-mode controller will automatically switch to the simple nonlinear controller from step three until the system state returns to region 12. We prove that the dualmode controller does indeed yield global asymptotic stability and give conditions under which this control structure achieves stability even robustly.

2

The

local

Assumption 2 guarantees that the state decays to zero asymptotically once it enters region 12. Note that we need to assume invariance of 12 and asymptotic stability in 12 for the nonlinear controlled system. For practical control problems it is not always easy to find such a region. For the case that the local controller is designed on the basis of LQ-theory we exemplary give a systematic approach on how to construct such a region in Section 2.1. 2.1

Construction

u = kTx

with k(0) = 0.

(2)

Typically, but not necessarily, a linear state feedback controller k(x) = k T" x can be used. Because state feedback (2) is employed in the dualmode configuration for which we want to proof global asymptotic stability we need to make two assumptions on the region 12 in which the local controller is intended to operate.

Assumption 1 Region f~ is invariant with respect to the nonlinear closed loop consisting of system (1) and state feedback (2), i.e.

f(x, k(x)) ~ 12

(5)

is determined as the solution of a linear quadratic optimal control problem on the basis of the Jacobian linearization A := °o-~(0 ) andb := g(O) of (1) with objective function J := fo°° ( x T Q x + r u 2 ) d t .

The local controller is intended to deliver asymptotic stability and desired performance in an operating regime in which the usual operation of the process takes place. This controller is not intended to stabilize or to achieve desired performance outside of this region. Therefore a large number of well developed controller design methods can be used to address this local control problem. Because we are dealing with a local problem, linear techniques (like for example linear Hoo methods, LQmethods, pole placement methods, but also classical methods commonly applied in industrial practice) can be used, even though the so designed linear controllers may fail to operate appropriately far away from the operating point. In the following we assume that the local controller is given by a state feedback law of the form

W e 12.

f2

Proposition 3 Suppose the local controller

mode

u = k(x)

of the region

(6)

Then, (i) the Ljapunov equation (AK + ~I) T P + P (AK + aI) = - (Q + rllkl[ 2) (7)

with AK := A + bk T admits a unique symmetric positive definite solution P for any 1% E

[0,-Am=(AK)). (ii) For any region 12c~ with

ao :=

(8)

with a E (0, 0o) such that in 12~ I%" )~min (P)

L~ <

IlPll

'

(9)

with L ¢ : = s u p { l ~ ]xE12a,x 5 0 ) and ~(x) := f ( x ) + g ( x ) k T x - A K x the following two properties hold: i. V = x T P x is a Ljapunov function for the nonlinear closed loop with system (1) and controller (5). 2. Region f~a is a subregion of the region of attraction F of the nonlinear controller system, i.e. 12~ C F and f~a is invariant for the nonlinear system (1) controlled with the local controller (5).

(3) Proof: The proof is based on the same idea that is for

Assumption 1 guarantees that if at some time instance t the state x(t) enters region 12, it will never leave the region again.

example used in (Chen and AllgSwer, 1997) and (Michalska and Mayne, 1993) and is omitted here for brevity. From Proposition 3 the procedure to determine the

Assumption 2 Region 12 belongs to the region of at- region 12 is as follows: traction of the operating point x = 0 of the closed-loop system 5= = f ( x , k(x)), i.e. x(t) ~ 0 for t --+ c~ if x(t) E 12 for some time t.

(4)

Step 1 Based on the Jacobi-linearized system (A, b) solve a LQ optimal control problem with objective function (6) to Tget a locally stabilizing linear state feedback u = k x.

PSE '97-ESCAPE-7 Joint Conference S t e p 2 Choose a constant ~ • [ 0 , - A m ~ ( A K ) ) and solve the Lyapunov equation (9) to get a positive definite and symmetric P. S t e p 3 Find the largest possible c~ • (0, c~) in (8) such that inequality (9) is satisfied in f ~ .

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4.2 f and ~ are globally Lipschitz at (0,0), i.e. for some unknown constant M], M~ > 0 we have

IIf(y,z)-f(0,o)ll _< M i

3

The

interlock

(12)

= h(u, z ) .

Z

V(y,z) • ~xZ~ n-x

mode and

The purpose of the interlock controller is to bring the system state x back to the region ~ if, for example, a disturbance causes the state of the controlled loop to leave ~. The interlock controller must not be restricted to a subregion of the state space but has to ensure that x(T) • 12 for some finite time T for arbitrary initial conditions x(0) • ]Rn. The design of such a feedback controller is a very difficult task if we require additionally that the transition takes place in an optimal fashion, i.e. that the controller exhibits desired performance in addition to stability. For the cases in which such a global (nonlinear) controller can be designed it makes no sense to apply the dual-mode concept introduced in this paper. For these cases the global controller should be used for all x • ]R'~ including region f~. For the majority of cases for which such a globally optimally performing controller can not be found we now introduce a simple adaptive feedback law that guarantees global asymptotic stability. However, we will see that due to its simple structure this controller will not always yield optimal performance. This adaptive control method is based on the theory of high-gain adaptive stabilization of nonlinear systems (AllgSwer and Ilchmann, 1997) which is an extension of earlier work developed for linear systems (see for example (Mareels, 1984; Willems and Byrnes, 1984). See also (Allg6wer et al., 1997) for an extension to adaptive A-stabilization and A-tracking.

3.1

A d a p t i v e high-gain stabilization of nonlinear processes

In this section we briefly review the theory of adaptive high-gain stabilization of nonlinear systems adapted appropriately to allow for the design of the interlock controller. We consider nonlinear system (1) together with a fictitious, i.e. not necessarily natural, output function y • IR with y = ~'(x) (10)

II~(y,z)-~(0,o)ll O, independent of z,

IIh(y,z)-h(o,z)ll < MhllYll V(y,z) • ~ x ~ "-1. Also, h(O, .) is globally Lipschitz, i.e. for some unknown constant M h > O, lift(0, z) - £(0, ~)II -< M h IIz - zll V (Z,~) • ~n--1 X /~n--1

4.4 The zero dynamics of system (1), (10) are uniformly exponentially converging towards 7 = O, i.e. there exist (unknown) M, ~ > 0 such that the solution of il(t) = h(O,7(t)),

7(0) = 7o

satisfies

117(t)l[ _< M e -~t 11701[ for all t >_0, 7o • ~ m . 4.5 ~(', ") is uniformly bounded away from zero and uniformly bounded from above, i.e. there exist al,a2 with 0 < ffl < if2 or ffl < o'2 < 0 such that (7"1

__< ~(y, Z) __< a2

V (y, Z) E Lt~ X ~n--1.

a l , a2 need not be known, only existence is assumed.

From an application point of view, Asumptions 4.2 and 4.3 can be considered as "technical assumptions". Assumption 4.5 means that .~ is either 'positive' or 'negative'. It is merely required that .~ will not change sign anywhere. This is a rather strong assumption that essentially excludes the presence of any singular points in where we assume -y(0) = 0. We make the following as- the whole state space. The strongest assumptions are 4.1 sumptions about nonlinear system (1) with output (10): and 4.4, that require the system to have relative degree one and to be globally minimum phase. For a system (1) Assumption 4 with given natural output function y = ~(x) these as4.1 System (1) with output (10) has strong relative sumptions are very strong and only a restricted number degree one. Furthermore, there exists a global of practical control systems will satisfy them. Note howequilibrium-preserving diffeomorphism T with ever, that output function 7(x) is to our free disposition here. That means we can choose 7(x) such that these [ Y= T]( X ) z (11) assumptions are satisfied. In essence these assumptions are comparable to the ones needed to proof global statransforming system (1) with output (10) into bility of sliding-mode controllers.

Byrnes-Isidori normalform

= ](y, z) + ~(y, z)u

T h e o r e m 5 Suppose g > O, al positive, and 7(x) is

chosen such that Assumptions 4.1-4.5 are satisfied for

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PSE '97-ESCAPE-7 Joint Conference

system (1) with this 7(~c). For any such system the ap- particular choice of 5 > 0. 5 only determines the "speed" plication of the nonlinear adaptive state feedback strategy of the gain adaptation and has thus a very transparent u(~) = - k v .

meaning.

~(~)

dkp = 5. Ih,(z)ll z dt

(13)

kp(O) = kpo

4

Global stability m o d e controller

of the

dual-

permits for arbitrary initial conditions kpo • IR, x • ~'~ a solution to the closed-loop system that exists for all The dual-mode controller consisting of local state feedtimes and has the following properties: (i) limt.-+~ kp(t) = kpoo exists and is finite;

back controller (2) and adaptive high-gain control law (13) is represented by the following equations:

(ii) x(.) • Lq(O, cx~) for all q • [2,~] ; (iii) l i m t - ~ x = 0 . Proof: The proof is a straightforward adaptation of the general proof given in (AllgSwer and Ilchmann, 1997) to the present case. •

k(x) u=

kv ' 7 ( x )

for x • f~ for x C f ~

kp(0)

kp0 (14)

dk___ZP= S 0 dt ~ 6.17(x)l 2

for :r • f~ for x ¢ f~.

T h e o r e m 7 Suppose system (1), feedback law u = k(x) Theorem 5 essentially states that the state of the and region f~ in (14) do satisfy Assumptions 1 and 2. closed loop with controller (13) will decay to zero for Furthermore suppose that system (1) together with the arbitrary initial conditions (global asymptotic stability) fictitious output function y = 7(x) with 7 as in (14) and that the controller gain kp does not grow unbounded. does satisfy Assumptions 4.1-4.5. Then the closed loop Nonlinear adaptive state feedback controller (13) can be consisting of system (1) and controller (14) permits for viewed as a simple P-controller with a proportional gain arbitrary initial conditions kpo • ~ and x(O) = Xo • ~ n that is adapted on-line when the function 7(x) is intera solution that exists for all times and has the following preted as output. Note, that the gain kp grows as long as properties: h,(x)l ~ 0, namely until a large enough level is reached to stabilize the closed loop. (i) limt--+ookp(t) = kp~ exists and is finite; Remark 6 (ii) x(.) e Lq(O, oo) for all q • [2, c¢] ;

• For the case al < 0 the sign of the controller gain has to be changed, i.e. u = +kp. 7(x). No other modifications are required. • Note, that controller (13) robustly stabilizes a whole class of systems, namely all systems for which Assumptions 4.1-4.5 are satisfied for the particular function ~/(x) chosen.

(iii} l i m t ~ x = 0 .

Proof: By assumption the closed loop is asymptotically stable for initial conditions x0 • f~ and kp = 0. Furthermore, invariance of f~ ensures that x • f~ Vt and thus (i)-(iii) are satisfied for x0 • fL For initial conditions x0 ~ f~ the adaptive high-gain feedback law is applied initially. By Theorem 5 we know that the solutions exist 3.2 The adaptive high-gain interlock for all times and that l i m t - ~ x = 0. This implies that controller for some finite time T the state will enter region f~ for a In the context of the dual-mode controller, feedback finite value of kp(T). As for initial conditions Xo • f~ the law (13) is only applied if the current state is outside remaining part of the closed loop trajectory also satisfies region fL The property, that the system state x decays (i)-(iii) and thus these properties are satisfied globally. to zero if this controller were applied everywhere, then assures, that the system state reaches region f~ in finite The closed loop is not only nominally but also robustly time independent of the initial state x(0) in the dual- stable. mode approach. The design of the interlock controller involves two T h e o r e m 8 The dual-mode controller (14) will globally steps: First, a function 7(x) needs to be found (which stabilize any system can be interpreted as a fictitious output), so that Assumptions 4.1-4.5 are satisfied. This is not always an = a(x) + b ( x ) u (15) easy task and further research to derive a systematic approach for the computation of 7(x) and conditions for with a(x) • ~'~ and b(x) • ~ n if Asumptions 1, 2 and the existence of such a fictitious output function 7(x) Assumptions 4.1-4.5 are satisfied with system (15). is needed. However, our experience shows that a suitable fictitious output can be found by inspection fairly Proof: The proof is an immediate consequence of the easily in most practical applications. The second step robustness of the adaptive high-gain controller and the only involves determination of a suitable value for the fact that f~ is assumed to be invariant and to belong to scalar tuning parameter 6. The global asymptotic sta- the region of attraction for system (15) controlled by the • bility property of this controller is independent of the local controller.

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PSE '97-ESCAPE-7 Joint Conference

5

Example

with "design parameter" fl satisfies Assumptions 4.1-4.5: With the transformation [y, Z] y = T ( X , S) and

To exemplify the approach outlined in this paper we consider control of a standard bio-chemical reactor where T(X,S)= [~S)+_X,flX~-~-D] (23) some simple microbial growth process is assumed to take place. The reactor state is characterized by the biomass concentration X and the substrate concentration S and system (16) is transformed to Byrnes-Isidori normal the dynamics can be described by two simple differential form. It is also easily seen that this transformation is equations nonsingular in A. Furthermore, a simple computation dX shows that d---t = I~(S)X - D X dS dt

k#(S)X

DS+

-

(16) DS/

(24)

dy = f u n c t i o n ( y , X ) - 0^~ DS l . dt Ub' T-1

representing material balances for the biomass and sub- Thus the reactor has relative degree one and negative strate respectively. The specific growth rate #(S) is high-frequency gain for all values of the states X and S given by the following law, that assumes a substrate in- because o°-~ss < 0 VS. The zero dynamics are given by hibition for increasing substrate concentration: dZ dt

(17)

# ( S ) = lZm,,~e - B s .

We consider the state-feedback case where the manipulated input is chosen to be the concentration of the substrate feed Sf. The parameters of this reactor are: D = 0. 2 [,/zma= = 0.382 ,k = 2, B = 0.242. g

(18)

fiZ

-

(25)

and are globally exponentially stable for fl > 0. This analysis shows that system (16) with output function (22) can be globally (in A) stabilized by the simple adaptive controller presented in Section 3. Using linear theory a linear state feedback controller

The value of the steady state input is S / s = 5t~. This reactor is a modified version of a reactor considered in (Bastin and Dochain, 1990). System (16) exhibits two steady states at

SI = S f s + [ k l k2]'

S-Ss

can be designed on the basis of the Jacobi-linearization of (16) so that desirable closed loop behavior is achieved in X~ = 1.161~ X,2 = 0[ (19) a neighborhood of the operating point. This controller is used as the local controller in the dual-mode approach. The second equilibrium characterizesan undesired wash- Figure 1 exemplary shows the dynamics of the closed out condition in the reactor where all microbial life loop with initial conditions X(0) 1.5~, S(0) = 3t~ and ceases. Steady state (19) constitutesthe operating point with controller parameters where we want to operate the reactor. However, a simkl = 0.2225 k2 = -0.0619 (27) ple analysis shows that this operating point is unstable. Thus control is essentialfor the operation of this reactor. However, for initial conditions further away from the opAlso note that the subregion A of the state space with

A = {(s,s): x > 0}

(20)

1.5

is an invariant manifold of the open-loop reactor as well as the controlled reactor. This is independent of the controller used because lt'

dX

d--y=0

' 0

forX=0

(21)

independent of the value of S and Sy. We first investigate the applicability of the dual-mode approach presented, i.e. the existence of a (fictitious) output function y = 7(X, S) such that system (16) with output y satisfies Assumptions 4.1-4.5. First note, that the global conditions can be replaced here by conditions that have to be satisfied only for region A as this region is an invariant manifold of the controlled system. This implies that the states of the controlled system can never leave this region for initial conditions (X(0), S(0)) E A. It is then easy to verify that output function y = It(S) + f i X ~ X s A

D

(22)

i

10

20

30

,

40

50

60

70

80

time [hi 3

.

.

.

.

lin~ [hl

Figure 1: Dynamic behavior of bioreactor (16) with controller (26), (27) and initial conditions X(0) = 1.5 t~, S(0) = 3t~. erating point the closed loop becomes unstable with this controller. This is for example the case for initial conditions X(0) = 1.5~,S(0) = 1.5~. For many practical applications, region f~ can be found by inspection as the desired region of operation of the local controller (the

PSE '97-ESCAPE-7 Joint Conference

S 160

region where performance is achieved) is usually much smaller than the region of attraction. On the basis of a number of closed-loop simulations a candidate matrix P is determined very easily for this second order system: P =

I -1°6 °1 10

~/~ i ~ D t

o

~

By numerical optimization the parameter a is found according to

1o

i

........ . . . . . . . . . . . . , . . . . . . . . . . . . ~. . . . . . . . . . . .

i .....

i

i

i

i

i

i

I

20 ,

30

40 i i ~ PI

so

6o

7o ,

so

50

60

70

:

switching

:

,

i

0

10

20

215

........... 4

,

i

u

/

i ............ i............

1o ,

~ 0b J

(29)

max d x T p x

\

.......... i .......... i .......... i ~01

"

i ......

i 30

40

80

i

along closed-loop trajectories and subject to x T P x = c~. i I ~ For the region for which d x T p x < 0 the closed loop is sW ...I.. ......... .,I asymptotically stable and invariant. Here it turns out 0 ~ 0 10 20 30 40 50 60 70 80 time [hi that the maximal value is a = 4.5 and a = 4 is chosen so that Figure 2: Dynamic behavior of bioreactor (16) with = _< 4} (30) dual-mode controller for initial conditions X(0) = 1.5~,S(0) = 1.5~. with P as in (28). We now apply the dual-mode controller (14) consisting Due to the "safety-jacket" character of the interlock of linear state feedback (26) with parameters (27) (that is applied when the state is within region ~) and the mode this control strategy is very attractive for industrial control problems where potential failure of conadaptive high-gain stabilization law trollers leading to instabilities cannot be tolerated. The proposed dual-mode control structure may thus lead to Ss = Ss, + k~. (t~(S) + # X -xX' (31) the application of more advanced controller design techniques for the inner region as the "safety jacket" will dk_.2, p ~X DI 2 dt = l # ( S ) + f l X , always guarantee that this controller is only applied in its region of validity. (that is applied when the system state is outside region 12). For small disturbances, that do not drive Acknowledgement: The authors are indebted to the system state outside 12, the control loop perforC. Hong and A. Rehm of the University of Stuttgart for their mance is only governed by the local linear controller expert advice and help with the reactor example. Special that is designed to deliver good performance locally. For thanks go to A. Ilchmann of the University of Exeter for larger disturbances, the system state might be driven valuable suggestions and for his involvement during the early outside region 12 and thus potentially outside the region of stability of this controller. The adaptive control stage of this work. law (31) will then automatically generate a gain kp high enough to guarantee that the state is brought back to R e f e r e n c e s the operating regime 12 of the local controller. Figure 2 shows the closed-loop behavior with the dual-mode con- AllgSwer, F., Ashman, J., and Ilchmann, A. (1997). Highgain adaptive A-tracking for nonlinear systems. Autotroller and /7 = 0.5 and kvo = 0 for initial conditions matica, 33(5). X(0) = 1.5~, S(0) = 1.5~ that are outside region 12 in AllgSwer, F. and Doyle III, F. (1996). Nonlinear process (30). For this initial condition the local controller alone control: Which way to the promised land? In Kantor, J. and Garcia, C., editors, Chemical Process Control could not stabilize the system. As seen from Figure 2 CPC V. the dual-mode controller first brings the state back to AllgSwer, F. and Ilchmann, A. (1997). High-gain adaptive region 12 using the adaptive high-gain controller mode stabilization for nonlinear systems. In Gesellschaft ]iir Angewandte Mathematik und Meehanik, Annual Meet(interlock controller mode) and then switches to the loing, Regensburg. cal controller mode that is designed to achieve stability Bastin, G. and Dochain, D. (1990). On-line estimation and and performance in this region. adaptive control of bioreactors. Elsevier, Amsterdam.

D)

Chen, H. and AllgSwer, F. (1997). A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability, accepted for Automatica. 6 Conclusions Marcels, I. (1984). A simple selftuning controller for stably invertible systems. Syst. Control Lett., 4:5-16. We proposed a state feedback nonlinear dual-mode con- Michalska, H. and Mayne, D. (1993). Robust receding horizon trol scheme, consisting of a local mode and an interlock control of constrained nonlinear systems. IEEE 7Yans. mode. The controller switches between the two modes Automat. Contr., AC-38(11):1623-1633. depending on the system states being inside or outside Willems, J. and Byrnes, C. (1984). Global adaptive stabilization in the absence of information on the sign of the some pre-calculated region 12. It has been shown that for high frequency gain. In Lect. Notes in Control and Inf. a large class of nonlinear systems global robust stability Sciences 62, pages 49-57. Springer-Verlag, Berlin. is achieved.