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Integral control by variable sampling based on steady-state data
Automatica 39 (2003) 135 – 140
www.elsevier.com/locate/automatica
Brief Paper
Integral control by variable sampling based on steady-state data a , Stuart Townleyb;∗ ' Necati Ozdemir a Department
of Mathematics, Faculty of Science and Art, Balkesir University, 10100 Balkesir, Turkey b School of Mathematical Sciences, University of Exeter, Exeter, Devon, UK
Received 13 March 2000; received in revised form 16 April 2002; accepted 16 August 2002
Abstract In this paper we consider integral control algorithms with convergent adaptive sampling for multivariable in2nite-dimensional systems. Steady-state gain information is used in choosing suitable integrator gains and we also consider robustness with respect to error in measuring the steady-state gain. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Integral control; Steady-state gain matrix; Adaptive sampling; In2nite-dimensional systems
1. Introduction There has been much interest over the last 25 years in low-gain integral and proportional plus integral control. Indeed, the following principle has become well established (see Davison, 1976; Lunze, 1985; Morari, 1985, and the references therein; Logemann & Townley, 1997a): closing the loop around a stable, continuous-time, single-input single-output plant, with transfer function G(s), compensated by a pure integral controller k=s, will result in a stable closed-loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that |k| is su
There are several key issues in integral control: (1) In Logemann and Townley (1997a) it was shown that if a low-gain continuous-time integral controller achieves asymptotic tracking of constant vector valued reference signals, then necessarily G(s) is analytic in some right half plane, that is the system to be controlled is well-posed; (2) if the open-loop system is stable, then the steady-state gain, G(0), can be obtained experimentally and this information can be used as a basis for choosing the integrator gain and (3) for tracking of constant reference signals it is necessary that G(0) is invertible. Two approaches have been developed by which knowledge of G(0) can be used in the design of integral controllers either detailed steady-state data from the plant is used o@-line to determine the gain or else information on G(0) is used crudely and combined with on-line adaptive tuning. For example, if is chosen so that G(0) has eigenvalues in C+ , then convergent adaptation of a scalar gain k in an I -controller u˙ = ke is achieved via k = −p
with p ∈ (0; 1] and ˙ = r − y2 :
See e.g. Cook (1992) or Logemann and Townley (1997a). The 2rst of these approaches is quite complicated, whilst the second does not make full use of the available information, nor does it account for error in measuring G(0). We adopt a sampled-data integral control approach with 2xed integrator gain but convergent adaptive sampling. Adapting the sampling period allows us to choose the integrator gain based only on knowledge of the steady-state
0005-1098/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 1 8 2 - 6
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
136
gain matrix. We show how to select the gain robustly with respect to error in measuring the steady-state gain. We illustrate the results by a simple two-input, two-output example of a di@usion equation with point actuators and spatially distributed sensors. We consider throughout the following class of continuoustime, in2nite-dimensional m-input, m-output systems:
Applying variation of constants to (1), with input given by (2a), gives
x n+1 = T (n )x n + (T (n ) − I )A−1 Bun
(3a)
x(t) ˙ = Ax(t) + Bu(t);
(1a)
un+1 = un + K(r − Cx n ):
(3b)
(1b)
If we apply the change of coordinates
x(0) ∈ X;
y(t) = Cx(t):
In (1), X is a Hilbert space, with inner product ·; · and norm · ; A is the generator of an exponentially stable semigroup T (t); t ¿ 0 on X so that, in particular, there exists M ¿ 1 and w ¿ 0 so that T (t)x 6 M e−wt x for all x ∈ X . The input operator B is potentially unbounded but we assume A−1 B is bounded, whilst we assume that the output operator C is bounded. As is necessary for tracking of constant reference signals, we assume invertibility of the steady-state gain matrix G(0) := −CA−1 B ∈ Rm×m : Remark 1. (a) The class of systems encompassed by (1) is large. Unbounded input operators B are allowed provided that A−1 B is bounded. We need the output operator C to be bounded because the output y(·), which is sampled directly, needs to be continuous. If C was not bounded, then usually the free output y(·) would not be continuous so that sampling would require pre-2lters. We do not persue this aspect here. (b) We emphasize that, whilst our results are valid for a large class of in2nite-dimensional systems, they are new even in the 2nite-dimensional case. 2. Adaptive integral control of multivariable systems by adaptive sampling and based on steady-state data For stable systems given by (1) a non-adaptive, sampled-data integral controller with integrator gain K and, possibly variable, sampling period takes the form u(t) = un
for t ∈ [tn ; tn+1 )
(2a)
un+1 = un + K(r − Cx(tn )):
(2b)
with Here y(tn ) = Cx(tn ) is the sampled output at the sampling time tn . Typically, tn =n where is a 2xed sampling period. Our key idea is to use variable sampling times tn+1 = tn + n
with t0 = 0
and an adaptive sampling period n . This idea is not without precedent. Indeed variable and adaptive sampling has been used in a high-gain adaptive control context, see Owens (1996) and Ilchmann and Townley (1999).
x(tn+1 ) = T (n )x(tn ) + (T (n ) − I )A−1 Bun : Let x n := x(tn ). Then
zn = x n + A−1 Bun
and
v n = u n − ur
with ur := G(0)−1 r to (3) then zn+1 T (n ) − A−1 BKC = vn+1 −KC
−A−1 BKG(0)
I − KG(0)
zn vn
: (4)
We see clearly in (4) how the gain K, the steady-state gain G(0) and the variable sampling period n inKuence the closed-loop dynamics. In the following theorem we show that by choosing K on the basis of knowledge only of the steady-state gain we can achieve asymptotic tracking of constant reference signals for an otherwsie unknown system via convergent adaptation of the sampling period. Theorem 2. Let r ∈ Rm be an arbitrary constant reference signal. De7ne u(t) by (2) with n+1 = n + r − y(tn )2 :
n = tn+1 − tn = f( n );
(5)
In (5) f : [0; ∞) → [0; ∞) is monotone, f(0) = 0 and f( ) → ∞ as → ∞. Let K ∈ Rm×m be chosen so that the zeros of det( ( − I ) + KG(0)) have modulus less than one. If the sampled input u(t) given by (2), with sampling times tn given by (5), is applied to (1), then for each x(0) ∈ X; u0 ∈ Rm and 0 ¿ 0 (a) (b) (c) (d) (e)
limn→∞ n = ∞ ¡ ∞, limn→∞ n = ∞ ¡ ∞, limt→∞ u(t) = ur := G(0)−1 r, limt→∞ x(t) = xr := −A−1 Bur , limt→∞ y(t) = r.
Proof. Let 0 AK := 0
0 I
−
A−1 B I
K(C G(0)):
(6)
AK is the part of the operator in the right-hand side of (4) not depending on n . We claim that AK has spectral radius less one. To see this 2rst observe that since Im(AK ) is 2nite-dimensional AK is a compact operator. Hence !(AK ) consists only of eigenvalues (possibly accumulating at zero). Now let be a non-zero eigenvalue of (AK ). First note that
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
137
= 1 is not possible. To see this, multiply the eigenvalue– eigenvector equation −1 v1 0 0 v1 A B = − K(C G(0)) 0 I v2 v2 I
for all n ¿ N0 . Now Czn + G(0)vn = Cx n − r and using the boundedness of C we can 2nd M ¿ 0 so that
by (C G(0)) on the left. Using CA−1 B + G(0) = 0 this gives Cv1 = 0. But K(Cv1 + G(0)v2 ) = 0 so
Vn+1 − Vn 6 − M r − y(tn )2
KG(0)v2 = 0: Then either v2 = 0 or else det(KG(0)) = 0, neither of which can hold. Indeed, if v2 = 0, then v1 = 0 which is not possible, and det(KG(0)) = 0 does not hold by the assumption on K. So = 1. Next multiplying the eigenvalue–eigenvector equation −1 0 0 v1 v1 A B − = K(C G(0)) 0 I v2 v2 I by K[(1= )C 1=( −1)G(0)] on the left and setting K(Cv1 + G(0)v2 ) = v we obtain 1 −1 1 I +K CA B + G(0) v = 0: (7) −1 Now G(0) = −CA−1 B; G(0) is invertible and v = 0 (since otherwise v1 = 0 so that v1 = 0 and (1 − )v2 = 0 so that v2 = 0). Then (7) implies that det( ( − I ) + KG(0)) = 0: By the choice of K this shows that | | ¡ 1 as required. Hence, the spectral radius of AK is less than one and AK is power stable. Therefore, we can 2nd P = P ∗ ¿ 0 so that A∗K PAK − P = −I: Let Vn =
zn vn
;P
zn vn
zn 2 + vn 2 ¿ 2M r − Cx n 2 : This gives which, using (5), yields Vn+1 − Vn 6 − M ( n+1 − n )
for all n ¿ N0 :
It follows that 1 N 6 N0 + VN ¡ ∞: M 0 This proves that n is bounded and therefore converges to ∞ , and consequently n converges to ∞ . This proves (a) and (b). Then (5) implies that {r − Cx n } = {Czn + G(0)vn } ∈ ‘2 . Now zn+1 = T (n )zn − A−1 BK(Czn + G(0)vn ) = : T (n )zn + wn ;
(8)
−1
where wn = −A BK(Czn + G(0)vn ). Clearly, {wn } ∈ ‘2 . Taking norms in (8) gives n
zn+1 6 M e−wtn+1 z0 + M (e−w1 )n−i wi : i=0
In the last inequality we used the fact that n =f( n ), with n non-decreasing, gives n ¿ 1 for all n ¿ 1. So, {zn }∞ n=0 is an ‘2 -sequence plus a convolution of an ‘1 and an ‘2 sequence. It follows that zn ∈ ‘2 so that lim zn = 0:
(9)
n→∞
Then (Czn + G(0)vn ); zn ∈ ‘2 and invertibility of G(0) implies limn→∞ vn =0, which using the de2nition of vn gives limn→∞ un = ur and so (c) holds. Then (9) gives
:
Then computing Vn+1 − Vn along solutions of (4) gives 2 zn Vn+1 − Vn = − vn zn T (n )zn +2 ; PAK 0 vn T (n )zn T (n )zn + ;P : 0 0 Suppose that n is not bounded. Then n = f( n ) is not bounded and so for large enough n, the second and third terms are small. More precisely, it is easy to show that there exists N0 ∈ N so that for all n ¿ N0 , T (n ) is small enough to give Vn+1 − Vn 6 − 12 (zn 2 + vn 2 )
lim x n = −A−1 Bur = xr :
n→∞
For t ∈ (tn ; tn+1 ) we have x(t) = T (t − tn )x n + (T (t − tn ) − I )A−1 Bun ; y(t) = Cx(t): It follows that lim x(t) = xr
t→∞
and
lim y(t) = r;
t→∞
so that (d) and (e) hold and the proof is complete. Example 3. Consider a di@usion process (with di@usion coe
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
We suppose a two-dimensional output obtained by spatially distributed sensing on *-neighbourhoods of two points xc1 ; xc2 ∈ (0; 1). Speci2cally, the output is given by: xc +* xc1 +* 2 1 z(t; x) d x; z(t; x) d x : y(t) = 2e xc1 −* xc2 −* With these inputs u(t) ∈ R2 and outputs y(t) ∈ R2 , this diffusion system can be represented in the form of (1). Now the form of the transfer function G(s) for this system, and so also the steady-state gain G(0), depends on the relative positions of the actuators (xb1 ; xb2 ) and sensors (xc1 ; xc2 ). Signi2cantly in this context of integral control, G(0) is only invertible for certain relative positions of xb1 ; xb2 ; xc1 and xc2 . If xb1 ¡ xb2 ¡ xc1 ¡ xc2 , then G(0) is singular. If, however, xb1 ¡ xc1 − *;
xc1 + * ¡ xc2 − *;
3.5 Output Y1 (.) Output Y2 (.)
3
Output Y (tn)
138
2.5 2 1.5 1 0.5 0 0
20
40
xc2 + * ¡ xb2
then G(0) is invertible with 1 xb1 (1 − xc1 ) xc1 (1 − xb2 ) : G(0) = a xb1 (1 − xc2 ) xc2 (1 − xb2 )
60
80
100
120
Time tn Fig. 1. Output y(tn ), against tn , for the system described in Example 3.
2
For purposes of illustration, we adopt the following values: xb1 = 14 ;
xc2 = 58 ;
* = 0:01:
This gives 1 G(0) = 32
xb2 = 34 ;
50
30
30
50
xc1 = 38 ;
:
In this example we assume that G(0) is known and choose the gain K so that the zeros of det( ( − 1) + KG(0)) have modulus less than one. One choice is 0:2728 −0:1654 K= : −0:1654 0:2728 In the simulations we assume zero steady-state initial conditions for z and un , a stepped-reference (1; 2)T t ¡ 45; r(t) = (2; 3)T t ¿ 45
and an adaptive sampling period n = log( n + 1) with 0 = 1. In Fig. 1 we show the plot of the two outputs and in Fig. 2 the corresponding two inputs. In this simulation
the adaptive sampling period n = log( n + 1) converges to 1.774. In producing the simulations we used MATLAB and a truncated eigenfunction approximation of order 10 to model the di@usion process. In practice, the steady-state gain G(0) is obtained by step response experiments. Typically, this will lead to approximate measurements of G(0), and the value of G(0) available for design purposes will be a perturbation of the true value. This uncertainty in the value of G(0) would be due to measurement noise or else to the use of 2nite-time, as opposed to steady-state experiments.
1.5 Input U ( tn)
a = 0:1;
Input U1(.) Input U2(.)
1
0.5
0
-0.5 0
20
40
60 Time tn
80
100
120
Fig. 2. Input un applied, via idealised hold to the system described in Example 3.
Denote the measured value of G(0) by Gexpt (0). It is not our intention to develop a full theory. To simply illustrate what could be achieved consider the case where G(0) = Gexpt (0) + , and , is unknown but , ¡ ) for some ) ¿ 0. This is a set-up of the so-called stability radius, see Hinrichsen and Pritchard (1986). Of course, in this context of integral control it is necessary that G(0) = Gexpt (0) + , is invertible for all , ¡ ). This means that necessarily ) 6 Gexpt (0)−1 −1 : In Theorem 2 we chose K so that 0 I 0 + ,(I 0) −KGexpt (0) I K
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
is Schur. It is easy to see that this is guaranteed if 1 : , ¡ min |z|=1 (Gexpt (0) + z(z − 1)K −1 )−1
3.5 Output Y1 (.) Output Y2 (.)
3
In order to allow for the maximum experimental error (i.e maximum ) ¿ 0) we choose K to maximise the right-hand side of (10). Now clearly for any K the right-hand side of (10) is not greater than Gexpt (0)−1 −1 (just choose z = 1). Hence, the maximum possible ) ¿ 0 satis2es
2.5
Output Y (tn)
(10)
2 1.5 1
max min (Gexpt (0) + z(z − 1)K −1 )−1 −1 K
139
|z|=1
0.5
6 Gexpt (0)−1 −1 :
0
In fact, equality is achieved in the above by a large class of integrator gains K ∈ Rm×m .
0
20
40
60 Time tn
80
100
120
Fig. 3. Output y(tn ) for the system described in Example 5.
Theorem 4. 1 1 = −1 −1 ) Gexpt (0)−1 |z|=1 (Gexpt (0) + z(z − 1)K
max min K
and K achieves the maximum if K −1 = Gexpt (0)H;
H = HT ¿ 0
2
and
min (H ) ¿ 3: 1.5
This result is proved in the appendix.
so that in this case G(0) − Gexpt (0) = 0:0156 and −1 = 0:639. Hence Theorem 4 applies. We 1=Gexpt choose H = 4:5I; so 0:2186 −0:1303 −1 K = (Gexpt (0)H ) = : −0:1303 0:2186
As in Example 3 we use f( ) = log( + 1). In the simulations we assume zero steady-state initial conditions for z, un with r(t) as in Example 3 and 0 = 1. In Figs. 3 and 4 we show the plot of the output and corresponding input. In this simulation, the adaptive sampling period n converges to 1.801. Comparing Fig. 1 with 3, we see that both the response times and overshoot are similar even though the latter is obtained from an algorithm using only approximate steady-state gain data.
Input U (tn)
Example 5. Let us reconsider Example 3. This time we assume that G(0) can only be obtained from steady-state experiments so that K has to be designed on the basis of an experimentally measured steady-state gain Gexpt (0). To illustrate the e@ectiveness of our controller in coping with such experimental error in measuring G(0) we simulate experimental conditions by using for Gexpt (0) the steady-state gain for an eighth order truncated eigenfunction approximation of the di@usion process. This leads to a measured steady-state gain 1:576 0:939 Gexpt = ; 0:939 1:576
Input U1 (.) Input U2 (.)
1
0.5
0
-0.5 0
20
40
60
80
100
120
Time tn Fig. 4. Input un for the system described in Example 5.
3. Conclusions We have considered sampled-data integral control for a large class of in2nite dimensional systems. Motivated by existing results in the literature we focused on the following aspects: Simple robust use of readily available steady-state data and adaptation of controller parameters. The parameter we chose for adaptation was the sampling period. Using the sampling period in this way allowed us to determine, quite simply, appropriate integrator gains based on knowledge of step response data. We investigated robustness of the choice of K with respect to uncertainty in experimental measurement of the steady-state gain.
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
140
Appendix. Proof of Theorem 4. We know that 1 1 max min 6 : K |z|=1 (Gexpt (0) + z(z − 1)K −1 )−1 Gexpt (0)−1 Therefore, to show equality we need to 2nd K so that for all z with |z| = 1 1 1 ¿ −1 −1 (Gexpt (0) + z(z − 1)K ) Gexpt (0)−1 or equivalently (Gexpt (0) + z(z − 1)K −1 )−1 6 Gexpt (0)−1 : Now using K
−1
(A.1)
= Gexpt (0)H and estimating we have
(Gexpt (0) + z(z − 1)K −1 )−1 −1 (0)(z(z − 1)H + I )−1 : 6 Gexpt
So (A.1) holds if (z(z − 1)H + I )−1 6 1
for all z
with |z| = 1: (A.2)
Now (A.2) holds if z(z − 1)H + z( M zM − 1)H T + z(z − 1)z( M zM − 1)HH T ¿ 0 (A.3) for all z with |z| = 1. But H = H T ¿ 0. So we can choose / with /T / = I , so that /T H/ = D and D := diag(di )m i=1 . Then (A.3) becomes z(z − 1)di + z( M zM − 1)di + z(z − 1)z( M zM − 1)d2i ¿ 0:
(A.4)
Now z = ei1 , for 1 ∈ [0; 22] so that Re
z ei1 cos 21 − cos 1 = Re −i1 = zM − 1 e −1 2 − 2 cos 1
(2 cos 1 + 1) : 2 Then (A.4) becomes di −(2 cos 1+1) ¿ 0 for all 1 ∈ [0; 22] and i = 1; : : : ; m. This is true if di ¿ 3. So we need −1 min (H ) ¿ 3 and so K = H −1 Gexpt (0) achieves the maximum if H ¿ 3I . =−
References Cook, P. A. (1992). Controllers with universal tracking properties. Proceedings of the international IMA conference on control: modelling, computation, information, Manchester. Davison, E. J. (1976). Multivariable tuning regulators: The feedforward and robust control of a general servomechanism problem. IEEE Transactions on Automatic Control, 21, 35–47. Hinrichsen, D., & Pritchard, A. J. (1986). Stability radius for structured perturbations and the algebraic Riccati equation. Systems and Control Letters, 8, 105–113. Ilchmann, A., & Townley, S. (1999). Adaptive sampling control of high-gain stabilizable systems. IEEE Transactions on Automatic Control, 44, 1961–1966.
Logemann, H., & Ryan, E. P. (2000). Time-varying and adaptive integral control of in2nite-dimensional regular systems with input nonlinearities. SIAM Journal of Control and Optimization, 38, 1940–1961. Logemann, H., Ryan, E. P., & Townley, S. (1998). Integral control of in2nite-dimensional linear systems subject to input saturation. SIAM Journal of Control and Optimization, 36, 1940–1961. Logemann, H., Ryan, E. P., & Townley, S. (1999). Integral control of linear systems with actuator nonlinearities: Lower bounds for the maximal regulating gain. IEEE Transactions on Automatic Control, 44, 1315–1319. Logemann, H., & Townley, S. (1997a). Low-gain control of uncertain regular linear systems. SIAM Journal of Control and Optimization, 35, 78–116. Logemann, H., & Townley, S. (1997b). Discrete-time low-gain control of uncertain in2nite-dimensional systems. IEEE Transactions on Automatic Control, 42, 22–37. Logemann, H., & Townley, S. (2001). Adaptive low-gain integral control of multivariable well-posed linear systems. SIAM Journal of Control and Optimization, to appear. Lunze, J. (1985). Determination of robust multivariable I-controllers by means of experiments and simulation. System Analysis Modelling Simulation, 2, 227–249. Morari, M. (1985). Robust stability of systems with integral control. IEEE Transactions on Automatic Control, 30, 574–577. Owens, D. H. (1996). Adaptive stabilization using a variable sampling rate. International Journal of Control, 63, 107–119. ' Ozdemir, N. (2000). Robust and adaptive sampled data I—control. Ph.D. thesis, University of Exeter, UK. ' Ozdemir, N., & Townley, S. (1998). Adaptive low-gain control of in2nite dimensional systems by means of sampling time adaptation. Methods and models in automation and robotics, Miedzyzdroje, Poland (pp. 63– 68).
Necati Ozdemir was born in Balikesir, Turkey on 23rd July 1970. He received the B.Sc. degree in mathematics from Marnara University, Turkey in 1992, the M.Sc. degree in mathematics from the University of Warwick, UK, in 1996, and the Ph.D. degree in mathematics from the University of Exeter, UK, in 2000. He is currently a lecturer in mathematics at the University of Balikesir, Turkey. His research interests are in Functional Analysis and Sampled-data Control.
Stuart Townley was born in Widnes, UK on 2nd September 1961. He received the B.Sc. degree in mathematics, 1983, and the Ph.D. in engineering, 1987, both from the University of Warwick. After post doctoral fellowships in mathematics at the University of Warwick (1986 –1988) and University of Bath (1988–1990), he was appointed to a Lectureship in mathematics at the University of Exeter. He was promoted in 1996 to a Readership in Dynamics and Control, and in 1999 to Professor of Applied Mathematics. His research interests are in Robust Control, Adaptive Control, In2nite-dimensional systems, Stochastic Stability and Nonlinear Dynamics. He has held numerous UK Research Council and EU-funded grants and has supervised several Ph.D. students. He currently serves on the editorial board of Dynamics and Control.
www.elsevier.com/locate/automatica
Brief Paper
Integral control by variable sampling based on steady-state data a , Stuart Townleyb;∗ ' Necati Ozdemir a Department
of Mathematics, Faculty of Science and Art, Balkesir University, 10100 Balkesir, Turkey b School of Mathematical Sciences, University of Exeter, Exeter, Devon, UK
Received 13 March 2000; received in revised form 16 April 2002; accepted 16 August 2002
Abstract In this paper we consider integral control algorithms with convergent adaptive sampling for multivariable in2nite-dimensional systems. Steady-state gain information is used in choosing suitable integrator gains and we also consider robustness with respect to error in measuring the steady-state gain. ? 2002 Elsevier Science Ltd. All rights reserved. Keywords: Integral control; Steady-state gain matrix; Adaptive sampling; In2nite-dimensional systems
1. Introduction There has been much interest over the last 25 years in low-gain integral and proportional plus integral control. Indeed, the following principle has become well established (see Davison, 1976; Lunze, 1985; Morari, 1985, and the references therein; Logemann & Townley, 1997a): closing the loop around a stable, continuous-time, single-input single-output plant, with transfer function G(s), compensated by a pure integral controller k=s, will result in a stable closed-loop system which achieves asymptotic tracking of arbitrary constant reference signals, provided that |k| is su
There are several key issues in integral control: (1) In Logemann and Townley (1997a) it was shown that if a low-gain continuous-time integral controller achieves asymptotic tracking of constant vector valued reference signals, then necessarily G(s) is analytic in some right half plane, that is the system to be controlled is well-posed; (2) if the open-loop system is stable, then the steady-state gain, G(0), can be obtained experimentally and this information can be used as a basis for choosing the integrator gain and (3) for tracking of constant reference signals it is necessary that G(0) is invertible. Two approaches have been developed by which knowledge of G(0) can be used in the design of integral controllers either detailed steady-state data from the plant is used o@-line to determine the gain or else information on G(0) is used crudely and combined with on-line adaptive tuning. For example, if is chosen so that G(0) has eigenvalues in C+ , then convergent adaptation of a scalar gain k in an I -controller u˙ = ke is achieved via k = −p
with p ∈ (0; 1] and ˙ = r − y2 :
See e.g. Cook (1992) or Logemann and Townley (1997a). The 2rst of these approaches is quite complicated, whilst the second does not make full use of the available information, nor does it account for error in measuring G(0). We adopt a sampled-data integral control approach with 2xed integrator gain but convergent adaptive sampling. Adapting the sampling period allows us to choose the integrator gain based only on knowledge of the steady-state
0005-1098/03/$ - see front matter ? 2002 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 0 2 ) 0 0 1 8 2 - 6
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
136
gain matrix. We show how to select the gain robustly with respect to error in measuring the steady-state gain. We illustrate the results by a simple two-input, two-output example of a di@usion equation with point actuators and spatially distributed sensors. We consider throughout the following class of continuoustime, in2nite-dimensional m-input, m-output systems:
Applying variation of constants to (1), with input given by (2a), gives
x n+1 = T (n )x n + (T (n ) − I )A−1 Bun
(3a)
x(t) ˙ = Ax(t) + Bu(t);
(1a)
un+1 = un + K(r − Cx n ):
(3b)
(1b)
If we apply the change of coordinates
x(0) ∈ X;
y(t) = Cx(t):
In (1), X is a Hilbert space, with inner product ·; · and norm · ; A is the generator of an exponentially stable semigroup T (t); t ¿ 0 on X so that, in particular, there exists M ¿ 1 and w ¿ 0 so that T (t)x 6 M e−wt x for all x ∈ X . The input operator B is potentially unbounded but we assume A−1 B is bounded, whilst we assume that the output operator C is bounded. As is necessary for tracking of constant reference signals, we assume invertibility of the steady-state gain matrix G(0) := −CA−1 B ∈ Rm×m : Remark 1. (a) The class of systems encompassed by (1) is large. Unbounded input operators B are allowed provided that A−1 B is bounded. We need the output operator C to be bounded because the output y(·), which is sampled directly, needs to be continuous. If C was not bounded, then usually the free output y(·) would not be continuous so that sampling would require pre-2lters. We do not persue this aspect here. (b) We emphasize that, whilst our results are valid for a large class of in2nite-dimensional systems, they are new even in the 2nite-dimensional case. 2. Adaptive integral control of multivariable systems by adaptive sampling and based on steady-state data For stable systems given by (1) a non-adaptive, sampled-data integral controller with integrator gain K and, possibly variable, sampling period takes the form u(t) = un
for t ∈ [tn ; tn+1 )
(2a)
un+1 = un + K(r − Cx(tn )):
(2b)
with Here y(tn ) = Cx(tn ) is the sampled output at the sampling time tn . Typically, tn =n where is a 2xed sampling period. Our key idea is to use variable sampling times tn+1 = tn + n
with t0 = 0
and an adaptive sampling period n . This idea is not without precedent. Indeed variable and adaptive sampling has been used in a high-gain adaptive control context, see Owens (1996) and Ilchmann and Townley (1999).
x(tn+1 ) = T (n )x(tn ) + (T (n ) − I )A−1 Bun : Let x n := x(tn ). Then
zn = x n + A−1 Bun
and
v n = u n − ur
with ur := G(0)−1 r to (3) then zn+1 T (n ) − A−1 BKC = vn+1 −KC
−A−1 BKG(0)
I − KG(0)
zn vn
: (4)
We see clearly in (4) how the gain K, the steady-state gain G(0) and the variable sampling period n inKuence the closed-loop dynamics. In the following theorem we show that by choosing K on the basis of knowledge only of the steady-state gain we can achieve asymptotic tracking of constant reference signals for an otherwsie unknown system via convergent adaptation of the sampling period. Theorem 2. Let r ∈ Rm be an arbitrary constant reference signal. De7ne u(t) by (2) with n+1 = n + r − y(tn )2 :
n = tn+1 − tn = f( n );
(5)
In (5) f : [0; ∞) → [0; ∞) is monotone, f(0) = 0 and f( ) → ∞ as → ∞. Let K ∈ Rm×m be chosen so that the zeros of det( ( − I ) + KG(0)) have modulus less than one. If the sampled input u(t) given by (2), with sampling times tn given by (5), is applied to (1), then for each x(0) ∈ X; u0 ∈ Rm and 0 ¿ 0 (a) (b) (c) (d) (e)
limn→∞ n = ∞ ¡ ∞, limn→∞ n = ∞ ¡ ∞, limt→∞ u(t) = ur := G(0)−1 r, limt→∞ x(t) = xr := −A−1 Bur , limt→∞ y(t) = r.
Proof. Let 0 AK := 0
0 I
−
A−1 B I
K(C G(0)):
(6)
AK is the part of the operator in the right-hand side of (4) not depending on n . We claim that AK has spectral radius less one. To see this 2rst observe that since Im(AK ) is 2nite-dimensional AK is a compact operator. Hence !(AK ) consists only of eigenvalues (possibly accumulating at zero). Now let be a non-zero eigenvalue of (AK ). First note that
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
137
= 1 is not possible. To see this, multiply the eigenvalue– eigenvector equation −1 v1 0 0 v1 A B = − K(C G(0)) 0 I v2 v2 I
for all n ¿ N0 . Now Czn + G(0)vn = Cx n − r and using the boundedness of C we can 2nd M ¿ 0 so that
by (C G(0)) on the left. Using CA−1 B + G(0) = 0 this gives Cv1 = 0. But K(Cv1 + G(0)v2 ) = 0 so
Vn+1 − Vn 6 − M r − y(tn )2
KG(0)v2 = 0: Then either v2 = 0 or else det(KG(0)) = 0, neither of which can hold. Indeed, if v2 = 0, then v1 = 0 which is not possible, and det(KG(0)) = 0 does not hold by the assumption on K. So = 1. Next multiplying the eigenvalue–eigenvector equation −1 0 0 v1 v1 A B − = K(C G(0)) 0 I v2 v2 I by K[(1= )C 1=( −1)G(0)] on the left and setting K(Cv1 + G(0)v2 ) = v we obtain 1 −1 1 I +K CA B + G(0) v = 0: (7) −1 Now G(0) = −CA−1 B; G(0) is invertible and v = 0 (since otherwise v1 = 0 so that v1 = 0 and (1 − )v2 = 0 so that v2 = 0). Then (7) implies that det( ( − I ) + KG(0)) = 0: By the choice of K this shows that | | ¡ 1 as required. Hence, the spectral radius of AK is less than one and AK is power stable. Therefore, we can 2nd P = P ∗ ¿ 0 so that A∗K PAK − P = −I: Let Vn =
zn vn
;P
zn vn
zn 2 + vn 2 ¿ 2M r − Cx n 2 : This gives which, using (5), yields Vn+1 − Vn 6 − M ( n+1 − n )
for all n ¿ N0 :
It follows that 1 N 6 N0 + VN ¡ ∞: M 0 This proves that n is bounded and therefore converges to ∞ , and consequently n converges to ∞ . This proves (a) and (b). Then (5) implies that {r − Cx n } = {Czn + G(0)vn } ∈ ‘2 . Now zn+1 = T (n )zn − A−1 BK(Czn + G(0)vn ) = : T (n )zn + wn ;
(8)
−1
where wn = −A BK(Czn + G(0)vn ). Clearly, {wn } ∈ ‘2 . Taking norms in (8) gives n
zn+1 6 M e−wtn+1 z0 + M (e−w1 )n−i wi : i=0
In the last inequality we used the fact that n =f( n ), with n non-decreasing, gives n ¿ 1 for all n ¿ 1. So, {zn }∞ n=0 is an ‘2 -sequence plus a convolution of an ‘1 and an ‘2 sequence. It follows that zn ∈ ‘2 so that lim zn = 0:
(9)
n→∞
Then (Czn + G(0)vn ); zn ∈ ‘2 and invertibility of G(0) implies limn→∞ vn =0, which using the de2nition of vn gives limn→∞ un = ur and so (c) holds. Then (9) gives
:
Then computing Vn+1 − Vn along solutions of (4) gives 2 zn Vn+1 − Vn = − vn zn T (n )zn +2 ; PAK 0 vn T (n )zn T (n )zn + ;P : 0 0 Suppose that n is not bounded. Then n = f( n ) is not bounded and so for large enough n, the second and third terms are small. More precisely, it is easy to show that there exists N0 ∈ N so that for all n ¿ N0 , T (n ) is small enough to give Vn+1 − Vn 6 − 12 (zn 2 + vn 2 )
lim x n = −A−1 Bur = xr :
n→∞
For t ∈ (tn ; tn+1 ) we have x(t) = T (t − tn )x n + (T (t − tn ) − I )A−1 Bun ; y(t) = Cx(t): It follows that lim x(t) = xr
t→∞
and
lim y(t) = r;
t→∞
so that (d) and (e) hold and the proof is complete. Example 3. Consider a di@usion process (with di@usion coe
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
We suppose a two-dimensional output obtained by spatially distributed sensing on *-neighbourhoods of two points xc1 ; xc2 ∈ (0; 1). Speci2cally, the output is given by: xc +* xc1 +* 2 1 z(t; x) d x; z(t; x) d x : y(t) = 2e xc1 −* xc2 −* With these inputs u(t) ∈ R2 and outputs y(t) ∈ R2 , this diffusion system can be represented in the form of (1). Now the form of the transfer function G(s) for this system, and so also the steady-state gain G(0), depends on the relative positions of the actuators (xb1 ; xb2 ) and sensors (xc1 ; xc2 ). Signi2cantly in this context of integral control, G(0) is only invertible for certain relative positions of xb1 ; xb2 ; xc1 and xc2 . If xb1 ¡ xb2 ¡ xc1 ¡ xc2 , then G(0) is singular. If, however, xb1 ¡ xc1 − *;
xc1 + * ¡ xc2 − *;
3.5 Output Y1 (.) Output Y2 (.)
3
Output Y (tn)
138
2.5 2 1.5 1 0.5 0 0
20
40
xc2 + * ¡ xb2
then G(0) is invertible with 1 xb1 (1 − xc1 ) xc1 (1 − xb2 ) : G(0) = a xb1 (1 − xc2 ) xc2 (1 − xb2 )
60
80
100
120
Time tn Fig. 1. Output y(tn ), against tn , for the system described in Example 3.
2
For purposes of illustration, we adopt the following values: xb1 = 14 ;
xc2 = 58 ;
* = 0:01:
This gives 1 G(0) = 32
xb2 = 34 ;
50
30
30
50
xc1 = 38 ;
:
In this example we assume that G(0) is known and choose the gain K so that the zeros of det( ( − 1) + KG(0)) have modulus less than one. One choice is 0:2728 −0:1654 K= : −0:1654 0:2728 In the simulations we assume zero steady-state initial conditions for z and un , a stepped-reference (1; 2)T t ¡ 45; r(t) = (2; 3)T t ¿ 45
and an adaptive sampling period n = log( n + 1) with 0 = 1. In Fig. 1 we show the plot of the two outputs and in Fig. 2 the corresponding two inputs. In this simulation
the adaptive sampling period n = log( n + 1) converges to 1.774. In producing the simulations we used MATLAB and a truncated eigenfunction approximation of order 10 to model the di@usion process. In practice, the steady-state gain G(0) is obtained by step response experiments. Typically, this will lead to approximate measurements of G(0), and the value of G(0) available for design purposes will be a perturbation of the true value. This uncertainty in the value of G(0) would be due to measurement noise or else to the use of 2nite-time, as opposed to steady-state experiments.
1.5 Input U ( tn)
a = 0:1;
Input U1(.) Input U2(.)
1
0.5
0
-0.5 0
20
40
60 Time tn
80
100
120
Fig. 2. Input un applied, via idealised hold to the system described in Example 3.
Denote the measured value of G(0) by Gexpt (0). It is not our intention to develop a full theory. To simply illustrate what could be achieved consider the case where G(0) = Gexpt (0) + , and , is unknown but , ¡ ) for some ) ¿ 0. This is a set-up of the so-called stability radius, see Hinrichsen and Pritchard (1986). Of course, in this context of integral control it is necessary that G(0) = Gexpt (0) + , is invertible for all , ¡ ). This means that necessarily ) 6 Gexpt (0)−1 −1 : In Theorem 2 we chose K so that 0 I 0 + ,(I 0) −KGexpt (0) I K
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
is Schur. It is easy to see that this is guaranteed if 1 : , ¡ min |z|=1 (Gexpt (0) + z(z − 1)K −1 )−1
3.5 Output Y1 (.) Output Y2 (.)
3
In order to allow for the maximum experimental error (i.e maximum ) ¿ 0) we choose K to maximise the right-hand side of (10). Now clearly for any K the right-hand side of (10) is not greater than Gexpt (0)−1 −1 (just choose z = 1). Hence, the maximum possible ) ¿ 0 satis2es
2.5
Output Y (tn)
(10)
2 1.5 1
max min (Gexpt (0) + z(z − 1)K −1 )−1 −1 K
139
|z|=1
0.5
6 Gexpt (0)−1 −1 :
0
In fact, equality is achieved in the above by a large class of integrator gains K ∈ Rm×m .
0
20
40
60 Time tn
80
100
120
Fig. 3. Output y(tn ) for the system described in Example 5.
Theorem 4. 1 1 = −1 −1 ) Gexpt (0)−1 |z|=1 (Gexpt (0) + z(z − 1)K
max min K
and K achieves the maximum if K −1 = Gexpt (0)H;
H = HT ¿ 0
2
and
min (H ) ¿ 3: 1.5
This result is proved in the appendix.
so that in this case G(0) − Gexpt (0) = 0:0156 and −1 = 0:639. Hence Theorem 4 applies. We 1=Gexpt choose H = 4:5I; so 0:2186 −0:1303 −1 K = (Gexpt (0)H ) = : −0:1303 0:2186
As in Example 3 we use f( ) = log( + 1). In the simulations we assume zero steady-state initial conditions for z, un with r(t) as in Example 3 and 0 = 1. In Figs. 3 and 4 we show the plot of the output and corresponding input. In this simulation, the adaptive sampling period n converges to 1.801. Comparing Fig. 1 with 3, we see that both the response times and overshoot are similar even though the latter is obtained from an algorithm using only approximate steady-state gain data.
Input U (tn)
Example 5. Let us reconsider Example 3. This time we assume that G(0) can only be obtained from steady-state experiments so that K has to be designed on the basis of an experimentally measured steady-state gain Gexpt (0). To illustrate the e@ectiveness of our controller in coping with such experimental error in measuring G(0) we simulate experimental conditions by using for Gexpt (0) the steady-state gain for an eighth order truncated eigenfunction approximation of the di@usion process. This leads to a measured steady-state gain 1:576 0:939 Gexpt = ; 0:939 1:576
Input U1 (.) Input U2 (.)
1
0.5
0
-0.5 0
20
40
60
80
100
120
Time tn Fig. 4. Input un for the system described in Example 5.
3. Conclusions We have considered sampled-data integral control for a large class of in2nite dimensional systems. Motivated by existing results in the literature we focused on the following aspects: Simple robust use of readily available steady-state data and adaptation of controller parameters. The parameter we chose for adaptation was the sampling period. Using the sampling period in this way allowed us to determine, quite simply, appropriate integrator gains based on knowledge of step response data. We investigated robustness of the choice of K with respect to uncertainty in experimental measurement of the steady-state gain.
( N. Ozdemir, S. Townley / Automatica 39 (2003) 135 – 140
140
Appendix. Proof of Theorem 4. We know that 1 1 max min 6 : K |z|=1 (Gexpt (0) + z(z − 1)K −1 )−1 Gexpt (0)−1 Therefore, to show equality we need to 2nd K so that for all z with |z| = 1 1 1 ¿ −1 −1 (Gexpt (0) + z(z − 1)K ) Gexpt (0)−1 or equivalently (Gexpt (0) + z(z − 1)K −1 )−1 6 Gexpt (0)−1 : Now using K
−1
(A.1)
= Gexpt (0)H and estimating we have
(Gexpt (0) + z(z − 1)K −1 )−1 −1 (0)(z(z − 1)H + I )−1 : 6 Gexpt
So (A.1) holds if (z(z − 1)H + I )−1 6 1
for all z
with |z| = 1: (A.2)
Now (A.2) holds if z(z − 1)H + z( M zM − 1)H T + z(z − 1)z( M zM − 1)HH T ¿ 0 (A.3) for all z with |z| = 1. But H = H T ¿ 0. So we can choose / with /T / = I , so that /T H/ = D and D := diag(di )m i=1 . Then (A.3) becomes z(z − 1)di + z( M zM − 1)di + z(z − 1)z( M zM − 1)d2i ¿ 0:
(A.4)
Now z = ei1 , for 1 ∈ [0; 22] so that Re
z ei1 cos 21 − cos 1 = Re −i1 = zM − 1 e −1 2 − 2 cos 1
(2 cos 1 + 1) : 2 Then (A.4) becomes di −(2 cos 1+1) ¿ 0 for all 1 ∈ [0; 22] and i = 1; : : : ; m. This is true if di ¿ 3. So we need −1 min (H ) ¿ 3 and so K = H −1 Gexpt (0) achieves the maximum if H ¿ 3I . =−
References Cook, P. A. (1992). Controllers with universal tracking properties. Proceedings of the international IMA conference on control: modelling, computation, information, Manchester. Davison, E. J. (1976). Multivariable tuning regulators: The feedforward and robust control of a general servomechanism problem. IEEE Transactions on Automatic Control, 21, 35–47. Hinrichsen, D., & Pritchard, A. J. (1986). Stability radius for structured perturbations and the algebraic Riccati equation. Systems and Control Letters, 8, 105–113. Ilchmann, A., & Townley, S. (1999). Adaptive sampling control of high-gain stabilizable systems. IEEE Transactions on Automatic Control, 44, 1961–1966.
Logemann, H., & Ryan, E. P. (2000). Time-varying and adaptive integral control of in2nite-dimensional regular systems with input nonlinearities. SIAM Journal of Control and Optimization, 38, 1940–1961. Logemann, H., Ryan, E. P., & Townley, S. (1998). Integral control of in2nite-dimensional linear systems subject to input saturation. SIAM Journal of Control and Optimization, 36, 1940–1961. Logemann, H., Ryan, E. P., & Townley, S. (1999). Integral control of linear systems with actuator nonlinearities: Lower bounds for the maximal regulating gain. IEEE Transactions on Automatic Control, 44, 1315–1319. Logemann, H., & Townley, S. (1997a). Low-gain control of uncertain regular linear systems. SIAM Journal of Control and Optimization, 35, 78–116. Logemann, H., & Townley, S. (1997b). Discrete-time low-gain control of uncertain in2nite-dimensional systems. IEEE Transactions on Automatic Control, 42, 22–37. Logemann, H., & Townley, S. (2001). Adaptive low-gain integral control of multivariable well-posed linear systems. SIAM Journal of Control and Optimization, to appear. Lunze, J. (1985). Determination of robust multivariable I-controllers by means of experiments and simulation. System Analysis Modelling Simulation, 2, 227–249. Morari, M. (1985). Robust stability of systems with integral control. IEEE Transactions on Automatic Control, 30, 574–577. Owens, D. H. (1996). Adaptive stabilization using a variable sampling rate. International Journal of Control, 63, 107–119. ' Ozdemir, N. (2000). Robust and adaptive sampled data I—control. Ph.D. thesis, University of Exeter, UK. ' Ozdemir, N., & Townley, S. (1998). Adaptive low-gain control of in2nite dimensional systems by means of sampling time adaptation. Methods and models in automation and robotics, Miedzyzdroje, Poland (pp. 63– 68).
Necati Ozdemir was born in Balikesir, Turkey on 23rd July 1970. He received the B.Sc. degree in mathematics from Marnara University, Turkey in 1992, the M.Sc. degree in mathematics from the University of Warwick, UK, in 1996, and the Ph.D. degree in mathematics from the University of Exeter, UK, in 2000. He is currently a lecturer in mathematics at the University of Balikesir, Turkey. His research interests are in Functional Analysis and Sampled-data Control.
Stuart Townley was born in Widnes, UK on 2nd September 1961. He received the B.Sc. degree in mathematics, 1983, and the Ph.D. in engineering, 1987, both from the University of Warwick. After post doctoral fellowships in mathematics at the University of Warwick (1986 –1988) and University of Bath (1988–1990), he was appointed to a Lectureship in mathematics at the University of Exeter. He was promoted in 1996 to a Readership in Dynamics and Control, and in 1999 to Professor of Applied Mathematics. His research interests are in Robust Control, Adaptive Control, In2nite-dimensional systems, Stochastic Stability and Nonlinear Dynamics. He has held numerous UK Research Council and EU-funded grants and has supervised several Ph.D. students. He currently serves on the editorial board of Dynamics and Control.