Adaptive internal model control: the discrete-time case

INTERNATIONAL JOURNAL OF ADAPTIVE CONTROL AND SIGNAL PROCESSING Int. J. Adapt. Control Signal Process. 2001; 15:15}36

Adaptive internal model control: the discrete-time case Guillermo J. Silva and Aniruddha Datta*R Department of Electrical Engineering, Texas A & M University, College Station, TX 77843-3128, U.S.A.

SUMMARY This paper considers the design and analysis of a discrete-time H optimal robust adaptive controller based  principle of adaptive control is used to on the internal model control structure. The certainty equivalence combine a discrete-time robust adaptive law with a discrete-time H internal model controller to obtain  a discrete-time adaptive H internal model control scheme with provable guarantees of stability and  robustness. The approach used parallels the earlier results obtained for the continuous-time case. Nevertheless, there are some di!erences which, together with the widespread use of digital computers for controls applications, justi"es a separate exposition. Copyright  2001 John Wiley & Sons, Ltd. KEY WORDS:

internal model control; discrete time; H -optimal control; certainty equivalence principle 

1. INTRODUCTION The internal model control (IMC) structure, in which the controller implementation includes an explicit plant model in parallel with the plant, is a very popular one in process control applications [1}3]. The plants encountered in process control are usually open-loop stable and for such plants, the IMC structure provides a convenient paradigm for not only ensuring the stability of the controlled plant, but also for facilitating the easy adjustment of design parameters to address the typical tradeo!s that are encountered in control system design. Furthermore, as discussed in References [1, 2], the IMC structure is at the heart of several industrially popular model predictive control schemes such as model algorithmic control (MAC) [4] and dynamic matrix control (DMC) [5]. Inherent in the IMC structure is the assumption that a plant model is available to be used as part of the controller. However, in many instances no such model may be available a priori or the

* Correspondence to: Aniruddha Datta, Department of Electrical Engineering, Texas A & M University, College Station, TX 77843-3128, U.S.A. R E-mail: [email protected]

Copyright  2001 John Wiley & Sons, Ltd.

Received 18 September 1998 Revised 25 February 2000 Accepted 12 April 2000

16

G. J. SILVA AND A. DATTA

plant characteristics may change with time due to aging, etc. and this is where adaptation, learning, etc. become important within the context of internal model control. The type of approach used to construct a plant model on-line will very much depend on the assumptions made about the unknown plant to be controlled. For instance, if the nominal plant is assumed to be single-input single-output (SISO), linear, time-invariant and "nite-dimensional but with unknown parameters, then techniques from adaptive parameter estimation can be used to obtain a plant model on-line which can then be used to design and implement the IMC scheme. There are several instances of such adaptive IMC schemes in the literature [6, 7]. However, these schemes are all empirically designed and do not provide any guarantees of stability, robustness, etc. beyond what is observed in simulations. Since the theory of parameter adaptive control did make signi"cant strides during the same period (see Reference [8] and the references therein), one is led to the unavoidable conclusion that these advances in adaptive system theory did not "nd their way into the realm of industrial applications. In an e!ort to narrow down this gap between the theory and applications of adaptive control, we recently focussed our attention on the design and analysis of adaptive IMC schemes for SISO, linear, time-invariant, continuous-time plants, with nominal parts that are "nite-dimensional [9]. By exploiting and adapting existing continuous-time results from the adaptive control literature, it was shown that continuous-time adaptive IMC schemes could indeed be designed for SISO plants to provide theoretically provable guarantees of stability and robustness. A comprehensive and tutorial exposition of this work and other related results for the continuous-time case can be found in Reference [10]. In this paper, we consider the analogous results for the discrete-time case. Given the widespread use of digital computers and microprocessors for controls applications, such a study is certainly warranted. For clarity of presentation, we will speci"cally focus on a discrete-time adaptive design based on pointwise minimizing an H performance index. However, as in the  continuous-time case [10], the results can be extended in a straightforward fashion to handle other adaptive IMC designs such as those based on pointwise pole placement or pointwise model reference control. In the context of discrete-time adaptive IMC, it is appropriate to refer to the earlier work of Ydstie [11], who used the IMC implementation for his extended horizon adaptive control (EHAC) scheme. To the authors' knowledge, this is the only instance in the literature where an attempt was made to theoretically establish the stability of an adaptive IMC scheme. This attempt was somewhat successful in the sense that ideal-case stability was established by exploiting the parameter convergence property (not to the true values) of the &pure' least-squares algorithm, in the absence of modelling errors. However, signi"cant advances in adaptive control since then have now made it possible to design and analyse discrete-time control schemes in a systematic fashion and in the presence of modelling errors. Our objective in this paper will be to present an example of such a design and analysis. The paper is organized as follows. In Section 2, we present a known result on the design of a discrete-time H -optimal controller for a continuous-time stable plant when the parameters of  the plant are known [3]. The resulting controller structure is then combined with a discrete-time robust adaptive law in Section 3 to obtain a discrete-time robust adaptive H -optimal control  scheme. Section 4 contains the stability and robustness analysis for this scheme. In Section 5, we present simulation examples to demonstrate the e$cacy of the proposed adaptive IMC designs. Section 6 concludes the paper by summarizing the main results and outlining the topics for future research. Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

17

Figure 1. The IMC con"guration.

2. DISCRETE-TIME H OPTIMAL INTERNAL MODEL CONTROL: KNOWN  PARAMETERS In this section, we consider the problem of designing a discrete-time H -optimal controller for  a continuous-time stable plant by making use of the IMC structure. To this end, we consider the discrete-time IMC con"guration for a stable plant P(s) as shown in Figure 1. Here h (s) denotes M a zero-order hold and c(s) is an analog anti-aliasing pre-"lter. Now, if we denote the sampling time of the discrete-time system by ¹, then the continuous plant P(s) can be discretized as follows: P(z)"ZL\+h (s)P(s)c(s), M where Z, L denote the z and Laplace transform operations, respectively. The role of the stable pre-"lter c(s) is to cut o! high-frequency components from the analog signals before sampling, whenever that is necessary to avoid aliasing. The IMC Controller consists of a stable discretetime &IMC parameter' Q(z) and a discrete-time model P(z) of the plant which is usually referred to as the &internal model'. It can be shown [3, 9] that if the plant P(s) is stable and the internal model is an exact replica of the plant (along with the zero-order hold and the anti-aliasing pre-"lter), then the stability of the IMC parameter Q(z) is equivalent to the internal stability of the con"guration in Figure 1. Indeed, the IMC parameter is really the discrete-time Youla parameter [12] that appears in a special case of the YJBK parametrization of all stabilizing controllers [9]. Because of this, internal stability is assured as long as Q(z) is chosen to be any Schur stable rational transfer function. Thus by restricting ourselves to the set of all Schur stable rational transfer functions, Q(z) could be chosen to minimize a given performance index. As a particular example, one can choose Q(z) to minimize the l -norm of the tracking error r(k)!y(k), provided  r-y3l . This leads to an H optimization problem as shown below.   Now, by de"nition, the tracking error e(k) is given by e(k)"r(k)!y(k) N e(k)"(1!P(z)Q(z))[r(k)] (from Figure 1) Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

18

G. J. SILVA AND A. DATTA

 1 L N "e(k)"" "[1!P(e U)Q(e U)]R(e U)" dw (using Parseval's theorem) 2n \L I



N #e# "#[1!P(z)Q(z)]R(z)#   where R(z) denotes the z-transform of r(k) and #.(z)# denotes the standard H norm of   a discrete-time transfer function. Thus, minimizing the l norm of e(k) is mathematically equi valent to choosing Q(z) to minimize #[1!P(z)Q(z)]R(z)# . The following theorem gives  the analytical expression for the minimizing Q(z). The detailed derivation can be found in Reference [3]. ¹heorem 2.1. Let P(z) be the discrete-time stable plant to be controlled and let R(z) be the z-transform of the external input signal r(k). Let a , j"1,2, h be the zeros of P(z) outside the unit circle and de"ne H the Blashcke product* F [1!(a )\](z!a ) H H P (z)"z\, “  (1!a )[z!(a )\] H H H so that P(z) can be rewritten as P(z)"P (z) P (z)  + where P (z) is minimum phase, and the positive integer N is chosen so that P (z) is semi-proper, + + i.e. its numerator and denominator have the same degree. Similarly, let b , j"1,2, h be the zeros H P of R(z) outside the unit circle and de"ne the Blashcke product h P [1!(b )\](z!b ) H H R (z)"z!NP “  (1!b )[z!(b )\] H H H

so that R(z) can be rewritten as R(z)"R (z)R (z)  + where R (z) is minimum phase, and the positive integer N is chosen so that R (z) is semi-proper. + P + Then the Q (z) which minimizes #[1!P(z)Q(z)]R(z)# is given by &  Q (z)"z[P (z)R (z)]\[z\P\ (z)R (z)] & + +  + *

(1)

where [.] denotes that after a partial fraction expansion, only the strictly proper and stable * (including poles at z"1) terms are retained.

* Here (.) denotes complex conjugation. Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

19

Remark 2.1. In the formulation of the discrete-time H optimal control problem no attention was paid to  the intersample behaviour. As pointed out in Reference [3], experience has shown that the H  optimal controller can lead to unacceptable intersample rippling. Thus, a modi"cation is necessary to obtain an acceptable Q(z). The intersample rippling produced by the optimal Q (z) in (1) is caused by the presence of poles & close to (!1, 0)[3]. Due to this fact, one can modify the optimal Q (z) to obtain & Q(z)"Q (z)Q (z)B(z) & !

(2)

Here Q (z) cancels all the poles of Q (z) with negative real parts and substitutes them with poles ! & at the origin while B(z) is a transfer function selected to preserve the system type. If we let i , G i"1,2, o be the poles of Q (z) with negative real parts, then a possible choice for Q (z) is & ! M z!i H Q (z)"z\M “ ! 1!i H H

(3)

If, in addition, the system is of Type 1,* then, a possible choice for B(z) is B(z)"1. Remark 2.2. The discrete-time IMC controller Q(z) in (2) is usually augmented with a discrete-time IMC "lter F(z), primarily for the purpose of achieving an optimal compromise between the con#icting design objectives of performance and robustness [3]. The design of such a "lter is carried out in an ad hoc manner (subject to preserving the system type) and does not qualitatively a!ect the design or analysis of the corresponding adaptive schemes. Thus, for clarity of presentation, in this paper we will omit the "lter F(z). It should, of course, be pointed out that in the continuous-time case considered in Reference [10], such a choice was not possible since otherwise the optimal Q turned out to be improper.

3. ADAPTIVE H OPTIMAL INTERNAL MODEL CONTROL  In order to implement the IMC-based H -optimal controller of the last section, the plant must be  known a priori so that the &internal model' can be designed and the IMC parameter Q(z) calculated. When the plant itself is unknown, the IMC-based controller cannot be implemented. In this case, the natural approach to follow is to retain the same controller structure as in Figure 1, with the internal model being adapted on-line based on some kind of parameter estimation mechanism and the IMC parameter Q(z) being updated to pointwise solve the discrete-time H -norm minimization problem for the estimated internal model. This is the standard certainty  equivalence approach of adaptive control and results in what is called a discrete-time Adaptive

* Other system types can also be handled as in Reference [3]. Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

20

G. J. SILVA AND A. DATTA

H -optimal internal model control scheme. Although certainty equivalence adaptive IMC  schemes have been empirically studied in the literature [6, 7], our objective here is to develop a discrete-time adaptive H -optimal IMC scheme with provable guarantees of stability and  robustness. To this end, we assume that the stable discrete-time plant to be controlled is described by the transfer function Z (z) P(z)"  #k* (z) ? R (z) 

(4)

where R (z) is a monic Schur polynomial of degree n, Z (z) is a polynomial of degree l(n,   Z (z)/R (z) represents the modelled part of the plant, k* (z) is a strictly proper stable additive   ? uncertainty, and k'0 is a small positive constant. The robust discrete-time adaptive H -optimal  IMC scheme is obtained by combining a robust discrete-time adaptive law with the IMC scheme considered in the last section. Let us now present the design of the robust adaptive law. 3.1. Design of the robust adaptive law We start with the plant equation



y(k)"



Z (z)  #k* (z) [u(k)], k'0 ? R (z) 

(5)

where u(k), y(k) are the plant input and output signals, respectively. This equation can be rewritten as R (z)[y(k)]"Z (z)[u(k)]#k* (z)R (z)[u(k)]   ?  Let "(z) be an arbitrary monic Schur polynomial of degree n. Filtering both sides of the previous equation by 1/"(z) we obtain "(z)!R (z) Z (z) k* (z)R (z)  [y(k)]#  [u(k)]# ?  [u(k)] y(k)" "(z) "(z) "(z)

(6)

The above equation can be rewritten as y"h*2 #kg

(7)

where h*"[h*2, h*2]2, h*, h* are vectors containing the coe$cients of ["(z)!R (z)] and Z (z),       respectively; "[ 2 , 2]2, "(a (z)/"(z))[y], (a (z)/"(z))[u],    L\  J a (z)"[zL\, zL\,2, 1]2 L\ a (z)"[zJ, zJ\,2, 1]2 J Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

21

* (z)R (z)  [u] gO ? "(z)

(8)

and

Equation (7) is in the form of the linear parametric model with modelling error for which a large class of robust adaptive laws can be developed. In particular, we will consider the dynamically normalized gradient adaptive law with parameter projection [13] h(k#1)"Pr[h(k)#ce(k) (k)], h(0)3C

(9)

y(k)!yL (k) e(k)" m(k)

(10)

yL (k)"h2(k) (k)

(11)

m(k)"m (k)# 2(k) (k) Q

(12)

m (k#1)"d (m (k)!1)#u(k)#y(k)#1, m (0)"1 Q  Q Q

(13)

where c3(0, 0.5) is an adaptive gain; C is a known compact convex set in RL>J> containing h*; Pr[)] is the standard discrete-time projection operator which guarantees that the parameter estimate h(k) does not exit the set C and d 3(0, 1) is a constant chosen so that * (z), 1/"(z) are  ? analytic in "z"*(d . This choice of d , of course, necessitates some a priori knowledge about the   stability margin of the unmodelled dynamics, an assumption which has by now become fairly standard in the robust adaptive control literature [8]. The discrete-time robust adaptive H -optimal IMC scheme is obtained by replacing the  internal model in Figure 1 by that obtained from Equation (11), and the IMC parameter Q(z) by the time-varying operator which implements the certainty equivalence version of the controller structure considered in the last section. The design of this certainty equivalence controller is discussed next. 3.2. Design of the certainty equivalence control law We "rst outline the steps involved in designing a general certainty equivalence adaptive IMC scheme. Thereafter, the additional requirements that must be satis"ed while using an H optimal  control law will be discussed. Step 1. First use the parameter estimate h(k) obtained from the robust adaptive law (9)}(13) to generate estimates of the numerator and denominator polynomials for the modelled part of the plant:* ZK (z, k)"h2 (k)a (z)   J RK (z, k)""(z)!h2 (k)a (z)   L\ * In the rest of this paper, the &hats' denote the time-varying polynomials/frozen time &transfer functions' that result from replacing the time-invariant coe$cients of a &hat-free' polynomial/transfer function by their corresponding time-varying values obtained from adaptation and/or certainty equivalence control. Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

22

G. J. SILVA AND A. DATTA

Figure 2. Robust adaptive IMC scheme.

Step 2. Using the frozen time modelled part of the plant PK (z, k)"ZK (z, k)/RK (z, k), calculate    the appropriate QK (z, k) using the non-adaptive results such as the ones developed in Section 2 for H optimal control.  Step 3. Express QK (z, k) as QK (z, k)"QK (z, k)/QK (z, k) where QK (z, k) and QK (z, k) are time-varying L B L B polynomials with QK (z, k) being monic and of degree n . B B Step 4. Choose " (z) to be an arbitrary monic Schur polynomial of degree equal to n .  B Step 5. The certainty equivalence control law is given by an !1 (z) an (z) u"q2 (k) B [u]#q2 (k) B [r!em] B L " (z) " (z)  

(14)

where q (k) is the vector of coe$cients of " (z)!QK (z, k); q (k) is the vector of coe$cients of B  B L QK (z, k); L an (z)"[znB, znB!1,2, 1]2 B

and an !1 (z)"[znB!1, znB!2,2, 1]2 B

The robust adaptive IMC scheme that results from combining the control law (14) with the robust adaptive law (9)}(13) is schematically depicted in Figure 2. We now proceed to discuss the additional requirements that must be met for an adaptive H design. First, for adaptive H optimal control, QK (z, k) in Step 2 of the certainty equivalence   design is obtained by replacing P (z), P (z) on the right-hand side of (2) with PK (z, k), PK (z, k), i.e. +  +  QK (z, k)"z[PK (z, k)R (z)]\[z\PK \ (z, k)R (z)] QK (z, k)BK (z, k) + +  + * ! Copyright  2001 John Wiley & Sons, Ltd.

(15)

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

23

where PK (z, k) is the Blaschke product containing the zeros of ZK (z, k) outside the unit circle,   PK (z, k) is the minimum phase portion of ZK (z, k)/RK (z, k); QK (z, k), BK (z, k) play similar roles as +   ! Q (z), B(z) in (2); and [.] denotes that after a partial fraction expansion, only the strictly proper ! * and stable (including poles at z"1) terms are retained. Now, our stability analysis to be presented in the next section requires that QK (z, k) be pointwise stable. For this to be possible, it is clear from (15) that for every "xed k, ZK (z, k) should not have  any zeros on the unit circle. To guarantee such a property for ZK (z, k), the projection set C in  (9)}(13) is chosen so that ∀h3C, the corresponding Z (z)"h2 a (z) has no zeros on the unit circle.   J By restricting C to be a subset of a Cartesian product of closed intervals, discrete-time analogues of known results from parametric robust control e.g. Reference [14] can be used to ensure that C satis"es such a requirement. Also, when the projection set C cannot be speci"ed as a single convex set, discrete-time analogues of known results from hysteresis switching using a "nite number of convex sets [15] can be used. We next come to the second requirement that has to be met for an adaptive H design. As will  be seen in the next section, the stability analysis requires that the leading coe$cient of ZK (z, k) be  not allowed to pass through zero. This feature can be built into the adaptive law by assuming some knowledge about the sign and a lower bound on the absolute value of the leading coe$cient of Z (z). Projection techniques, appropriately utilizing this knowledge, are by now standard in  the adaptive control literature [8] and we will, therefore, assume that for IMC-based H -optimal  adaptive control, the set C has been suitably chosen to guarantee that the estimate h(k) obtained from (9)}(13) actually satis"es both of the required properties discussed above. Remark 3.1. The actual construction of the set C to satisfy the two properties may not be straightforward especially for higher-order plants. However, this is a well-known problem that arises in implementing any certainty equivalence adaptive control scheme based on the estimated plant model and is really not a drawback speci"cally associated with the IMC design methodology. Although from time to time, a lot of possible solutions to this problem have been proposed in the adaptive literature, it would be fair to say that, by and large, no completely satisfactory solution is currently available.

4. STABILITY AND ROBUSTNESS ANALYSIS In this section, we analyse the stability and robustness properties of the adaptive IMC scheme presented in the last section. We "rst introduce some de"nitions [13]. De,nition 4.1. For any signal x: Z>PRL, x denotes the truncation of x to the discrete-time interval [0, k] I and is de"ned as



x(i) if i)k, x (i)" I 0 otherwise Copyright  2001 John Wiley & Sons, Ltd.

(16)

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

24

G. J. SILVA AND A. DATTA

De,nition 4.2. For any signal x: Z>PRL, and for any 0)d)1, k*0, #x #B is de"ned as I 






I  #x #B " dI\Gx2 (i)x(i) I  G

(17)

The #( . ) #B is an exponentially weighted l norm of the sequence x(i) truncated to 0)i)k. Note I   that when d"1 and k"R, #( . ) #B becomes the usual l norm and will be denoted by #( . )# . It I    can be shown that #( . ) #B satis"es the usual properties of the vector norm. I  De,nition 4.3. Consider the sequences x: Z>PRL, y: Z>PR> and the set



D(y)" x:Z>PRL



k #N!1 



k"k 

x2(k)x(k))c

k #N!1 





k"k



y(k)#c 



for some c , c *0 and ∀k *0, N*1. We say that x is y-small in the mean if x3D(y).    We now present some lemmas and a theorem that can be used for the design and the analysis of discrete-time adaptive control schemes. The proofs of these results are analogous to their continuous-time counterparts presented in Reference [8] and are, therefore, omitted. Consider a linear time-invariant (LTI) causal system described by the discrete-time convolution of two sequences u, h : Z>PR, i.e. I y(k)"u * hO h(k!i)u(i) G

(18)

where u, y are the input, output signals, respectively, of the system and h is its impulse response. Let H(z) be the z-transform of the input}output operator h(.), i.e. H(z) is the transfer function associated with the system. Then we can state the following two lemmas. ¸emma 4.1. If h3l , then u3D(k) implies that y3D(k) and y3l for any "nite k*0.   ¸emma 4.2. Let H(z) in (18) be a proper rational transfer function. If H(z) is analytic in "z"*(d for some d3(0, 1] and u3l then ∀k*0 C #y #B )#H(z)#B #u #B I   I 

(19)

where #H(z)#B O sup "H((de U)".  w3[0, 2n] Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

25

If, in addition, H(z) is strictly proper then "y(k)")#zH(z)#B #u #B  I\ 

(20)

where 1 #H(z)#B O  (2n



L



"H((de U)" dw



¸emma 4.3. Consider the linear time-varying system given by x(k#1)"A(k)x(k)#B(k)u(k), x(0)"x  y(k)"C(k)x(k)#D(k)u(k)

(21)

where x3RL, y3RP, u3RK, and the matrices A, B, C, D3l . If the state transition matrix '(k, j)  of (21) satis"es #'(k, j)#)j dI\H, ∀k'j   for some j '0, d 3[0, 1) and u3l , then for any d3(d , 1) we have   C  j #x #B ) #u #B #e I  (1!d (d(d!d ) I  I   where e is an exponentially decaying to zero term due to x and j"cj , where c is the l bound I    for B(k). ¹heorem 4.1. Consider the linear time-varying system x(k#1)"A(k)x(k)

(22)

Let the elements of A(k) be bounded sequences and assume that A1. "j (A(k))")p ∀k*0 and for i"1, 2,2, n where p 3(0, 1) is some constant. G Q Q Furthermore, de"ne *A(k)OA(k#1)!A(k). If any one of the following conditions #K!1  (i) kk"k #A(k#1)!A(k)#)kK#a , i.e. (#*A#)3D(k), or   k #K!1  (ii) k"k #A(k#1)!A(k)#)kK#a , i.e. #*A#3D(k), or   (iii) #A(k#1)!A(k)#)k, or (iv) #A(k#1)!A(k)#3l  where a 3R> and k*0 is satis"ed ∀k *0 and ∀K*1, then there exists a k*'0 such that   ∀k3[0, k*), the equilibrium state x "0 of (22) is uniformly asymptotically stable (u.a.s.) in the C large.

Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

26

G. J. SILVA AND A. DATTA

The following lemma plays a key role in the stability and robustness analysis of the H -optimal  discrete-time adaptive control scheme of this paper. Its proof relies heavily on the requirement that the leading coe$cient of ZK (z, k) be not allowed to pass through zero.  ¸emma 4.4. Suppose that the leading coe$cient of ZK (z, k) is non-zero for every k. Then at any "xed time k,  the coe$cients of QK (z, k) and QK (z, k) are continuous functions of the estimate h(k). B L Proof. Since the leading coe$cient of ZK (z, k) is not allowed to pass through zero, then, for any  "xed time k, the roots of ZK (z, k) are continuous functions of h(k). Hence, the coe$cients of the  numerator and denominator of the function [PK (z, k)]\"[PK (z, k)][PK (z, k)]\ +   are continuous functions of h(k). Moreover, [z\(PK (z, k))\R (z)] is the sum of the residues of  + * z\(PK (z, k))\R (z) at the poles of R (z) and possibly at the origin, which clearly depends  + + continuously on h(k) (through the factor [PK (z, k)]\). Thus the numerator and denominator  polynomial coe$cients of QK (z, k) depend continuously on h(k). Now, from (3) we know that & QK (z, k) contains the poles of QK (z, k) with negative real parts. Thus, QK (z, k) is also continuous ! & ! function of h(k). Finally, for a system of type m, BK (z, k) is a polynomial de"ned as [3] K\ BK (z, k)" b (h(k))z\H H H where the coe$cients b "0,2, m!1 depend continuously on the estimate h(k). Thus, from (15) H it follows that the coe$cients of QK (z, k) and QK (z, k) depend continuously on h(k). B L Remark 4.1. Lemma 4.4 is important because it translates the slow variation of the estimated parameter vector h(k) to slow variation of the controller parameters. Since the stability and robustness analysis of most adaptive schemes rely on results from the stability of slowly time-varying systems, establishing continuity of the controller parameters as a function of the estimated plant parameters (which are known to vary slowly) is an important ingredient of the analysis. Remark 4.2. From the proof of Lemma 4.4, it is clear that the only property of the adaptive law that has to be exploited is the fact that the leading coe.cient of ZK (z, k) is not allowed to pass through zero.  Thus, for a given plant parametrized by h, we can use exactly the same arguments as before to conclude that the coe$cients Q (z) and Q (z) are continuous functions of h, where Q (z), Q (z) are B L B L appropriately de"ned for the non-adaptive H -optimal controller. Thus if h belongs to a compact  set C, then the image of C under the mapping that produces the coe$cients of Q (z), Q (z) will also B L be compact. This fact will play an important role later on in our stability and robustness analysis. Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

27

The following theorem describes the stability and robustness properties of the discrete time adaptive H IMC scheme.  ¹heorem 4.2. Consider the discrete-time plant (5) subject to the robust adaptive IMC control law (9)}(13), where r(k) is the bounded external signal to be tracked. Then, k*'0 such that ∀k3[0, k*), all the signals in the closed-loop system are uniformly bounded and the error y!yL 3D(kg/m). Proof. The proof is based on combining the properties of the robust adaptive law (9)}(13) with the properties of the discrete-time IMC-based H -optimal control structure. We "rst analyse the  properties of the adaptive law. From (7), (10) and (11), we obtain (h*2!h2) #kg e" m De"ning hOh!h* we have !h2 #kg e" m

(23)

Consider the Lyapunov-like function h2(k)hI (k) . Hence, h, h, and d 3(0, 1) such that the state transition matrix '(k, k ) corresponding to    the homogeneous part of (27) satis"es #'(k, k )#)c d(k!k) ∀k*k *0     Copyright  2001 John Wiley & Sons, Ltd.

(28)

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

30

G. J. SILVA AND A. DATTA

Now, suppose that the polynomial " (z) is given by  " (z)"znB#j znB!1#2#jn B   Then from the identity u"[" (z)/" (z)][u], we can rewrite the control input u as   an (z) u"v2(k)X#q2 (k) B [r!em] L " (z) 

(29)

where v(k)"[jn !bn (k), jn !1!bn !1 (k),2, j !b (k)]2. Also, using (29) in the plant equation B B B B   (5), we obtain




y"



Z (z)  #k* (z) ? R (z) 



an (z) v2(k)X#q2 (k) B [r!em] L " (z) 

(30)

Now let d3(max[d , d ], 1) be chosen such that 1/R (z), 1/" (z) are analytic in "z"*(d and     de"ne the "ctitious normalizing signal m (k) by D m (k)"1##u #B ##y #B D I\  I\ 

(31)

Now, we can take truncated exponentially weighted l norms on both sides of (29) and (30) and  make use of Lemmas 4.2 and 4.3, while observing that v(k), q (k), r(k)3l , to obtain* L  #u #B )c#c#(em) #B I\  I\ 

(32)

#y #B )c#c#(em) #B I\  I\ 

(33)

These together with (31) imply that m (k))c#c#(em) #B D I\ 

(34)

Squaring both sides of (34) we obtain I\ m (k))c#c dI\\G[e(i)m(i)][e(i)m(i)] D G I\ )c#c dI\\Ge(i)m(i)m (i) (since m(k))cm (k)). D D G * In the rest of this proof &c' is the generic symbol for a positive constant. For the qualitative presentation here the exact values of these constants are not important. Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

31

ADAPTIVE INTERNAL MODEL CONTROL

Using the discrete Bellman}Gronwall lemma [16] we have





I\ I\ m (k))c# cdI\\G “ (1#cd\e( j)m( j)) e(i)m(i) D G HG> #ce(k!1)m(k!1)





I\ cd\ I\ I\\G )c# cdI\\G 1# e( j)m( j) e(i)m(i) k!1!i G HG> #ce(k!1)m(k!1)

(35)

(using the inequality relating arithmetic and geometric means [17]). Now, since em3D(kg/m), it follows that I\ I\ g( j) e( j)m( j))c k #c   m( j) HG> HG> I\ N e( j)m( j))(k!1!i)c k*#c (using (26))    HG> I\ c 1  e( j)m( j))c k*# N   k!1!i k!1!i HG> Thus from (35), we obtain








I\ c  m (k))c# cdI\\G 1#cd\ c k*#   k!1!i D G #ce(k!1)m(k!1)

I\\G

e(i)m(i)





I\ I\\G cd\ "c# cdI\\G [1#cd\k*]I\\G 1#  (k!1!i)(1#cd\k*)  G ;e(i)m(i)#ce(k!1)m(k!1) I\ )c# cdI\\G[1#cd\k*]I\\Ge(cd\)/(1#cd\k*) e(i)m(i)  G (using (1#)V)e∀x'0) V Now, if k is &small enough' then we have [1#cd\k*]d(1  Thus, using Lemma 4.1 together with the fact that em3D(kg/m), it follows that k*3(0, k*)  such that ∀k3[0, k*), m 3l . Since m(k)4cm (k), ∀k3Z>, we conclude that m3l . Since /m, D  D  g/m3l , it follows that , n3l . Now from (23), we see that em"!hI 2 #kg is also bounded.   Thus from (27), we obtain X3l . From (29) and (30), we can now conclude that u, y3l . This   establishes the boundedness of all the closed-loop signals in the adaptive IMC scheme. Finally we Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

32

G. J. SILVA AND A. DATTA

note that y!yL "em"(em) ) m and since em3D(kg/m), m3l , it follows that y!yL 3D(kg/m). This completes the proof.  Remark 4.3. The robust adaptive IMC scheme of this section recovers the performance properties of the ideal case if the modelling error disappears, i.e. we can show that if k"0 then y(k)!yL (k)P0 as kPR. This is a consequence of the use of parameter projection as the robustifying modi"cation in the adaptive law, and follows immediately from the fact that k"0Ny!yL 3l .  5. SIMULATION EXAMPLES In this section, we present two examples to illustrate the steps involved in designing the adaptive H IMC schemes of this paper.  Example 1. Consider the following continuous-time plant with additive uncertainty: 0.01 !s#1 # P(s)" s#3s#2 s#3 For a sampling time of ¹"0.5 s, the corresponding discrete-time model (4) is given by Z (z) !0.1612z#0.2856 0.259 P(z)"  #k* (z)" #0.01 ? R (z) z!0.9744z#0.2231 z!0.2231  Choosing "(z)"z!0.1z!0.02, the linear parametric model for this plant becomes y"h*2 #kg where h*"[0.8744,!0.2431,!0.1612, 0.2856]2



"



z 1 z 1 2 [y], [y], [u], [u] "(z) "(z) "(z) "(z)

0.2590(z!0.9744z#0.2231) g" [u] (z!0.2231)(z!0.1z!0.02) and k"0.01 Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

33

Figure 3. Adaptive H IMC simulation for Example 1. 

Now choosing c"0.4, d "0.36, h(0)"[0.8,!0.3,!0.1, 0.25]2, and all other initial conditions  equal to zero, we implemented the robust adaptive law (9)}(13). In order to ensure that ZK (z, k)  has no zeros on the unit circle and its degree does not drop, we chose C as C"[0.4, 1.2]; [!1, !0.1];[!0.18,!0.05];[0.2, 1]. Let h(k)"[h (k), h (k), h (k), h (k)]2 denote the estimate of h* obtained from the robust     adaptive law (9)}(13). Then the numerator and denominator of the estimated modelled part of the plant are given by ZK (z, k)"h (k)z#h (k)    RK (z, k)"z!(0.1#h (k))z!(0.02#h (k))    Thus, for a step input, the IMC parameter QK (z, k) obtained from (15) is given by QK (z, k) (1!t)/[h (k)#h (k)] [z!(0.1#h (k))!(0.02#h (k))]     QK (z, k)" L " QK (z, k) z(z!t) B where t"!h (k)/h (k). We now choose " (z)"z!0.1z!0.02 and implement the certainty    equivalence control law (14). Figure 3 shows the response of the resulting closed-loop system to a step input occurring at t"2 s. From this "gure it is clear that y(k) asymptotically tracks r(k) Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

34

G. J. SILVA AND A. DATTA

quite well and the intersample behaviour is acceptable. Note that in this case QK (z, k) does not & have any poles with negative real parts so that QK (z, k) and BK (z, k) can be omitted from (15). ! Example 2. Now consider the following discrete-time plant with multiplicative uncertainty





z!2 z#0.1 Z (z) 1#0.01 P(z)"  [1#k* (z)]" K R (z) z!0.7z#0.1 z!0.4 

Choosing "(z)"z!0.1z!0.02, the linear parametric model for this plant becomes y"h*2 #kg where h*"[0.6,!0.12, 1,!2]2



"



z 2 1 z 1 [y], [y], [u], [u] "(z) "(z) "(z) "(z)

(z#0.1)(z!2) g" [u] (z!0.4)(z!0.1z!0.02) and k"0.01 Now choosing c"0.3, d "0.36, h(0)"[0.5,!0.2, 1.3,!1.8]2, and all other initial conditions  equal to zero, we implemented the robust adaptive law (9)}(13). In order to ensure that ZK (z, k)  has no zeros on the unit circle and its degree does not drop, we chose C as C"[0.3, 1]; [!1,!0.05];[0.5, 1.5];[!3, !1.7]. Let h(k)"[h (k), h (k), h (k), h (k)]2 denote the estimate of h* obtained from the robust     adaptive law (9)}(13). Then the numerator and denominator of the estimated modelled part of the plant are given by ZK (z, k)"h (k)z#h (k)    RK (z, k)"z!(0.1#h (k))z!(0.02#h (k))    Thus, for a step input, the IMC parameter QK (z, k) obtained from (15) is given by QK (z, k) (1!t)/[h (k)#h (k)] [z!(0.1#h (k))!(0.02#h (k))]     QK (z, k)" L " QK (z, k) z(z!t) B where t"!h (k)/h (k). We now choose " (z)"z!0.1z!0.02 and implement the certainty    equivalence control law (14). Figure 4 shows the response of the resulting closed-loop system to a discrete-time step input occurring at k"20. From this "gure, it is clear that y(k) asymptotically tracks r(k) quite well even in the presence of the multiplicative uncertainty. Note that in this case k* (z) is a Schur stable uncertainty such that (Z (z)/R (z))* (z) is strictly proper. K   K Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

ADAPTIVE INTERNAL MODEL CONTROL

35

Figure 4. Adaptive H IMC simulation for Example 2. 

6. CONCLUDING REMARKS In this paper, we have presented the design and analysis of an IMC-based robust discrete-time time H -optimal adaptive control scheme. By combining in a certainty equivalence fashion  a robust discrete-time adaptive law with a robust discrete-time H -optimal control structure, we  obtained a discrete-time adaptive IMC scheme with provable guarantees of stability and robustness. Given the generality of the certainty equivalence approach the results can be easily extended to handle other kinds of adaptive IMC schemes such as those based on partial pole-placement or model reference control. In view of the immense popularity of IMC in the process control industries, it is our hope that the results presented here will "nd use in industrial applications. The results of this paper are applicable to only single-input single-output, linear, time-invariant plants. The extension to the multivariable case is a topic for further investigation.

REFERENCES 1. GarcmH a CE, Morari M. Internal model control } 1. A unifying review and some new results. Industrial and Engineering Chemistry Process, Design and Development 1982; 21:308}323. 2. GarcmH a CE, Prett DM, Morari M. Model predictive control: theory and practice } a survey. Automatica 1989; 25:335}348. 3. Morari M, Za"riou E. Robust Process Control. Prentice-Hall: Englewood Cli!s, NJ, 1989. 4. Richalet JA, Rault A, Testud JL, Papon J. Model predictive heuristic control: applications to an industrial process. Automatica 1978; 14:413}428. 5. Cutler CR, Ramaker BL. Dynamix matrix control } a computer control algorithm. A.I.Ch.E. National Meeting, Houston, Texas, 1979; also Proceedings of Joint Automatic Control Conference, San Francisco, California, 1980. Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36

36

G. J. SILVA AND A. DATTA

6. Takamatsu T, Shioya S, Okada Y. Adaptive internal model control and its application to a batch polymerization reactor. IFAC Symposium on Adaptive Control of Chemical Processes, Frankfurt am main, 1985. 7. Soper RA, Mellichamp DA, Seborg DE. An adaptive nonlinear control strategy for photolithography. Proceedings of the American Control Conference 1993. 8. Ioannou PA, Sun J. Robust Adaptive Control. Prentice-Hall: Englewood Cli!s, NJ, 1996. 9. Datta A, Ochoa J. Adaptive internal model control: design and stability analysis. Automatica 1996; 32(2):261}266. 10. Datta A. Adaptive Internal Model Control. Springer: London, 1998. 11. Ydstie BE. Auto tuning of the time horizon. Proceedings of the 2nd IFAC =orkshop on Adaptive Control, Lund, Sweden, 1986; 215}219. 12. Youla DC, Jabr HA, Bongiorno JJ. Modern Wiener-Hopf design of optimal controllers } Part II: The multivariable case. IEEE ¹ransactions on Automatic Control 1976; AC-21:319}338. 13. Datta A. Robustness of discrete-time adaptive controllers: an input-output approach. IEEE ¹ransactions on Automatic Control 1993; AC-38:1852}1857. 14. Soh CB. On a root distribution criterion for interval polynomials. IEEE ¹ransactions on Automatic Control 1992; AC-37(12):1977}1978. 15. Middleton RH, Goodwin GC, Hill DJ, Mayne DQ. Design issues in adaptive control. IEEE ¹ransactions on Automatic Control 1988; AC-33:50}58. 16. Desoer CA, Vidyasagar M. Feedback Systems: Input-Output Properties. Academic Press: New York, 1975. 17. Rudin W. Real and Complex Analysis. McGraw-Hill: New York, 1986.

Copyright  2001 John Wiley & Sons, Ltd.

Int. J. Adapt. Control Signal Process. 2001; 15:15}36