Robust control of processes subject to saturation nonlinearities

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14. No. 415. pp. 343-358, rights

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CONTROL OF PROCESSES SUBJECT SATURATION NONLINEARITIES P. J. CAMPO and M.

16 October

1989;

receivedfor

1990

$3.00 + 0.00

Pergamon

Press plc

TO

MORARI~

Chemical Engineering, 210-41, California Institute of Technology, (Received

a

publication

Pasadena, CA 91125, U.S.A.

29 November

1989)

Abstract-Motivated by current practice, a two-step design technique for saturating systems is studied. First an “optimal” (for example in the H” sense) linear controller is designed neglecting saturation. Then a saturation compensation scheme (anti-windup) is designed which provides graceful degradation of closed-loop performance in the face of saturation. The focus in this paper is on the second step, and obtaining general results and insights applicable to any (linear) system subject to saturation. A design technique is developed which results in effective saturation compensation for a given multivariable plant and linear compensation design. For particular controller choices the resulting saturation compensator is shown to be equivalent to proven techniques including anti-rest windup and internal model control (IMC). Tools are developed for robust stability and performance analysis of nonlinear systems. Well-known structured singular value robustness tests for linear systems are extended to a class of nonlinear systems. Sufficient conditions are developed which guarantee closed-loop stability for all plants in a structured uncertainty set and for all nonlinearities of a specified form. These tests result in simple conditions on the initial linear controller design which must be satisfied in order to guarantee robust stability of the saturating plant. In some instances this requires that the original linear design be detuned. A procedure for performing this detuning is outlined. A promising single-step procedure for the synthesis of optimal robust linear controllers for saturating systems is also outlined. While this approach lacks the simplicity of the two-level decomposition, it appears to have promise for situations where the impact of the saturation on the closed loop is severe.

1. INTRODUCTION We consider in this paper systems which are subject to actuator saturations but are otherwise linear. Such saturations are present in every physical system and are the dominant nonlinearity, in terms of closedloop performance limitations, in many practical situations. While linear control theory is not formally applicable to saturating systems, the standard controller design procedure is to neglect the saturation, develop a linear design. and add some problemspecific scheme to deal with stability and performance degradation caused by actuator saturations (e.g. windup). For single input-single ouptut (SISO) systems this approach has been quite successful and saturation compensation is relatively well understood. For multiple input-multiple output (MIMO) systems. however, this is not the case and few workable schemes have been reported. Although optimal trajectories for saturating systems can be determined using nonlinear optimal control theory, the resulting bang-bang control laws involving complicated switching surfaces are very difficult to implement. In addition these systems can be very sensitive to model uncertainties. Since a full nonlinear robust control theory is not available, and actuator saturations are relatively simple nonlinearitAuthor

ties, the two-level decomposition of the design problem seems justified. In this paper we generalize this approach and extend it to the MIMO case. Specifically we outline the performance and stability problems introduced by saturations, develop a general method for the design of saturation compensation, and use the results to quantify the limitations on the initial linear design imposed by saturations. Our focus in the development is on obtaining general results and insights which can be applied to any (linear) system subject to actuator saturation, as opposed to results for a specific problem or case study. Since we will be interested in obtaining global stability results we will for the most part restrict consideration to open-loop stable plants. If the plant is not open-loop stable, there is always an external input “large enough” to keep the system in saturation, effectively opening the feedback loop. With no feedback the plant will demonstrate its open-loop characteristics, namely instability. With certain additional assumptions, e.g. on the size of external inputs to the system or on the size of certain internal signals, this condition can be relaxed. 2.

WINDUP

A common performance degradation phenomenon in saturating systems is known as “windup” or “integrator windup”. We consider the system shown

to whom all correspondence should be addressed. 343

344

P. J.

r *K

1. Classical

MORARI

dl

e

I Fig.

and M.

CAMP0

“I

I feedback control saturation.

structure

with

actuator

Fig. 1 where P(S) is the linear time invariant (LTI) plant, and K(s) is an LTI controller determined to be satisfactory in the absence of saturations. The block between the plant and the controller represents the actuator saturation and is modelled as:

in

fi = sat(u),

(1)

where &

=

141G 1, i sig:;u,)

(2)

Iu,I> 1.

Windup occurs when an actuator becomes saturated, effectively breaking the feedback loop. While the controller output u remains above the saturation limit we have: u(s) = K(s)[r(s)

- d(s) - P(S)ti(S)]

+ P(s)K(s)]-‘[r(s)

-d(s)].

Feedback

system

with saturation

compensation,

(4)

The effect of these “wound-up” states is a significant transient which must decay (unwind) after the system returns from saturation. This transient is most pronounced when there are slow dynamics in the controller, driven by the error while the system is in saturation, which then unwind slowly after the return to the linear regime. While windup has been widely observed and discussed, it has rarely been defined. In its strictest sense (integrator) windup has been used to refer to windup of the integral term of classical single input-single ouput PI or PID controllers (Buckley, 1971; Krikelis, 1980; Krikelis and Barkas, 1984; Astrom and HIgglund, 1988; Kapasouris and Athans, 1985). In its broadest use windup has been used to describe any performance degradation which occurs as a result of saturation (Doyle et al., 1987; Astrijm and Wittenmark, 1984; Prett, 1988). Motivated by the above discussion, we adopt for the purpose of this paper the following definition: Definition l-Windup occurs when the states of the controller are driven by the error while the actuator is in saturation. While this definition is certainly broader than strict integrator windup [windup of a single state in K(s) corresponding to the integrator], it is not as all

R.

encompassing (as we will see) as the broadest possible definition suggested above. Before developing a general method for dealing with windup (as defined) we review several standard approaches which have proven successful in some applications. 2. I. Anti-windup The classical approach of “turning off”, or modifying, error integration during saturation can be understood using Fig. 2 where the anti-windup block R(s) is given by: R(s)

=;L

(5)

(3)

and the states [for example the integral term of a proportional-integral (PI) controller] “windup”; i.e. for given external inputs, r(s) and d(s), they obtain values significantly different than they would in the absence of saturation when: u(s) = K(s)[l

Fig. 2.

We note that the system shown in Fig. 2 is not internally stable and therefore a realization of this configuration could not be used in practice. Equivalent configurations, which are internally stable, and give rise to somewhat more complicated block diagrams can be found in Buckley (1971), Astrom and Hagglund (1988) and Astrom and Wittenmark (1984). For simplicity we will ignore this internal stability problem and refer to the otherwise equivalent Fig. 2 in the following discussion. When the saturation is not active ti = u and the additional block R(s) has no effect so that closedloop performance (for small inputs) is determined by the design of K(s). During saturation we have: u(s) = [I + R(s)] -‘K(s)[r(~) + [I + R(s)]-‘[R(s)

=

--$ +

K(s)fr(s)

-d(s)] - K(s)P(s>]C(s>

(6)

- d(s)1 (7)

and the effect of R(s) is to remove an integrator from K(s) and K(s)&‘(s) and replace it with a first-order lag. If the parameter u is small then this effectively removes slow dynamics (integrator) and replaces them with fast dynamics (high bandwidth lag) which are much less susceptible to windup. These fast dynamics are still driven by the error while the system is in saturation, but they unwind quickly when the system returns to the linear regime and therefore have a less adverse impact on the system response. While successful in preventing windup in its narrowest sense, this simple approach is not adequate in

Control of processes subject to saturation nonlinearities

Fig. 3. The internal model control (IMC)

line computation and do not lend themselves to simple analysis. As such it is difficult to develop insight into the limitations posed by saturations on linear design by studying them. Indeed it is not clear at this point that it is necessary to introduce nonlinearities (in addition to the existing saturation) in the closed loop in order to provide adequate saturation compensation.

structure

all cases. As demonstrated by Doyle ef al. (1987), the controller need not include an integrator for windup to be observed. Any relatively slow dynamics in K(s) will result in undesired effects on the response for a substantial period of time after the actuators have returned from saturation. Additional limitations of this approach are a lack of a genera1 method for selecting the appropriate value of a [Astrom (1987) suggests making a “proportional to the integral time”], and a lack of stability guarantees [it is not difficult to see from (7) that if K(s) has righthalf plane poles, the saturating system will be unstable]. Another approach which has been suggested, and which guarantees closed-loop stability when there is no mode1 error, is the use of the internal model control (IMC) structure lsee Morari and Zafiriou (1989) and references therein] shown in Fig. 3. This corresponds to selecting R(s) = K(s)&s) and we have from (6) (when P = P): u(s) = K(s)[l

+ P(s)K(s)]-‘[r(s)

= Q(s)]r(s)

-

-d(s)]

(8) (9)

&)I,

where the IMC controller, Q(S), is defined by: Q(S) = K(s)[l+

P(s)K(s)]-‘.

(10)

2.2. Anri-windup

= P(s)sat(u(s)} =

JYs)sattQ(s)[r(s)

(12)

With P(s) stable (by assumption), stability of Q(s) is necessary and sufficient for internal stability in the absence of saturation [see Morari and Zafiriou (1989)]. Consequently, stability of the linear system implies stability of the nonlinear system. Nonlinear performance, however, is often excessively sluggish with the IMC implementation. This is clear from (9) which holds both in saturation and in linear operation. The IMC controller Q(S) never “sees” the effect of the saturation on the plant output y(s), and U(S) is only a function of the set point r(s) and disturbance d(s). Other, more elaborate, nonlinear schemes to deal with saturations have been proposed [e.g. model predictive control (Garcia et al., 1989), which involves solving a (simplified) optima1 control problem on-line, and the approach of Kapasouris (1988)]. While some of these techniques have been successful in practical applications, they require extensive on-

a sfate space perspective

d = Au + Be,

(13)

u = Cv + De,

(14)

where v E LP X ’ is the state vector of the controller. We denote the transfer function matrix obtained from this realization as: A

K(s) =

c

B D

7

It-1

(15)

in block diagram form in Fig. 4. Clearly with this realization, the state of the controller v is driven (only) by the error signal and we can expect significant windup resulting from saturations whenever A includes slow dynamics. Following Astrom and Wittenmank (1984) we can restructure this realization to achieve a controller with anti-windup properties. By multiplying (14) by n E@Xm, and substracting from (13) we obtain: which is represented

(11)

- +)I).

from

In order to develop a general anti-windup scheme, which extends trivially to MIMO systems, we adopt a state space perspective. In this section we outline a state space construction of a linear saturation compensator designed to avoid windup (Definition 1). We begin with a minimal state space realization of the m x p transfer function matrix K(s) given by:

Thus y(s)

345

d=(A-HC)v+(B--HD)e+Hu,

(16)

u=Cv+De.

(17)

rather than using the controller output u to drive the states in (16) we use the actual plant input ti. Thus we have (shown schematically in Fig. 5): Now

ti=(A-HC)v+(B-HD)e+HG,

(18)

u = Cv + De,

(19)

22= sat(u).

(20)

Astrom argues that by selection of H we can ensure that A - HC has all of its eigenvalues in the open

Fig.

4. Block

diagram of K(s) compensation.

without

saturation

P. J. CAMm and M. MORARI

346

left-half plane. In fact since (A, C) is observable (by minimality) we can arbitrarily assign the eigenvalues of A - UC and make the dynamics driven by the error as fast as desired. This approach begs the question-“What is the ‘optimal’ assignment of these eigenvalues?” The answer to this question is very simple and comes directly from Definition 1 which states: windup DCCU~Swhen the states of the controller are driven by the error whiIe the system is in saturation. With the controller parametrization (18-20) it is clear that we can avoid windup by selecting H = BD --I so that: ti = (A u=Cv

BD -‘C)v

+ BD ~‘~2,

(21)

-!-De,

(22)

1.2= sat(u).

(23)

With this parametrization the error has no effect on the states of the controller. Instead the states are updated based on ti, the plant input. We note that the realization (21-23) is only meaningful when a left inverse of D, denoted D-’ exists. We will assume throughout the sequel the existence of such a D -‘. In certain circumstances this may require modification of a prespecified K(s) at high frequency to ensure a left invertible D term. The parametization (21-23) is exactly the “conditioned controller” introduced by Hanus et al. (1987). While we arrive at the same anti-windup compensation, our development and its interpretation are completely different than the treatment involving “realizable references” they present. Example

I

In order to demonstrate the effectiveness of the saturation compensator we will consider a simple SISO example. The plant is given by:

and the controller K(s)

=

by: (5s + 1)(6.3s (s + 1)’ -

(6.3s

+ I)

(25)

+ l)e-‘““’

This controller was obtained via the IMC procedure [see Chap. 4 of Morari and Zafiriou

design (1989)]

and is based on an integral square controller for the output disturbance:

d(s) =

5. Block

diagram of the closed-loop saturation compensation.

system with

’

optimal

+ 1)’

The input constraints for this problem are 1~1 < 1.2. The linear responses (no saturation) to the designed disturbance (26) and a pulse disturbance of magnitude 0.5 and duration 10 arc shown in Figs 6a and b, respectively. Without saturation compensation, the system limit cycles in response to both of these disturbances as shown in Figs 6c and d. Instability with no saturation compensation is not unexpected and simply serves to underscore the dangers of ignoring the impact of saturations. The system also limits cycles when classical antiwindup (5) is applied (for any 0: 2 0). Inability of classical anti-windup to maintain stability demonstrates the limitations of a narrow definition of windup. Although an integrator in K(s) is removed while in saturation, other dynamics in K(s) cause instability. The IMC implementation is stable but somewhat sluggish, as shown in Figs 6e and f. With IMC, the controller output u is given by (9) both in the linear regime and during saturation, and is independent of the plant output y. Hence the controller does not the effect of the saturation, resulting in a “see” sluggish response. This sluggishness is most pronounced when the unconstrained input has a large peak and settles quickly. In this case the constrained system will come out of saturation quickly, before the plant output has reached its steady state value. With the controller output essentially constant the plant output approaches steady state with the open-loop dynamics of the plant. The response with the saturation compensator outlined in Section 2.2 is shown in Figs 6g and h. The system is stable (for any disturbance of bounded energy as we can show using results in Section 3) and provides a rapid response which closely resembles the unconstrained response (Figs 6a and b). The saturation compensator is able to preserve stability, as does IMC, but also keep the plant input saturated for a longer period than IMC allowing a faster response. 2.3.

Fig.

s(lOs

error

Relationship

between

saturation

compensators

We now consider the proposed saturation compensator from an input-output point of view. This allows us, using the block diagram of Fig. 2, to demonstrate that this anti-windup compensation is a generalization of both classical anti-windup and IMC. As we saw in Section 2.1, the classical antiwindup scheme corresponds to R(s) = (I/as)l. IMC corresponds to R(s) = K(s)P(s). From (21-23) and simple block diagram manipulations, it is easy to

Control of processes subject to saturation nonlinearities

347

output

.A 0.8 0.6 0.4 0.2 0 -0.2

0

Input

20

10

30

40

input

20 10 o-10

I -

-20 10

0

20

30

40

0

I

I

I

10

20

30

Time

Time for the unconstrained system, d(s) = I/s(lOs + I).

Fin. 6a. Examole

IAisturbance

40

resDonse

Fig. 6b. Example

l-pulse disturbance response for the unconstrained system.

output 1.5

output

I

1

1

0.5 0.5

0 0

-0.5 -0.5

I

I

0

30

I

I

60

90

I

120

150

-1

0

I‘

t

I

I

40

60

80

100

60

80

100

Input

2

1

1 0 0 -1

-22 0

-1 1 30

60

90

120

150

Time Fig.

response for the Idisturbance compensation, saturation system without d(s) = I/S(lOS + 1).

6c. Example

constrained

1

I

20

-2 0

20

40

Time

I

Fig. 4d. Example l-pulse disturbance response for the constrained system with no saturation compensation.

P. J. CAMPO and M. MORARI

348

output 1 0.80.6o-40*2O-0.2 0

I

I

I

10

20

30

-x1 0

40

10

20

30

40

30

40

Input

Input 2-

-22 0

10

20

30

0

40

10

20

Time

Time response Idisturbance Fig. 6e. Example IMC system the constrained using &S) = l/S(lOS + 1).

for the structure,

Fig. 6f. Example constrained

l-pulse

disturbance

system using the IMC

response for structure.

the

OutDut

-11

-0.2.1

0

0

10

20

30

10

Input

0

20

30

40

40

Input 2I

_j_fJJf-

-0.5

-1

-1.5 0

*

I

I

10

20

30

40

Time response for the Fig. 6.g. Example Idisturbance constrained system using saturation compensation (27). &S) = I/s(lOs + 1).

-22 0

10

20

30

40

Time Fig. 6h. Example l-pulse disturbance response for the constrained system using saturation compensation (27).

Control of processes subject to saturation nonlinearities determine that the proposed “conditioned (and Hanus’ to: R(s)

= K(s)D

saturation controller”)

-’

-

compensator correspond

1.

(27)

We note that for a purely proportional controller, K(s) = D = constant, which has no states to wind up, (27) provides R(s) = 0 as we would expect. For the PI controller: K(s) the corresponding (27) is:

=

k(r,s

windup

+ 1) T,S

(28)

’

compensation

given

by

Multivariable

E_mmple

=

Q(s)[l - Ws)Qb)l-‘,

consider

= p(s)Q(s).

the 2 x 2 plant:

P(s)

(31)

Selecting Q(s) = P ‘(0) provides integral action [T(O) = I] and a closed-loop bandwidth equal to the open-loop bandwidth (Morari and Zafiriou, 1989). The corresponding windup compensation is [using (27)] R(s) = K(s)P(s) which corresponds exactly to the IMC structure of Fig. 3. We conclude then that the IMC structure, which guarantees stability with respect to saturation, is the appropriate windup compensation only in a special case, namely when we have not used K(s) to modify the plant dynamics (bandwidth) but only to achieve integral action. This is consistent with simulation results where it is observed that IMC is excessively sluggish when saturation occurs in systems where K(s) is designed to speed up the cIosed loop. We have seen then that the proposed windup compensator (27) designed to avoid windup (Definition 1) is a generalization of well-known and succcssful strategies. Furthermore, the limitations of these traditional measures are clear from this discussion. Classical anti-windup avoids windup only for PI controllers and is inadequate when K(s) has poles in the right-half plane; IMC avoids windup only if the closed-loop dynamics are the same as the open-loop dynamics.

=

ii&[-t: -:-J

(32)

with both inputs limited by +- 15. The controller, selected on the basis of linear performance is: K(s)

lOs+l = ~ s

The linear response change

4

5

[ 3

4’

1

(33)

shown in Fig. 7a for a set point

r=

(30)

where Q(s) is an arbitrary proper, stable. transfer function matrix. With such a K(s), the (linear) closedloop transfer function b/r in Fig. 1) is: T(s)

2

&,

which is exactly the classic anti-windup strategy, “turn off the integrators during saturation”, used successfully for decades for PT controllers and openloop stable plants (Buckley, 1971; Krikclis, 1980: Astr6m and Hlgglund, 1988; Kaposouris and Athans, 1985). With P(s) stable, all controllers, K(s), which yield an internally stable closed-loop system (in the absence of saturations) are given by:

issues-directionality

In Section 2.2 we outlined a general windup compensation scheme which avoids windup in the states of K(s). For SISO plants this approach leads to graceful performance degradation when the system enters the nonlinear operating regime as a result of saturations. While our state space approach allows us to extend the windup compensation scheme to MIMO plants in a natural manner, windup compensation alone is often not adequate to ensure graceful performance degradation in the MIMO case. We demonstrate this with a simple example.

We

R(s) =

K(s)

2.4.

349

11 0.61

0.79

’

is decoupled with a first-order response in each output with time constant of 1 and no overshoot. The nonlinear response to this same set point change is shown in Fig. 7b. Both outputs overshoot significantly (approx. 500% at t = 4) then overcorrect and undershoot (approx. 100% at t = 8) before settling. With a broad definition this drastic performance deterioration would be assigned to “windup problems”. In fact only the smaller undershoot problem is the result of windup. This can be seen in Fig. 7c where we have included windup compensation (27) in the nonlinear simulation. The large initial overshoot is still present and the smaller undershoot (around t = 8) has been eliminated. It is clear then that relative to the large initial overshoot, windup is a relatively minor problem in this example. The problem demonstrated in the preceding example is unique to MIMO systems and results from the directional nature of the plant. In MIMO systems the plant gain is a function of the input direction. Since the saturation operates element by element on u to generate ri, the direction of Li is different than that of U. For example if

u=1; the resulting

l-2-l

1,

fi is ii=sat{u)=

: [1

P. J. CAMPO and M. MORARI

350

output8 5 3-

0.8

l-l-3-5

I 0

10

20

30

Inputs 201

2 -10 0

I

I

10

20

00 0

30

30

Time

Time Fig. 7a. Example 2-step

20

10

response for the unconstrained

- syst&.

Fig.

7b. Example 2-step response for the constrained system with no saturation compensation.

outputs

outputs 1

5

3

0.8

1

0.6

-1

0.1

-3

0.2

/ 0

0

20

10

30

Inputs

Inputs ‘“II

10

20

30

Time Fig.

7c. Example 2-step response for the constrained system with saturation compensation (27).

Time Fig. 7d. Example 2-step response for the constrained system with saturation compensation (27) and directionality compensation (35).

Control

of processes subject to saturation nonlinearities

and the direction of the controller output u is different than the plant input ti. If the saturation error u - ri corresponds to the high pfant gain direction, the difference in plant outputs corresponding to u and ri will be maximal. Correspondingly if u - ti is aligned with the low plant gain direction, the effect on the output will be relatively modest. Since the saturation operator acts element-byelement on u its structure is diagonal. It is well-known from robust linear control theory that some plant and controller combinations experience severe performance deterioration in the presence of diagonal input uncertainties. Specifically, ill-conditioned systems (those having large-scaled condition numbers) together with inverse-based (and consequently illconditioned) compensators as in the current example, have this property [see for example Skogestad and Morari (1987)]. Loosely speaking the diagonal operator disturbs the inversion so that P(s)sat K(s) z L(s), where L(s) is the d L (s) though P(s)K(s) desired loopshape. We can eliminate the directionality problem by adjusting all of the elements in u, when one of them becomes saturated, so that u and ii have the same direction [a similar approach was adopted in Doyle by inserting an et al. (1987)]. This can be achieved additional block in the loop as in Fig. 8. Here the block Rz is a nonlinear operator described by: u, = R2U

(34)

(35) where (36)

The purpose of R, is to scale back the controller output until its largest element has a magnitude of one. In this case the saturation will have no effect since its input JA’ always has \uil < 1 for all i. What we have effectively done then is replace the diagonal saturation operator by a scalar times identity operator. In this case, if we allow d, to be the scalar valued describing function appropriate for the composite operator, sat R,. we have: P(s)sat

Fig. 8.

and we see that (to a first approximation) the desired loop shape is only perturbed by a scalar factor. The impact on the closed loop is now not dependent on the direction of u but only on its magnitude. Using this approach to directionality compensation we return to Example 2 and simulate the response to the same set point change 0.61 r = [ 0.79

R,K(s)

c P(s)d2K(s)

= d,L(s)

(37)

Feedback system with saturation compensation and directionality compensation R,.

R

1 .

The response, shown in Fig. 76 is well-behaved, with no over- or undershoot characteristic of windup or directionality problems. It should be noted that this approach to directionality compensation is not necessarily optimal. Indeed it may happen that without directionality compensation u - ti is in the low plant gain direction. If information is available regarding the directional characteristics of the plant a constrained optimization can be performed to find a u’ which minimizes the effect of the saturation on the output error (e.g. minimizing the component of t( - u’ in the high gain plant direction). These schemes are typically very complicated and computationally intensive. We favor the simpler scheme because it is insensitive to the directionality of the plant (and hence requires no directional information), has provided very good results, is amenable to available analysis techniques, and is trivial to implement. Implicit in our assumption that K(s) is designed appropriately for the linear plant is the assumption that the output of K(s), u, is in the appropriate direction. With this in mind the simple directionality approach seems justified. 3.

11~ (IQ= max (~~1.

351

ANALYSIS

THEORY

In this section we present an approach to nonlinear systems analysis applicable to saturating systems. Since the theory has general applicability we will not discuss saturation nonlinearities per SC until the next section where we apply the genera1 results to the saturation compensation problem. The approach taken here originated with the work of Zames in the early 1960s (Zames, 1966), and is applicable to systems which include nonlinearities for which conic sector bounds can be obtained. The basic approach is to approximate nonlinear system components with linear ones and obtain norm bounds on the error involved in this approximation. The linear system is then studied subject to nonlinear perturbations within the specified norm bounds. If it can be shown that the linear system has certain properties (e.g. stability) for all perturbations within the norm bounds, then it is certain that the original nonlinear system has these properties as well. We begin with a few mathematical notions which are necessary for the subsequent development. In order to simplify the discussion we will present as few formal definitions and proofs as possible and refer the interested reader to relevant references [in particular,

P. J. CAMW and M. MORARI

352

r-e

much of this material is covered in Desoer and Vidyasagar (1975)]. We wit1 be concerned with signals which remain finite for all finite values of time. A mathematical characterization of the set of such functions is given by: Definition 2---L*, is the extended space valued functions x(r) with the property: Ilx(t)ll.=

1

=x*(t)x(t)dt LS 0

“’

of

< co,

vector Fig. 9. General

IlNx

/IT.~kll4lr~

interconnection analysis.

used for stability

(38)

for all T 2 0. System elements (blocks) are represented mathematically as operators which take inputs (signals in L,,) and produce outputs (signals in Lzp). The following is a formal definition of stability for system elements: Definition LAn operator N mapping said to be stable if there exists a constant that:

feedback

L, + L, is k -Z cc such

(39)

for all x E L,, and for all T 3 0. This definition corresponds to finite gain stability; input signals of bounded energy give rise to output signals of bounded energy.

and A is a (possibly nonlinear) operator in A defined by: A = {A = diag(A,,

block

diagonal

. . . , An)(Ai E Cone(0,

I)}.

(41)

Any feedback interconnection of linear and conebounded nonlinear blocks can be brought into this form. We will see examples of this in the next section. With these preliminaries we present the main result, a version of the multiloop circle criterion [see for example Safonov and Athans (19&l)]. Theorem AEA if: 1. M(s) 2.

f-The

system

in Fig.

9 is stable

for

all

is stable.

38 -E 1 ginf,,,

I/TM,,(s)T

‘I/_ G B.

where Definition Q--Given N, a possibly nonlinear and time varying operator, and two linear time invariant operators C and R, N is said to be inside Cone (C, R) if: IIN(-xII.G

HRxIIr,

(40)

for all x E L,, and for all T 2 0. A conic sector provides an LTI approximation to the input-output behavior of N. The cone center C provides an approximate output Cx for any input x. The cone radius R provides a measure of the error inherent in this approximation. For example the SISO memoryless nonlinearity N: x(t) + sat{x(t)} is inside Cone (4. $). The operator C: x(t) + $x(t) is our linear approximation to N, and R: x(t) -+4x(t) gives us a measure of the error in this approximation (as much as 100% in this case). We can replace any representation of all nonlinearities in Cone(C. R) with an equivalent representation in terms of all nonlinearities in Cone(O, I). Specifically y = Nx with N E Cone(C,R), if and only if, y = (C + fiR)x for some i? E Cone(O, I). This allows us to replace a nonlinear perturbation in Cone(C, R) with the LTI blocks C and R and a cone bounded nonlinearity in Cone(O, 1). This allows us to state all nonlinear stability results in terms of the Cone(0, I) and thereby simplifies the notation. Given these preliminaries we consider the general feedback interconnection of Fig. 9 where M is a linear time invariant operator with transfer function M(s)

(42) VAEA}.

9-={T]TAT-‘EA

(43)

Since a simple parametrization of the set Y is not available, the optimization problem implied in Case 2 is not tractable. We note, however, that the set: and

F’E{TITEF

TEW”“~“),

(44)

is characterized only by the structure of T. Specifically, 5’ consists of all block-diagonal constant matrices whose block structure is compatible with A in the sense that for each diagonal block in A the corresponding block in 5’ is diagonal, and for each full block in A there is a corresponding scalar times identity block in 9’. This simplification motivates: Corollary AoA if:

Z-The

system

1. M(s) is stable. 2. 38 < 13infr,.,.

in Fig.

IITM,,(s)T-‘11,

9 is stable

for all

< /?.

This simplification is significant and a complete solution to Condition 2 is available from state space structured singular value theory (Doyle and Packard, 1987; Packard, 1988). A significant advantage of this approach is that analysis of robustness with respect to uncertainties in the linear plant model is straightforward. As is standard in the robust control theory, we consider the

Control of processes subject to saturation nonlinearities nominal linear plant model subject to (possibly multiple) norm bounded LTI perturbations. The LTI uncertainty blocks are incorporated in the M - A framework (Fig. 9) in exactly the same manner as the cone bounded nonlinearities so that A is then a block diagonal operator in the set 8 defined by: 8 = {AlA = diag(A,,

. . , A., A,+, , . . , A,,,)],

(45)

are nonlinear operators each where A,....,A,, inside Cone(0, I) and A n+,r.. .1 A, are LTI operators satisfying 6(Aj) < 1 Vi = n + 1, . . , m. A straightforward extension of the structured singular value results (which handle LTI perturbations) provides: Z-The system in Fig. 9 is stable for all perturbations A E A if: Theorem

1. M(s)

= {TlT=diag(T

,,..

., T,,. T,+ ,,_..,

(45)

T,,)>,

with T,, . . . , T, E F and T,, , , . . . , T, arbitrary LTI operators which satisfy TiAi T;’ = A, Vb (A,) < 1 and Vi = n + 1, _ , tn. Again the simplification of T, , . . _, T, E .5-’ allows (relatively) straightforward evaluation of Condition 2 to assess robust stability. It should be noted that the conditions of Theorems 1 and 2 guaranteeing stability with respect to nonlinear perturbations are only sufficient, unlike the necessary and sufficient conditions provided by linear structured singular value theory. This conservatism and its impact on the analysis of saturation compensation are elaborated on in the next section. Before we move on to apply these analysis results to saturating systems we introduce the remaining definitions we will need. These are notions of passivity, or positive realness, for MIMO systems [see for example Desoer and Vidyasagar (1975) and Anderson and Vongpanitlerd (1973)].

M(s)

=

[21+

R + KP]-‘[R

-

-2P[21+R+KP]-‘[I+R]

Lemma

I-Z(s)

is strictly

in

passive if and only

if

3/!?< 13: ([]I - Z(s)]]J

+

z(~)l-‘ll,

=GB

(48)

and [I - Z(s)][f Further implications

+ Z(s)]-’

is stable.

of passivity are:

Lemma 2-Assuming that both Z(s) and Z-‘(s) are proper, Z(s) is strictly passive if and only if Z-‘(s) is strictly passive. Lemma 3-Z(s) strictly passive implies that Z(s) minimum phase (MP) and stable.

+ Z’(-io)

z cr,

VW E [O, co].

3.1. Application

is

of analysis theory

We will use the stability results outlined in the previous section to analyze the saturation compensation scheme developed in Section 2.2. Specifically we consider the stability of the system in Fig. 2. We stress again here that Fig. 2 is not appropriate for implementation, instead the system shown in Fig. 5 should be used with H chosen to correspond to a particular choice of R(s). The MIMO saturation nonlinearity li = sat(u) is a diagonal operator: iV = diag(n,, .

is

(47)

This is the standard notion of (strict) passivity which in the SISO case corresponds to the requirement that the Nyquist plot of Z(s) must remain in the (open) right-half plane.

, n,),

(49)

where each ni E Cone (f, 4). We can represent such a diagonal cone bounded linearity with the linear blocks C = {Z and R = ;Z and a cone bounded nonlinearity fi, with diagonal structure, in Cone(0, I). Rearranging Fig. 2 to obtain the standard framework of Fig. 9 provides: KP]

[21+R

+KP]-‘K

I-P[21+R+KP]-‘K

GA stable, proper, LTI system, Z(s), said to be strictly passive if 3~ > 0 3 : Definition

Z(jru)

The following Lemma characterizes passivity terms of an easily computed norm condition:

With these results we have completed the mathematical preliminaries necessary for robust stability analysis of general nonlinear systems. We now turn our attention to the particular nonlinearity of interest.

stable.

where: 3

353

1.

(50)

We can now apply Corollary 1 and Theorem 2 to evaluate nominal {no model uncertainty) and robust stability of saturating systems with saturation compensation R(s). Condition 1 of Corollary 1 requires M(s) to be stable. For R = 0, this is implied by stability of the linear (no saturation) closed loop when the controller gain is reduced by a factor of two.

P. J. CAMW and M. MORARI

354 Evaluating M,,(s) R = 0, we have: h4 ,,r.,.coaw(s) = =

when no compensation

KP]-‘KP

- KP[21+

2

TM,,

i~~i~~~!

KP]

+ KP)][f

of

Corollary

T-iIIm


< 1

(51) ’

(52)

+ (I + KP)]-‘.

(53)

1 which

inf [I[1 - T(I + KP)T-‘1 rE F’ x [I + T(I + KP)T-‘I-‘/I, which

is equivalent

T[1 + KP]T-’

requires in this

that case

< 1,

(54)


to:

strictly

MP

tiI+KP

passive for some T E .F’ (by Lemma

and stable (by

9[1-

QFJ-’

MP

*

[I -

QP]-‘Q

stable

-

K(s)

Lemma

l),

(55)

3),

(56)

and stable, (since

(57)

Q stable),

(58)

stable.

(59)

Thus we see that if the closed loop system with no saturation compensation is to be guaranteed stable with respect to saturation using Corollary 1, a necessary condition is that the controller be stable. This result is not surprising, as we pointed out in Section 2.1, K(s) stable is a necessary condition for closed loop stability if no saturation compensation is employed. Repeating this analysis for the proposed saturation compensation, R(s) = KD -’ - I, we find: M ,,RII/(~)=[l+

K(P

+D-‘)lp’[I-

K(P

-

D-‘)] (60)

= [D = [I -

Q][D

+ Q]-’

QD -‘][I

(61)

+ QD-‘I-‘,

(62)

Stability of M(s) requires that D + Q(s) be minimum phase. This requirement is not particularly restrictive. If we introduce a state space realization of Q(s):

Q(s)= [++$I. then [recall D = K(m)

[D + Q(s)]-~

=

(63)

= Q(a)]:

A -;BD-'C _jD-‘C

1iD-1 1’ 1 ;BD-’

(65)

so that D + Q(s) minimum phase requires only that the eigenvalues of A - +BD ‘C lie in the left-half plane. Condition 2 of Corollary I requires: inf Il[1T’E.P’

which

is equivalent

to:

TQ (s)D -‘T

-[21-t

= [I -(I Condition

is used,

TQD-‘T-‘][I


(66)

(67)

by Lemma 1. This implies (by Lemma 3) that Q(s) is minimum phase and stable. Since stability of Q(s) is necessary and sufficient for stability when there is no saturation, the only additional requirement is Q(s) MP [which implies K(s) MP for P(s) stable]. In terms of the state space realization of Q(s) (63), this condition is equivalent to the eigenvalues of A - BD ‘C and must he in the left-half plane. Since the complementary sensitivity function, T(s) = y(s)/r(s) = P(s)Q(s), it is rarely desirable to make Q(s) NMP since this would imply nonminimum phase behavior in T(s) (Morari and Zafiriou, 1989, pp. 58-59). While we require Q(s) minimum phase and D + Q(s) minimum phase for stability with saturation compensation, these conditions are less restrictive than if no compensation is employed. In contrast, for IMC, R(s) = K(s)P(s), no requirements other than linear stability need to be imposed. In this case M ,,n+-(~) is identically zero and Theorem 1 is satisfied trivially. Unfortunately IMC generally results in sluggish performance when saturation occurs. We consider next the impact on our stability analysis when directionality compensation is employed. The only modification to the above analysis involves the set of scaling matrices 9’. When no directionality compensation is employed. the saturation is a diagonal operator so that A has diagonal structure and 5 consists of nonsingular diagonal matrices. When directionality compensation is used, the series interconnection of R, and the saturation block is a scalar times identity operator and hence the corresponding A has scalar times identity structure. This implies that the set Y’ consists of arbitrary full invertible matrices in this case. Since we seek the infimum in Corollary 1 over a larger set when directionality compensation is used, a larger class of M,,(s) will satisfy the sufficient condition. This robustifying effect as a result of directionality compensation is not surprising. With directionality compensation we are guaranteed that the plant input will always be in the same direction as the controller output, only the magnitude of the actual plant input can be affected by saturation. With no directionality compensation both the direction and magnitude of the actual plant input are affected by saturation. To demonstrate the effect on the sufficiency test for nonlinear stability of the structure of elements in 5 imposed by including or not including directionality compensation we consider the following example. Example

3

In this example

+ TQD-‘T-‘-‘II,

I strictly passive,

P(s)

we consider 1 = ~ lOs+l

the 2 x 2 plant: 5

4

[ 4

3

1 ’

(68)

Control of processes subject to saturation nonlinearities

355

outDut8

0.8

0.6 0.4 0.2 0 -16

-0.2 0

5

10

15

I

I

I

20

Inputs

Inputs

6

6

1

3 1 o-

-6-9-12 0

5

10

15

20

0

I

I

I

5

10

15

Time

Time Fig.

20

10a. Example 3-pulse set point response for the unconstrained system.

Fig. lob. Example 3-pulse set point response for the constrained system with no saturation compensation.

outputs 0.8

_______-----____-_

0.6 0.4 0.2 0 -0.2

0

5

10

15

20

0

5

10

15

Inputs 6

-3 -6 -9 -12

1

gcp=-::

3 0

h

1

2

I

0

20

I

I

I

5

10

15

4 20

Time Fig. 1Oc. Example 3-pulse set response for the constrained system with saturation compensation (27).

-66 0

5

10

15

20

Time Fig. 1Od. Example 3--pulse set response for the constrained system with saturation compensation (27) and directionality compensation (35).

P. J. CAMPO and M. MORAN

356 with input magnitude limitations decentralized controller:

K(s)

Iu, 1-z 3, Iz+I < 10. A

lOs+l

1

s

[ 0

= ~

0 -1

1 ’

P(s)

is designed for P(S) neglecting saturation. The unconstrained response to a pulse set point change of magnitude +, = 0.6, r2 = 0.4, and duration 5 s is shown in Fig. 10a. The constrained response with no saturation compensation is shown in Fig. lob. The system is unstable; the manipulated variables are driven to their constraints and remain there indefinitely as the outputs move away from their set points. This is not surprising since: inf I/TM,, rt 6’

/I = 2.54,

T-’

(70)

violating Condition 2 of Corollary 1. Indeed since K(s) is unstable (it includes an integrator), we cannot expect nonlinear stability with no saturation compensation [consider (6) with R = 01. Rather than modify the controller K(s) we attempt to add saturation compensation and guarantee stability. Using the anti-windup compensator, R(s) = KD-’ - 1, with no directionality compenobtain sation we an M(s) which is stable. Unfortunately for diagonal matrices T: inf 11TM,, rt P

T

’ 11= 2.56,

(71)

so that Corollary 1 cannot be used to guarantee nonlinear stability. Indeed the pulse set point response, as shown in Fig. lOc, is unstable. Adding directionality compensation completes our saturation compensator design. With this modification: inf 11TM,, rE.P’

T-III

= 0.91,

(72)

(where we now include all nonsingular constant matrices in .F’) so that by Corollary 1 the system is guaranteed to be stable. This is confirmed for our pulse set point change by the response shown in Fig. 10d. This example demonstrates an important point for the design of decentralized controllers for saturating systems. Computing the relative gain array (RGA) for the plant in this example we find: RGA

=

-15 16

16 -15

some 3 x 3 and example:

1

systems)

or undesirable,

1 10s 1 4e’OS [ 5 = ___ +

3 4e10”

1’

for

(74)

has the same RGA as the plant (68), but pairing to avoid negative RGA elements would result in poor (linear) performance due to the off-diagonal delays in P(s). We generalize these observations with the following result. Theorem 3-With P(s) stable, saturation compensation R(s) = KD - ’ - I, no directionality compensation, and pairings with negative diagonal RGA elements, no diagonal controller exists which provides integral action and satisfies the conditions of Corollary 1. Proof-We adopt the notation RGA of P, and recall: A(P)

/i(P)

to denote

= P o(P-‘)r,

the

(75)

where 0 denotes element-by-element multiplication of two matrices. Using (75) it is not difficult to verify that: /t(P)

= /i(P-1)’

(76)

and that n (P) = n (S, PS, ). where S, and S, are any diagonal diagonal elements of n(P) satisfy: A,,(P)

det(P”) = P, ~ det(P)

’

(77) matrices.

Vi=l,...,n,

The

(78)

where P,, is the ith diagonal element of P and det(P”) is the determinant of the principle submatrix of P obtained by deleting the ith row and ith column of P. With P(s) stable, Q(0) = P(O)-’ is necessary and sufficient for integral action (Morari and Zaflriou 1989)], so that: A[P(O)]

= A[P(O)_y

(79)

=

nK2mT

(80)

=

A[TQ(O)D-‘T-‘I’,

(81)

where the last equality depends on both D = K(w) and Tbeing diagonal. From (78) and (8 1) it is clear that if any of the diagonal elements of _4 (P) are negative, then the determinant of some principle submatrix of TQ(O)D-‘T-’ must be negative. This implies that TQ(O)D-‘T-’ cannot be positive definite, or: TQ(O)D

(73)

and that the variable pairings chosen correspond to negative RGA elements. While it is generally not a good idea to pair variables with negative RGA elements for reasons of failure tolerance and ease of on-line controller tuning [see Grosdidier er al. (1985)], there may be situations where this is unavoidable (e.g.

larger

-‘T-l

so that (by Lemma inf #TM,, *e .F’

T-‘ll,

+ [TQ(O)D

-IT-‘I=

contradicts

(82)

1) = )z:,

II[I -

TQDm’Tp’]

x [I + TQD -‘T-‘]-‘ljy which

3 0,

Condition

# 1,

2 of Corollary

1.

(83)

q

Control

of processes

subject

We note that this proof does not go through if directionality compensation (34) is used since in this case the set of allowed scaling matrices T E F’ includes full, as well as diagonal constant matrices. This result simply adds another reason to avoid unfavorable pairings (corresponding to negative diagonal RGA elements), or employ multivariable controllers (for plants in which these pairings cannot be avoided). While Example 3 demonstrates the utility of our nonlinear stability test, Corollary 1, it must be stressed that this condition is not necessary for stability. Conservatism arises from several sources. The most significant problem is that we guarantee stability for all cone bounded nonlinearities, A E A, in addition to saturation which is a single nonlinearity in this set. By doing so we ignore all information about the saturation except its structure (diagonal) and its maximum and minimum gains (1 and 0, respectively). Other information such as memorylessness (a saturation produces no phase lag) is lost. Current research in the computation of the structured singular value for real perturbations promises to enable us to impose such a memorylessness constraint. Even if we were interested in guaranteeing stability for all such nonlinearities (perhaps to capture the effects of modelling errors for example) Corollary 2 remains conservative since we have used the set of constant scalings r’ rather than the more general 9. Nonetheless, these results have proven useful (as in the previous example), are the least conservative available, and have the important property that structured uncertainties in the linear plant can be handled as well.

4. SYNTHESlS

METHODS FOR SATURATING SYSTEMS

With our saturation compensation results in hand and a measure of the impact of saturations on the design of K(s), via the conditions elaborated in Section 3.1, we return to the general question of controller design for saturating systems. Ideally we would like to have a design procedure which produces a (generally) nonlinear controller for which some measure of nonlinear performance is guaranteed for all models in an uncertainty set of possible plants (robust performance). While recent advances in nonlinear analysis theory this synthesis problem remains are promising, unsolved. Relaxing our demands somewhat we might ask for a linear controller design which provides robust performance. The linear structured singular value (p) synthesis procedure could be employed using the M - A structure and including a performance block as is standard for linear systems. Unfortunately, since p-optimal controllers optimize performance for the worst-case perturbation, including zero gain for saturations, we can not expect to obtain performance

to saturation

nonlinearities

357

better than open loop. Clearly such a design methodology is too conservative to be useful. Further relaxing our demands, we may wish to develop a linear design method which optimizes nominal (linear) performance while guaranteeing nonlinear stability. Corollary 1 provides computable conditions on K(s) [equivalently Q(s)] which guarantee nonlinear stability. Here we adopt a weighted sensitivity performance measure as in N” optimal control. With saturation compensation, R(s) = KD-’ - I, we can pose the following optimal design problem:

Subject

to:

inf I(?“[I + K(P TE.9x [I -

+ D K(P

-

*)I- ’ D-‘)]T-‘11,

< j3 < 1.

(85)

Solutions to this problem would provide the optimal linear compensator K(s) which when coupled with saturation compensation would be guaranteed stable in the face of saturation. This couples the initial linear design problem to the subsequent saturation compensation design. Unfortunately this problem is intractable. Simplifying further by eliminating the infimum in the constraints, and introducing the definition Q(s) = K(s)[l + P(s)K(s)]-’ we obtain:

subject

to: (I(1 -

QD-9~

+ QD-~)-~II,

G B c 1.

(87)

This is a special case of a more general, and very meaningful, design problem, optimal nperforA similar design mance subject to l!Zm constraints. problem arises in the evaluation of the graph metric [see Vidyasagar (1985)]. In the more general setting, constraints could be included to not only guarantee stability margins, but also minimum levels for secondary performance objectives. These problems remain the subject of ongoing research. While we do not have techniques to obtain the solution to (84-85) we can generate optimal suboptimal designs. The obvious method is to select a linear design technique (e.g. p-synthesis, H”, IMC, loopshaping, LQG/LTR) and perform the following iteration: 1. Select values for the free parameters of the design technique. (Performance weights, loopshape, etc.). 2. Design K(s). 3. Evaluate (85) for the given design. 4. If (85) is satisfied stop, otherwise adjust free parameters and design a new K(s).

P. J. CAMPO and M. MOWI

358

We note that a feasible solution to (84-85) always exists since [with P(s) stable] K(s) = D[Z - P(s)D]-‘, where D is any constant matrix, is stabilizing and in this case (85) is satisfied with B = 0. A similar approach was proposed in Chen and Kuo (1988) using LQ optimal design, no saturation compensation, and evaluating the sufficient condition for nonlinear stability (54) with no scaling T. Clearly the development of a less conservative stability result and an improved saturation compensator has advanced the utility of this technique. 5.

CONCLUSIONS

In this paper we have outlined the factors which cause performance deterioration in nominally linear feedback systems when actuator saturation occurs. We developed a systematic procedure for the design of MIMO saturation compensation. It was shown that a simple linear windup compensator (a generalization of classical SISO integrator anti-windup, and LMC) coupled with a transparent (although nonlinear) directionality compensator produces graceful degradation of linear performance when saturations occur. The simplicity of this formulation stands in contrast to other complex nonlinear schemes. The simple form of this saturation compensator allows us to apply extensions of linear system theory to saturating systems, including tools for stability and performance analysis in the face of model uncertainty. Applications of these extensions allows the development of relatively simple tests which can guarantee nonlinear stability. While these preliminary extensions of linear system theory to simple nonlinear systems are very promising, substantial further work is needed. In order to further reduce “overdesign” of the linear K(s) to ensure robustness with respect to nonlinearities such as saturation, the conservativeness of the stability tests in Section 3.1 must be reduced. Promising approaches include, reducing the set of nonlinearities included in a particular norm bound, and increasing the set of allowable scalings F. Developments in the calculation of the structured singular value for real perturbations will allow us to consider only memoryless nonlinearities, and attempts to parametrize 5 in (43) promise consideration of more general scalings in the computation of the sufficient condition for nonlinear stability. An additional area of future work is the application of the nonlinear analysis tools of Section 3 to other common actuator nonlinearities. These include dead bands, rate saturations and hysteresis. Acknowfedgemenr-Partial financial support from the National Science Foundation is gratefully acknowledged.

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L

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