This paper considers linear control design for systems with input magnitude saturation. A general anti-windup scheme which optimizes nonlinear performance, applicable to MIMO systems, is developed. Several examples, including an ill-conditioned plant, show that the scheme provides graceful degradation of performance. The attractive features of this scheme are its simplicity and effectiveness

Kothare, Mayuresh V.

Morari, Manfred

Zheng, Alex

English

This paper considers linear control design for systems with input magnitude saturation. A general anti-windup scheme which optimizes nonlinear performance, applicable to MIMO systems, is developed. Several examples, including an ill-conditioned plant, show that the scheme provides graceful degradation of performance. The attractive features of this scheme are its simplicity and effectiveness. 1 Introduction  Of special interest and common occurrence are systems having control input saturation nonlinearities but which are otherwise linear. Windup problems were originally encountered when using PI/PID controllers for controlling such systems. However, it was recognized later that integrator windup is only a special case of a more general problem. As pointed out by Doyle et al [6], any controller with relatively slow or unstable modes will experience windup problems if there are actuator constraints. Windup is then interpreted as an inconsistency between the plant input and the states of ..

Alex Zheng

Mayuresh V. Kothare

Manfred Morari

CiteSeerX

Anti-Windup Design for Internal Model Control

Caltech Authors - Main

TECHNICAL MEMORANDUM NO. CIT-CDS 93-007 
May 6,1993 
"Anti-Windup Design for Internal Model Control" 
Alex Zheng, Mayuresh V. Kothare and Manfred Morari 
Control and Dynamical Systems 
California Institute of Technology 
Pasadena, CA 91125 
Anti-Windup Design for Internal Model Control 
Alex Zheng Mayuresh V. Kothare Manfred Morari* 
Chemical Engineering, 210-41 
California Institute of Technology 
Pasadena, CA 91 125 
May 6, 1993 
CIT-CDS Technical Memo 
Abstract 
This paper considers linear control design for systems with input magnitude saturation. A general 
anti-windup scheme which optimizes nonlinear performance, applicable to MIMO systems, is developed. 
Several examples, including an ill-conditioned plant, show that the scheme provides graceful degradation of 
performance. The attractive features of this scheme are its simplicity and effectiveness. 
1 Introduction 
Of special interest and common occurrence are systems having control input saturation nonlinearities but 
which are otherwise linear. Windup problems were originally encountered when using PI/PID controllers 
for controlling such systems. However, it was recognized later that integrator windup is only a special case 
of a more general problem. As pointed out by Doyle e t  a1 161, any controller with relatively slow or unstable 
modes will experience windup problems if there are actuator constraints. Windup is then interpreted as an 
inconsistency between the plant input and the states of the controller when the control signal saturates. The 
"conditioning technique" as an anti-windup technique was originally formulated by Hanus e t  a1 (8, 91 as an 
extension of the back calculation method of Fertik and Ross [7] to a general class of controllers. Astrom and 
Wittenmark [2, 11 proposed that an observer be introduced into the system to estimate the states of the 
controller in the face of constraints and hence restore consistency between the saturated control signal and 
the controller states. This observer-based approach represented a significant generalization of the existing 
anti-windup schemes. Walgama and Sternby [ll] have clearly exposed this inherent observer property in a 
large number of anti-windup schemes. Campo and Morari [3] have derived the Hanus conditioned controller 
independently as a special case of the observer-based approach. 
The Internal Model Control (IMC) structure [lo] (see Figure 1) was never intended to be an anti-windup 
scheme. Although stability of P and Q would guarantee global stability, provided that there is no plant- 
model mismatch, the performance suffers when there are actuator constraints. This is because the controller 
'To whom all correspondence should be addressed: phone (818)356-4186, fax (818)568-8743, e-mail: 
MMQIMC.CALTECH.EDU. 
Figure 1: The IMC interconnection 
(Q) is entirely unaware of the effect of its action. In particular, it does not know if and when the manipulated 
variable (u) saturates. This effect is most pronounced when the IMC controller has fast dynamics which are 
chopped'off by the saturation. Unless the IhlC controller is designed to optimize nonlinear performance, it 
will not give satisfactory performance for the saturating system. The focus of this paper is to identify this 
nonlinear performance. 
Assumptions and Notations We will assume that the plant is a linear time invariant and stable square 
system with n inputs and n outputs. For simplicity, we will use the same symbol to denote both the transfer 
function and the corresponding impulse response model. The meaning should be clear from context. P ,  
P, and Q denote the plant, the model of the plant, and the IMC controller, respectively. They are n by n 
sat(u1) I uyaa ui > uya= transfer matrices. For u E Rn, sat(u) = : , where sat(ui) = u; u r i n  5 u; < urea denotes sat(un) { ,,?in u; < ,,?in 
the input saturation function. For z E S n ,  Ix(t)ll = C:=, Iz;(t)l denotes the 1-norm. 
2 Problem Formulation 
Consider the IMC structure as shown in Figure 1. Define 
Thus y' corresponds to  the output of the constrained system. Because of the saturation constraints, y'(t) 
necessarily differs from y(t), the output for the unconstrained system. In general, we would like to keep y' as 
close to  y rn possible. Mathematically, we want to solve the following optimization problem instantaneously 
a t  each time t. 
9 1(f * ~ ) ( t )  - ( f  * yl)(t)ll = mjn I(fPQ * e)(t) - ( f P  * G)(t)li (2) 
where f is a filter such that fP  is biproper. If P is strictly proper, then S does not affect y' instantaneously 
and the minimization is meaningless. Since our ultimate goal is to minimize 1 y(t) - y' (t)ll, f must be diagonal 
in order not to  introduce any change in the output direction. 
The minimization is carried out continuously for t 2 0. It is important to realize that this instantaneous 
minimization differs from the minimization over a horizon. For the conventional IMC structure displayed 
in Figure 1, G(t) = sat(u(t)) = s a t ( ~ i  Q e ( r ) d ~ )  is completely determined for any given e(t). Thus, in 
general, the conventional IMC implementation does not solve optimization problem (2) which optimizes the 
Figure 2: Modified IMC Structure 
performance for the constrained system. In the next section, we will show that a modified IMC structure 
actually solve the optimization problem (2 )  instantaneously. 
3 Anti-windup Design 
3.1 IMC Structure 
Figure 2 shows the modified IMC structure where Q = (I + Q2)-lQ1. Assume that Q is biproper.' We have 
4 s )  = Qle(s)  - Q2fi(s) = Qle(s)  - (919-' - I)G(s) 2 (3) 
In the time domain, 
u( t )  - .ir(t) = (QI  * e)( t )  - (QIQ-' * 4) ( t )  
The following lemma states how f should be chosen such that the modified IMC structure shown in Figure 2 
solves the optimization problem (2). 
Lemma 1 Suppose that Q is &proper and that P = P. If fPI,=, is a diagonal nonaingulat matriz with 
finite elements and Ql = f PQ, then G(t) resulting from the modified IMC implementation (Figure 2) is the 
~olution of optimization problem (2). Furthermore, i f  g = D f  where D is a diagonal constant matriz, then 
the closed-loop responses with f and g are identical. 
Proof: Q1 = f P Q  * u( t )  - ti(t) = ( f P Q  * e ) ( t )  - ( f P  * i ) ( t )  = ( f  * y)(t)  - ( f  * d ) ( t )  = yj( t)  - 4 ( t ) .  We 
have 
ui( t )  - Gi(t) = yj; ( t )  - y;i ( t ) ,  i = 1,2, -.- ,n. ( 5 )  
Since fPI,=, is diagonal, $, j # i ,  do not affect 4 ,  instantaneously. Equations ( 5 )  can be solved in- 
dependently for each iri(t). Consider the first input, i.e. i = 1. When no saturation occurs at  t = t l ,  
Gl(t l )  = u l ( t l )  = sat(ul( t l ))  and Iyl,(tl) - 4 , ( t 1 ) l  = 0 is minimized. Suppose that saturation occurs at 
t = t2,i.e. u1(t2)  > uYaZ or u l ( t2 )  < u r i n ,  we want to show that Gl(t2)  = sat(ul(t2)) also minimizes 
lyjl ( t z )  - 4, (t2)I. Since til(t2) affects 4, ( t z )  linearly and %(t2), j = 2,3,..- , n, do not affect 4, ( t2) ,  
Iyfl(tz) - 4,(t2)( is a convex function of i l ( t z )  only. If dl(t2) = ul(t2) for which ly f l ( t2)  - @,(t2)( = 0 is 
not feasible, i.e. u l ( t z )  > ura= 
'Q is biproper if both Q and Q-' are proper. 
'Here zero initial condition is assumed. This is without loss of generality since Q is stable and nonzero initial conditions can 
be incorporated into e ( t ) .  
or ul ( t2) < u p n ,  then the optimal solution which minimizes 1 y!, (t2)-4, (ta)l must occur a t  the boundary, 
i.e. iil ( t2)  = sat(ul(t2)). Therefore, choosing 61 ( t )  = sat(u1 ( t ) )  minimizes Iyt, ( t )  - 6, (t)l for each t 1 0. 
Since ly,; ( t)  - Gi(t)l is minimized for each i, l y j ( t )  - 4(t)(l is minimized. 
If g = D f ,  Equations ( 5 )  become 
where D = diag{Dll, . . , D,,). Before saturation occurs, the system is unconstrained and ii(t) = u( t )  does 
not depend on D. Assume that system saturates for input 1 a t  t = t l ,  then f l ( t 1 )  = uya' or iil(t1) = uyin.  
As long as the right hand side of Equation ( 5 )  does not become zero for i = 1, input 1 stays saturated and 
i l ( t )  is constant during this period. Input 1 becomes unsaturated only if the right hand side of Equation (5) 
becomes zero for i = 1 which is not a function of Dll.  Therefore, the system comes out of the saturation a t  
the same time regardless of what Dll is. Similar arguments can be used when more than one input saturate. 
Therefore, the closed-loop response for f and g are identical. 
R e m a r k  1 If fPI.=, is not diagonal, then 4, ( t )  may also be affected by iij(t), j # i, instantaneowly. The 
convezity argument would not work since Iyli ( t )  - ( t ) (  is also affected by %(t) ,  j # i. 
Remark  2 f m w t  be diagonal in  order not to introduce any change in the output direction. However, f 
for which fPI,=, is diagonal may not be diagonal. To get around this problem, we can design a diagonal f 
for P such that f ~ l . = ,  is diagonal. P can be chosen arbitrarily close to P .  Qz  m w t  be strictly proper to 
be implementable. This can be achieved by choosing f appropriately. 
Remark  3 Q is wually minimum phase and always stable. If Q is minimum phaae and Q1 non-minimum 
phaue, then ( I  + Q2)-l  must be unstable. Therefore, Q1 must be minimum phaae and stable to guarantee 
internal stability of the closed-loop system. f m w t  be chosen such that fPQ is both minimum phase and 
stable. 
R e m a r k  4 For the modified IMC structure, the input is kept saturated for an optimal amount of time until 
lyf ( t )  - 4 (t)l becomes zero. T h w ,  in  general, the performance is greatly improved when f is appropriately 
chosen. 
Different controller factorizations can be obtained by choosing f differently. We discuss two special cases 
here. 
Case  1: f = P-'. The optimization problem ( 2 )  becomes mjn lu(t)- ti(t)ll. The solution corresponds to 
0 
the conventional IMC structure which "chops o f f  the control input resulting in performance deterioration. 
However, stability of the closed-loop system is guaranteed. 
Case  2: f i s  such t h a t  Q1 is a constant  matrix. The optimization becomes m@ IQl[e(t) - el( t )] l l ,  
I) 
where e l ( t )  = (Q-I * i i)(t) .  This factorization corresponds to the Model State Feedback proposed in [4] for 
SISO systems. The performance in this case is greatly improved, but stability of the closed-loop system is 
not guaranteed. If the dynamics of PQ are slow, however, minimizing the weighted controller input error 
( e ( t )  - e l ( t ) )  may not be a good way to optimize the nonlinear performance. After the system comes out 
of the nonlinear region, the controller takes no action to compensate for the effect of the error, e ( t )  - el( t ) ,  
introduced during the saturation. 
Figure 3: Classical F d k k  Structure 
d 
Figure 4: Classical Feedback Structure with Anti-windup 
In Case 1 f was chosen to guarantee stability while f was chosen to enhance performance in Case 2. 
Therefore, f can generally be tuned to trade off performance and stability of the constrained system. It 
should be pointed out that f in Caae 2 was not an extreme choice. 
3.2 Classical Feedback Structure 
For stable systems, the IMC structure shown in Figure 1 and the classical feedback structure shown in 
Figure 3 are equivalent. The results for the modified IhlC structure can be extended directly to the classical 
feedback structure to obtain the anti-windup structure shown in Figure 4. The controllers K1 and Kz are 
defined as follows: 
Hanus [8, 91 suggested the following 
where K = Q(I-  PQ) - l .  This factorization corresponds to f = Ii1Q-l P- ' .  Therefore, Hanus'a conditioning 
technique minimizes IK1[e(t) - et(t)]ll and represents the hIIhl0 extension of the Model State Feedback. 
4 Examples 
In this section, several examples are shown to demonstrate the effectivenw of the proposed method. 
Example 1 Consider the following plant: 
P ( s )  = 2 100s + 1 
The IMC controller designed for a step input is 
Case  1. Chooeing f = 2.5(208 + 1) gives 
Q1 = 2.5 
Case 2. Choosing f = 50(s + 1) gives 
Here f(oo) was chosen such that Q2 is strictly proper. The input is constrained between the saturation 
limits f 1. The responses to a unit step disturbance with the conventional IMC and the modified IMC 
implementations are shown in Figures 5 and 6 along with the unconstrained responses. The figures illustrate 
the sluggishness of performance of the conventional ILIC implementation when the closed loop dynamics are 
much faster than those of the open loop. For the conventional IMC implementation, the saturation effectively 
"chops off" the control input resulting in performance deterioration. The modified IMC implementation keeps 
the control signal saturated for an optimum length of time as discussed in Section 3 resulting in improved 
performance. f in Case 1 corresponds to minimizing (e( t)-e'( t ) l  while f in Case 2 corresponds approximately 
to minimizing l y ( t )  - b ( t ) l .  The control input in Case 2 stays saturated until y ( t )  s* d ( t )  while the control 
input in Case 1 stays saturated until e ( t )  = e l ( t ) .  In Case 1, the difference between y ( t )  and y'(t) resulting 
from the difference between e ( t )  and e l ( t )  during the saturation is not compensated as can be seen in Figure 
5. 
Example 2 This example is taken from [6] where the conventional anti-windup method did not result in a 
stable closed loop system. The plant is a fourth order lag-lead butterworth: 
where w l  = 0.2115, w2 = 0.0473,t1 = 0.3827 and €2 = 0.9239. 
The IMC controller is 
Choosing f = .w gives 
Solid: Unconstrained 
0.5 Dotted: Case 1 
Dotdash: Case 2 
Dashed: Conventional IMC with constr 
Time 
Figure 5: Example 1 - Plant Output Responses 
Time 
Figure 6: Example 1 - Coutroller Output Responses 
Time 
Figure 7: Example 2 - Plant Output Responses 
The input is constrained between the saturation limits f 1. Figure 7 shows the responses for a disturbance 
input with step of magnitude of 5 at time t = 0 and a switch to -5 at  t = 4. The performance improvement 
over the conventional IMC implementation is significant. Furthermore, the off-axis criterion (51 can be used 
to show that the closed-loop system is globally asymptotically stable. 
Example 3 Consider the following plant: 
Both inputs are constrained between the saturation limits f 1. A setpoint change of [0.63 0.79jT is applied. 
The IMC controller designed for a step input is 
Two values of f , one diagonal and one non-diagonal, are chosen to see how f (diagonal or not diagonal) 
affects the closed-loop performance. 
Case 1. 
Time 
3 
2 
1 -  
Figure 8: Example 3 - Plant Output Responses 
I I I :  1 I I 1 
:..-.. .. -. 
-. 
-.. 
1,. -. -. 
-.. i -... 
- .. i -... 
,i - ..  . .. 
- 
-. . -.
./' -.. ..-.-a - 
I 
. -.-.__ 
; -.-_ -. 
-.-.-__, 
i -.-.-.__ 
i ---.-._.__ -.-.-._. 
; -.-._. ---.-.*.- i 
' - 
Case  2. 
The responses for both case- and the conventional IMC implementation are shown in Figure 8. As we can 
see, choosing f to  be a diagonal nonsingular matrix is crucial to obtain good nonlinear performance. In Case 
1, some direction of the outputs is minimized. 
_--. 
_.-_-.-.-.- 
-.-- 
__-. 
_.-' 
5 Conclusions 
- 1 
-2 
-3 
We have proposed an anti-windup scheme which optilnizes the error between the outputs of the system 
generated by the constrained and unconstrained inputs, respectively. The method generalizes the Model 
State Feedback for SISO systems proposed in [-I] and Hanus's conditioning technique. In particular, the 
_.I.-.-' 
'. -.-' 
'. 
*.-' 
- .. -361id:. Unconstrained 
- 'i. _.-" 
'% 
- 
: -..-. 
L..' Dotted: Case 1 
'. 
'. 
. . 
- '  : 
\. ..' . ,.. : Dashed: Case 2 
't. \. Dotdash: Conventional IMC 
- '% '. 
'. 
- 
,' '. 
1, ?.-,-.: 
- I 1 1 I I 1 I 
0 20 40 60 80 100 120 140 160 180 200 
Model State Feedback corresponds to choosing f such that Q1 is constant; Hanus's conditioning technique 
corresponds to chming  f such that Q1 = K(oo). Furthermore, the derivation presented here elucidates the 
objectives behind the different methods and clearly shows the consequences of the various parameter choices. 
As shown by Example 3, the performance for Q1 = K ( w )  for MIMO systems may suffer when K(oo) is not 
diagonal.. Examples illustrate that this scheme provides graceful degradation of performance. The attractive 
features of the scheme are its simplicity and effectiveness. The filter f can be tuned to trade off performance 
and stability of the co nstrained system. 
6 Acknowledgement 
Partial support from the National Science Foundation is gratefully acknowledged. 
References 
[I] K. J. Astriim and L. Rundqwist. Integrator windup and how to avoid it. In Proceedings of the 1989 
American Control Conference, 1693-1698, 1989. 
[2] K. J. Astriim and B. Wittenmark. Computer Controlled Systems Theo+y and Design. Prentice-Ball, 
Inc., Englewood Cliffs, N.J., 1984. 
[3] P. J. Campo and M. Morari. Robust control of processes subject to saturation nonlinearities. Computers 
& Chemical Engineering, 14(4/5):343-358, 1990. 
[4] E. Coulibaly, S. Maiti and C. Brosilow. Internal hlodel Predictive Control. AIChE Annual Meeting, 
Miami, FL, 1992. 
[5] Y. S. Cho, K. S. Narendra. An Off-axis Circle Criterion for the Stability of Feedback Systems with a 
Monotonic Nonlinearity. IEEE Il.ansactions on Automatic Control, AC-13, 1968. 
[6] J. C. Doyle, R. S. Smith and D. F. Enns. Control of plants with input saturation nonlinearities. In 
Proceedings of the 1987 American Control Conference, 10341039, 1987. 
[7] H. A. Fertik and C. W. Ross. Direct digital control algorithms with anti-windup feature. ISA Thaw., 
6(4):317-328, 1967. 
[8] R. Hanus and M. Kinnaert. Control of constrained multivariable systems using the conditioning tech- 
nique. In Pmceedinga of the 1989 American Control Conference, pages 1711-1718, 1989. 
191 R. Hanus, M. Kinnaert and J. L. Henrotte. Conditio~li~lg technique, a general anti-windup and bumplesa 
transfer method. Automatics, 23(6):729-739, 1987. 
[lo] M. Morari and E. Zafiriou. Robust Pimess Control. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1989. 
[11] K. S. Walgama and J. Sternby. Inherent observer property in a class of anti-windup compensators. 
International Journal of Control, 52(3):705-724, 1900. 


TECHNICAL MEMORANDUM NO. CIT-CDS 93-007 
May 6,1993 
"Anti-Windup Design for Internal Model Control" 
Alex Zheng, Mayuresh V. Kothare and Manfred Morari 
Control and Dynamical Systems 
California Institute of Technology 
Pasadena, CA 91125 
Anti-Windup Design for Internal Model Control 
Alex Zheng Mayuresh V. Kothare Manfred Morari* 
Chemical Engineering, 210-41 
California Institute of Technology 
Pasadena, CA 91 125 
May 6, 1993 
CIT-CDS Technical Memo 
Abstract 
This paper considers linear control design for systems with input magnitude saturation. A general 
anti-windup scheme which optimizes nonlinear performance, applicable to MIMO systems, is developed. 
Several examples, including an ill-conditioned plant, show that the scheme provides graceful degradation of 
performance. The attractive features of this scheme are its simplicity and effectiveness. 
1 Introduction 
Of special interest and common occurrence are systems having control input saturation nonlinearities but 
which are otherwise linear. Windup problems were originally encountered when using PI/PID controllers 
for controlling such systems. However, it was recognized later that integrator windup is only a special case 
of a more general problem. As pointed out by Doyle e t  a1 161, any controller with relatively slow or unstable 
modes will experience windup problems if there are actuator constraints. Windup is then interpreted as an 
inconsistency between the plant input and the states of the controller when the control signal saturates. The 
"conditioning technique" as an anti-windup technique was originally formulated by Hanus e t  a1 (8, 91 as an 
extension of the back calculation method of Fertik and Ross [7] to a general class of controllers. Astrom and 
Wittenmark [2, 11 proposed that an observer be introduced into the system to estimate the states of the 
controller in the face of constraints and hence restore consistency between the saturated control signal and 
the controller states. This observer-based approach represented a significant generalization of the existing 
anti-windup schemes. Walgama and Sternby [ll] have clearly exposed this inherent observer property in a 
large number of anti-windup schemes. Campo and Morari [3] have derived the Hanus conditioned controller 
independently as a special case of the observer-based approach. 
The Internal Model Control (IMC) structure [lo] (see Figure 1) was never intended to be an anti-windup 
scheme. Although stability of P and Q would guarantee global stability, provided that there is no plant- 
model mismatch, the performance suffers when there are actuator constraints. This is because the controller 
'To whom all correspondence should be addressed: phone (818)356-4186, fax (818)568-8743, e-mail: 
MMQIMC.CALTECH.EDU. 
Figure 1: The IMC interconnection 
(Q) is entirely unaware of the effect of its action. In particular, it does not know if and when the manipulated 
variable (u) saturates. This effect is most pronounced when the IMC controller has fast dynamics which are 
chopped'off by the saturation. Unless the IhlC controller is designed to optimize nonlinear performance, it 
will not give satisfactory performance for the saturating system. The focus of this paper is to identify this 
nonlinear performance. 
Assumptions and Notations We will assume that the plant is a linear time invariant and stable square 
system with n inputs and n outputs. For simplicity, we will use the same symbol to denote both the transfer 
function and the corresponding impulse response model. The meaning should be clear from context. P ,  
P, and Q denote the plant, the model of the plant, and the IMC controller, respectively. They are n by n 
sat(u1) I uyaa ui > uya= transfer matrices. For u E Rn, sat(u) = : , where sat(ui) = u; u r i n  5 u; < urea denotes sat(un) { ,,?in u; < ,,?in 
the input saturation function. For z E S n ,  Ix(t)ll = C:=, Iz;(t)l denotes the 1-norm. 
2 Problem Formulation 
Consider the IMC structure as shown in Figure 1. Define 
Thus y' corresponds to  the output of the constrained system. Because of the saturation constraints, y'(t) 
necessarily differs from y(t), the output for the unconstrained system. In general, we would like to keep y' as 
close to  y rn possible. Mathematically, we want to solve the following optimization problem instantaneously 
a t  each time t. 
9 1(f * ~ ) ( t )  - ( f  * yl)(t)ll = mjn I(fPQ * e)(t) - ( f P  * G)(t)li (2) 
where f is a filter such that fP  is biproper. If P is strictly proper, then S does not affect y' instantaneously 
and the minimization is meaningless. Since our ultimate goal is to minimize 1 y(t) - y' (t)ll, f must be diagonal 
in order not to  introduce any change in the output direction. 
The minimization is carried out continuously for t 2 0. It is important to realize that this instantaneous 
minimization differs from the minimization over a horizon. For the conventional IMC structure displayed 
in Figure 1, G(t) = sat(u(t)) = s a t ( ~ i  Q e ( r ) d ~ )  is completely determined for any given e(t). Thus, in 
general, the conventional IMC implementation does not solve optimization problem (2) which optimizes the 
Figure 2: Modified IMC Structure 
performance for the constrained system. In the next section, we will show that a modified IMC structure 
actually solve the optimization problem (2 )  instantaneously. 
3 Anti-windup Design 
3.1 IMC Structure 
Figure 2 shows the modified IMC structure where Q = (I + Q2)-lQ1. Assume that Q is biproper.' We have 
4 s )  = Qle(s)  - Q2fi(s) = Qle(s)  - (919-' - I)G(s) 2 (3) 
In the time domain, 
u( t )  - .ir(t) = (QI  * e)( t )  - (QIQ-' * 4) ( t )  
The following lemma states how f should be chosen such that the modified IMC structure shown in Figure 2 
solves the optimization problem (2). 
Lemma 1 Suppose that Q is &proper and that P = P. If fPI,=, is a diagonal nonaingulat matriz with 
finite elements and Ql = f PQ, then G(t) resulting from the modified IMC implementation (Figure 2) is the 
~olution of optimization problem (2). Furthermore, i f  g = D f  where D is a diagonal constant matriz, then 
the closed-loop responses with f and g are identical. 
Proof: Q1 = f P Q  * u( t )  - ti(t) = ( f P Q  * e ) ( t )  - ( f P  * i ) ( t )  = ( f  * y)(t)  - ( f  * d ) ( t )  = yj( t)  - 4 ( t ) .  We 
have 
ui( t )  - Gi(t) = yj; ( t )  - y;i ( t ) ,  i = 1,2, -.- ,n. ( 5 )  
Since fPI,=, is diagonal, $, j # i ,  do not affect 4 ,  instantaneously. Equations ( 5 )  can be solved in- 
dependently for each iri(t). Consider the first input, i.e. i = 1. When no saturation occurs at  t = t l ,  
Gl(t l )  = u l ( t l )  = sat(ul( t l ))  and Iyl,(tl) - 4 , ( t 1 ) l  = 0 is minimized. Suppose that saturation occurs at 
t = t2,i.e. u1(t2)  > uYaZ or u l ( t2 )  < u r i n ,  we want to show that Gl(t2)  = sat(ul(t2)) also minimizes 
lyjl ( t z )  - 4, (t2)I. Since til(t2) affects 4, ( t z )  linearly and %(t2), j = 2,3,..- , n, do not affect 4, ( t2) ,  
Iyfl(tz) - 4,(t2)( is a convex function of i l ( t z )  only. If dl(t2) = ul(t2) for which ly f l ( t2)  - @,(t2)( = 0 is 
not feasible, i.e. u l ( t z )  > ura= 
'Q is biproper if both Q and Q-' are proper. 
'Here zero initial condition is assumed. This is without loss of generality since Q is stable and nonzero initial conditions can 
be incorporated into e ( t ) .  
or ul ( t2) < u p n ,  then the optimal solution which minimizes 1 y!, (t2)-4, (ta)l must occur a t  the boundary, 
i.e. iil ( t2)  = sat(ul(t2)). Therefore, choosing 61 ( t )  = sat(u1 ( t ) )  minimizes Iyt, ( t )  - 6, (t)l for each t 1 0. 
Since ly,; ( t)  - Gi(t)l is minimized for each i, l y j ( t )  - 4(t)(l is minimized. 
If g = D f ,  Equations ( 5 )  become 
where D = diag{Dll, . . , D,,). Before saturation occurs, the system is unconstrained and ii(t) = u( t )  does 
not depend on D. Assume that system saturates for input 1 a t  t = t l ,  then f l ( t 1 )  = uya' or iil(t1) = uyin.  
As long as the right hand side of Equation ( 5 )  does not become zero for i = 1, input 1 stays saturated and 
i l ( t )  is constant during this period. Input 1 becomes unsaturated only if the right hand side of Equation (5) 
becomes zero for i = 1 which is not a function of Dll.  Therefore, the system comes out of the saturation a t  
the same time regardless of what Dll is. Similar arguments can be used when more than one input saturate. 
Therefore, the closed-loop response for f and g are identical. 
R e m a r k  1 If fPI.=, is not diagonal, then 4, ( t )  may also be affected by iij(t), j # i, instantaneowly. The 
convezity argument would not work since Iyli ( t )  - ( t ) (  is also affected by %(t) ,  j # i. 
Remark  2 f m w t  be diagonal in  order not to introduce any change in the output direction. However, f 
for which fPI,=, is diagonal may not be diagonal. To get around this problem, we can design a diagonal f 
for P such that f ~ l . = ,  is diagonal. P can be chosen arbitrarily close to P .  Qz  m w t  be strictly proper to 
be implementable. This can be achieved by choosing f appropriately. 
Remark  3 Q is wually minimum phase and always stable. If Q is minimum phaae and Q1 non-minimum 
phaue, then ( I  + Q2)-l  must be unstable. Therefore, Q1 must be minimum phaae and stable to guarantee 
internal stability of the closed-loop system. f m w t  be chosen such that fPQ is both minimum phase and 
stable. 
R e m a r k  4 For the modified IMC structure, the input is kept saturated for an optimal amount of time until 
lyf ( t )  - 4 (t)l becomes zero. T h w ,  in  general, the performance is greatly improved when f is appropriately 
chosen. 
Different controller factorizations can be obtained by choosing f differently. We discuss two special cases 
here. 
Case  1: f = P-'. The optimization problem ( 2 )  becomes mjn lu(t)- ti(t)ll. The solution corresponds to 
0 
the conventional IMC structure which "chops o f f  the control input resulting in performance deterioration. 
However, stability of the closed-loop system is guaranteed. 
Case  2: f i s  such t h a t  Q1 is a constant  matrix. The optimization becomes m@ IQl[e(t) - el( t )] l l ,  
I) 
where e l ( t )  = (Q-I * i i)(t) .  This factorization corresponds to the Model State Feedback proposed in [4] for 
SISO systems. The performance in this case is greatly improved, but stability of the closed-loop system is 
not guaranteed. If the dynamics of PQ are slow, however, minimizing the weighted controller input error 
( e ( t )  - e l ( t ) )  may not be a good way to optimize the nonlinear performance. After the system comes out 
of the nonlinear region, the controller takes no action to compensate for the effect of the error, e ( t )  - el( t ) ,  
introduced during the saturation. 
Figure 3: Classical F d k k  Structure 
d 
Figure 4: Classical Feedback Structure with Anti-windup 
In Case 1 f was chosen to guarantee stability while f was chosen to enhance performance in Case 2. 
Therefore, f can generally be tuned to trade off performance and stability of the constrained system. It 
should be pointed out that f in Caae 2 was not an extreme choice. 
3.2 Classical Feedback Structure 
For stable systems, the IMC structure shown in Figure 1 and the classical feedback structure shown in 
Figure 3 are equivalent. The results for the modified IhlC structure can be extended directly to the classical 
feedback structure to obtain the anti-windup structure shown in Figure 4. The controllers K1 and Kz are 
defined as follows: 
Hanus [8, 91 suggested the following 
where K = Q(I-  PQ) - l .  This factorization corresponds to f = Ii1Q-l P- ' .  Therefore, Hanus'a conditioning 
technique minimizes IK1[e(t) - et(t)]ll and represents the hIIhl0 extension of the Model State Feedback. 
4 Examples 
In this section, several examples are shown to demonstrate the effectivenw of the proposed method. 
Example 1 Consider the following plant: 
P ( s )  = 2 
100s + 1 
The IMC controller designed for a step input is 
Case  1. Chooeing f = 2.5(208 + 1) gives 
Q1 = 2.5 
Case 2. Choosing f = 50(s + 1) gives 
Here f(oo) was chosen such that Q2 is strictly proper. The input is constrained between the saturation 
limits f 1. The responses to a unit step disturbance with the conventional IMC and the modified IMC 
implementations are shown in Figures 5 and 6 along with the unconstrained responses. The figures illustrate 
the sluggishness of performance of the conventional ILIC implementation when the closed loop dynamics are 
much faster than those of the open loop. For the conventional IMC implementation, the saturation effectively 
"chops off" the control input resulting in performance deterioration. The modified IMC implementation keeps 
the control signal saturated for an optimum length of time as discussed in Section 3 resulting in improved 
performance. f in Case 1 corresponds to minimizing (e( t)-e'( t ) l  while f in Case 2 corresponds approximately 
to minimizing l y ( t )  - b ( t ) l .  The control input in Case 2 stays saturated until y ( t )  s* d ( t )  while the control 
input in Case 1 stays saturated until e ( t )  = e l ( t ) .  In Case 1, the difference between y ( t )  and y'(t) resulting 
from the difference between e ( t )  and e l ( t )  during the saturation is not compensated as can be seen in Figure 
5. 
Example 2 This example is taken from [6] where the conventional anti-windup method did not result in a 
stable closed loop system. The plant is a fourth order lag-lead butterworth: 
where w l  = 0.2115, w2 = 0.0473,t1 = 0.3827 and €2 = 0.9239. 
The IMC controller is 
Choosing f = .w gives 
Solid: Unconstrained 
0.5 Dotted: Case 1 
Dotdash: Case 2 
Dashed: Conventional IMC with constr 
Time 
Figure 5: Example 1 - Plant Output Responses 
Time 
Figure 6: Example 1 - Coutroller Output Responses 
Time 
Figure 7: Example 2 - Plant Output Responses 
The input is constrained between the saturation limits f 1. Figure 7 shows the responses for a disturbance 
input with step of magnitude of 5 at time t = 0 and a switch to -5 at  t = 4. The performance improvement 
over the conventional IMC implementation is significant. Furthermore, the off-axis criterion (51 can be used 
to show that the closed-loop system is globally asymptotically stable. 
Example 3 Consider the following plant: 
Both inputs are constrained between the saturation limits f 1. A setpoint change of [0.63 0.79jT is applied. 
The IMC controller designed for a step input is 
Two values of f , one diagonal and one non-diagonal, are chosen to see how f (diagonal or not diagonal) 
affects the closed-loop performance. 
Case 1. 
Time 
3 
2 
1 -  
Figure 8: Example 3 - Plant Output Responses 
I I I :  1 I I 1 
:..-.. .. -. -. -.. 
1,. -. -. -.. i -... - .. i -... 
,i - ..  . .. - -. . -../' -.. ..-.-a - 
I 
. -.-.__ 
; 
-.-_ -. -.-.-__, 
i -.-.-.__ 
i ---.-._.__ -.-.-._. 
; -.-._. ---.-.*.- 
i ' - 
Case  2. 
The responses for both case- and the conventional IMC implementation are shown in Figure 8. As we can 
see, choosing f to  be a diagonal nonsingular matrix is crucial to obtain good nonlinear performance. In Case 
1, some direction of the outputs is minimized. 
_--. _.-_-.-.-.- -.-- __-. _.-' 
5 Conclusions 
- 1 
-2 
-3 
We have proposed an anti-windup scheme which optilnizes the error between the outputs of the system 
generated by the constrained and unconstrained inputs, respectively. The method generalizes the Model 
State Feedback for SISO systems proposed in [-I] and Hanus's conditioning technique. In particular, the 
_.I.-.-' '. -.-' '. 
*.-' - .. -361id:. Unconstrained - 'i. _.-" 
'% 
- 
: -..-. 
L..' 
Dotted: Case 1 
'. '. . . - '  : 
\. 
..' . ,.. : Dashed: Case 2 
't. \. Dotdash: Conventional IMC 
- '% '. 
'. 
- 
,' '. 
1, 
?.-,-.: 
- I 1 1 I I 1 I 
0 20 40 60 80 100 120 140 160 180 200 
Model State Feedback corresponds to choosing f such that Q1 is constant; Hanus's conditioning technique 
corresponds to chming  f such that Q1 = K(oo). Furthermore, the derivation presented here elucidates the 
objectives behind the different methods and clearly shows the consequences of the various parameter choices. 
As shown by Example 3, the performance for Q1 = K ( w )  for MIMO systems may suffer when K(oo) is not 
diagonal.. Examples illustrate that this scheme provides graceful degradation of performance. The attractive 
features of the scheme are its simplicity and effectiveness. The filter f can be tuned to trade off performance 
and stability of the co nstrained system. 
6 Acknowledgement 
Partial support from the National Science Foundation is gratefully acknowledged. 
References 
[I] K. J. Astriim and L. Rundqwist. Integrator windup and how to avoid it. In Proceedings of the 1989 
American Control Conference, 1693-1698, 1989. 
[2] K. J. Astriim and B. Wittenmark. Computer Controlled Systems Theo+y and Design. Prentice-Ball, 
Inc., Englewood Cliffs, N.J., 1984. 
[3] P. J. Campo and M. Morari. Robust control of processes subject to saturation nonlinearities. Computers 
& Chemical Engineering, 14(4/5):343-358, 1990. 
[4] E. Coulibaly, S. Maiti and C. Brosilow. Internal hlodel Predictive Control. AIChE Annual Meeting, 
Miami, FL, 1992. 
[5] Y. S. Cho, K. S. Narendra. An Off-axis Circle Criterion for the Stability of Feedback Systems with a 
Monotonic Nonlinearity. IEEE Il.ansactions on Automatic Control, AC-13, 1968. 
[6] J. C. Doyle, R. S. Smith and D. F. Enns. Control of plants with input saturation nonlinearities. In 
Proceedings of the 1987 American Control Conference, 10341039, 1987. 
[7] H. A. Fertik and C. W. Ross. Direct digital control algorithms with anti-windup feature. ISA Thaw., 
6(4):317-328, 1967. 
[8] R. Hanus and M. Kinnaert. Control of constrained multivariable systems using the conditioning tech- 
nique. In Pmceedinga of the 1989 American Control Conference, pages 1711-1718, 1989. 
191 R. Hanus, M. Kinnaert and J. L. Henrotte. Conditio~li~lg technique, a general anti-windup and bumplesa 
transfer method. Automatics, 23(6):729-739, 1987. 
[lo] M. Morari and E. Zafiriou. Robust Pimess Control. Prentice-Hall, Inc., Englewood Cliffs, N.J., 1989. 
[11] K. S. Walgama and J. Sternby. Inherent observer property in a class of anti-windup compensators. 
International Journal of Control, 52(3):705-724, 1900. 


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