ADAPTIVE DIGITAL CONTROL OF HAMMERSTEIN NONLINEAR ...

SIAM J. CONTROL OPTIM. Vol. 45, No. 6, pp. 2257–2276

c 2007 Society for Industrial and Applied Mathematics

ADAPTIVE DIGITAL CONTROL OF HAMMERSTEIN NONLINEAR SYSTEMS WITH LIMITED OUTPUT SAMPLING∗ FENG DING† , TONGWEN CHEN‡ , AND ZENTA IWAI§ Abstract. This paper is motivated by the practical control considerations that nonlinearity is abundant in industrial processes and output sampling rates are often limited due to hardware constraints. In particular, for a Hammerstein nonlinear sampled-data system in which the output sampling period is an integer multiple of the input updating period, we derive, by using a polynomial transformation technique, a mathematical model which is suitable for parameter estimation with dual-rate measurement data. Further, we present an adaptive control scheme for such a dual-rate nonlinear system; the parameter estimation–based adaptive algorithm can achieve virtually asymptotically optimal control and ensure that the closed-loop system is stable and globally convergent. The simulation results are included. Key words. Hammerstein systems, sampled-data systems, multirate modeling, self-tuning regulator, adaptive control, dual-rate systems, convergence properties, least squares methods AMS subject classifications. 93C40, 93E10, 93C10 DOI. 10.1137/05062620X

1. Introduction. This paper focuses on adaptive control problems of a class of nonlinear systems, namely, Hammerstein nonlinear systems, with limited output sampling frequencies; in particular, the Hammerstein systems of interest have output sampling periods which are integer multiples of input updating periods. These systems are sometimes referred to as dual-rate sampled-data systems. Such slow output sampled systems often arise in industry due to hard limits on sensoring devices; see the industrial processes described in, e.g., [26, 31, 34]. Without assuming knowledge of the nonlinear model involved, our main idea in this work is to perform closedloop adaptive control through an online identification scheme, which estimates the parameters of the nonlinear model by making use of only the available dual-rate data. In the identification area of nonlinear systems, there exists a large amount of research exploring different approaches; see, e.g., [3, 4, 5, 6, 35, 36, 37]. V¨ or¨ os presented a recursive half-substitution algorithm for discontinuous nonlinear systems [36]. For Hammerstein–Wiener nonlinear models, Bai reported a two-stage identification algorithm based on the recursive least squares and the singular value decomposition [3] and a blind identification approach [5]. Recently, Cerone and Regruto derived the parameter error bounds in the Hammerstein models [8] by assuming that the output measurement error was bounded; Ding and Chen developed an iterative method ∗ Received by the editors March 7, 2005; accepted for publication (in revised form) August 15, 2006; published electronically February 13, 2007. This research was completed while the second author visited Kumamoto University as a Guest Professor holding a JSPS Research Fellowship. Financial support from the Japan Society for the Promotion of Science (JSPS), the Natural Sciences and Engineering Research Council (NSERC) of Canada, and the National Natural Science Foundation of China (grants 60574051 and 60528007) is gratefully acknowledged. http://www.siam.org/journals/sicon/45-6/62620.html † Corresponding author. Control Science and Engineering Research Center, Southern Yangtze University, Wuxi, Jiansu 214122, People’s Republic of China ([email protected], [email protected]). ‡ Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada ([email protected]). § Department of Mechanical Engineering and Materials Science, Faculty of Engineering, Kumamoto University, 2-39-1 Kurokami, Kumamoto 860-8555, Japan ([email protected]).

2257

2258

FENG DING, TONGWEN CHEN, AND ZENTA IWAI

and a recursive least squares identification method for Hammerstein nonlinear autoregression and moving average with exogenous input (ARMAX) systems [12]. In this paper, we will extend the noniterative method of Chang and Luus [7] for Hammerstein models to present a parameter estimation–based adaptive algorithm for Hammerstein nonlinear dual-rate systems. In the field of adaptive control of linear dual-rate systems, Albertos, Salt, and Tormero studied adaptive control schemes for dual-rate systems [1]; Ishitobi, Kawanaka, and Nishi presented a least squares–based self-tuning control algorithm [28]; and Kanniah, Malik, and Hope proposed a control algorithm based on a parameterized model with its autoregression (AR) coefficients corresponding to the fast sampling rate and the moving average (MA) coefficients to the slow sampling rate [29]. Though these are earlier contributions in this area, the reported adaptive control schemes have two main limitations as follows (in view of the goal in our work): • First, the prediction and control were both based on the slow sampling rate; the desired fast-rate system performance could not be achieved in these schemes in design, and there might be poor intersample behavior even if the behavior at the slow sampling instants is acceptable. • Second, these schemes used linear models as the primary assumption; for nonlinear Hammerstein models, which capture a wide class of industrial systems and processes, new research is required. In order to overcome the two limitations for dual-rate Hammerstein models, we develop an estimation scheme to obtain fast-rate models which can be used to implement an adaptive control law updated at the fast rate, even if the output is sampled at a relatively slow rate. To the best of our knowledge, few contributions have addressed such adaptive control problems involving Hammerstein nonlinear dual-rate systems, especially the convergence problem of the algorithms involved, which are the focus of this work. Therefore, the objective of this paper is to extend the control scheme of dual-rate systems in [10] to the nonlinear case, in particular, in the following ways: • By a polynomial transformation technique, derive models suitable for dualrate identification of Hammerstein nonlinear sampled-data systems. • Based on the models derived, propose a parameter estimation–based adaptive control scheme. • Study the convergence properties of the proposed adaptive control algorithms. Briefly, the paper is organized as follows. Section 2 introduces the problem formulation related to Hammerstein nonlinear dual-rate systems and an adaptive control scheme. Section 3 uses a polynomial transformation technique to derive a suitable mathematical model and present an adaptive control algorithm based on parameter estimation. Sections 4 and 5 analyze the output tracking performance and global stability of the closed-loop systems under the proposed adaptive control. Section 6 presents an illustrative example demonstrating the effectiveness of the proposed algorithm. Finally, section 7 offers some concluding remarks. 2. Problem formulation. In many chemical processes [26, 31, 34], due to hardware sensoring constraints, only infrequent and scarce output sampling is available; but the input updating can be achieved by actuators and digital computers at relatively fast speeds [10, 11, 30]. This gives rise to a class of dual-rate systems in which the input updating rate is an integer multiple of the output sampling rate. This paper focuses on adaptive control of a class of Hammerstein nonlinear dual-rate sampleddata systems—a static nonlinearity f (·) followed by a linear dynamic subsystem G(z),

ADAPTIVE CONTROL OF HAMMERSTEIN NONLINEAR SYSTEMS

u(k)

- f (·)

u ¯(k) - G(z)

2259

y(k) -

Fig. 1. A Hammerstein nonlinear system.

as depicted in Figure 1, where u(k) is the system input, the inner variable u ¯(k) is the output of the nonlinear block (unmeasurable), y(k) is the system output but is available only every qth sampling instant, q being a positive integer (q > 1). Thus, the available input-output data set consists of • {u(k) : k = 0, 1, 2, . . . } at the fast rate, and • {y(kq) : k = 0, 1, 2, . . . } at the slow rate. Here, we refer to the unavailable intersample outputs, y(kq + j), j = 1, 2, . . . , q − 1, as the missing output samples, and to {u(k), y(kq)} as the dual-rate measurement data. Often the static nonlinear part in the Hammerstein model is assumed to be a polynomial of a known order m in the input as follows (see, e.g., [7, 22, 33]): u ¯(k) = c1 u(k) + c2 u2 (k) + · · · + cm um (k), or, more generally, to be a nonlinear function of a known basis (γ1 , γ2 , . . . , γm ) as follows [8]: (1) u ¯(k) = f (u(k)) = c1 γ1 (u(k)) + c2 γ2 (u(k)) + · · · + cm γm (u(k)) =

m

ci γi (u(k)),

i=1

where the ci ’s are unknown parameters. Of course, the nonlinearity considered here also covers cubic spline nonlinearity [9], support vector machines [23, 24], and piecewise linear functions with discontinuities [35, 36], in addition to the single-parameter nonlinearities [4]. The linear block is a time-invariant system (at the fast rate) with a known order n and takes the following real-rational form: (2)

y(k) =

B(z) B(z) u ¯(k), G(z) := A(z) A(z)

with A(z) = 1 + a1 z −1 + a2 z −2 + · · · + an z −n , B(z) = b1 z −1 + b2 z −2 + · · · + bn z −n . Here, z −1 represents a unit backward shift operator at the fast rate, z −1 u(k) = u(k − 1), and G(z) is the transfer function of the linear dynamical part with one unit delay. Notice that in the characterization of the Hammerstein model shown in Figure 1, f (u) and G(z) are actually not unique from the identification point of view. Any pair (cf (u), G(z)/c) for some nonzero and finite constant c would produce identical input and output measurements. In other words, none of the identification schemes can distinguish between (f (u), G(z)) and (cf (u), G(z)/c). Therefore, to get a unique parameterization, without loss of generality, one of the gains of f (u) and G(z) has to be fixed. There are several ways to normalize the gains [3, 8, 22]. Here, we adopt the assumption used in [5, 22]: The first coefficient of the function f (·) equals 1; i.e., c1 = 1. (In case c1 = 0, one can use any nonzero ci and normalize it to 1.)

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yr (k)

- Controller 6

u(k)

- f (·)

u ¯(k) - G(z)

-

- Param. Estimator -

ˆ ?θ Intersample Output Estimator

y(iq)

y(iq) b yˆ(iq + j)

6b b ? yf (k)

Fig. 2. The adaptive control scheme (j = 1, 2, . . . , q − 1).

For dual-rate sampled-data control systems, we expect that the control law be updated at the fast rate even if the output is sampled at the slow rate. To implement this, the adaptive control scheme we propose is shown in Figure 2, where yr (k) denotes a deterministic reference input or desired output signal. The control scheme makes use of a parameter estimator, or an identification algorithm, which generates online the estimates θˆ of the unknown system parameters based on the dual-rate data ˆ and thus the fast-rate model, and the input u(k), the inter{u(k), y(kq)}; based on θ, sample (missing) outputs yˆ are estimated—see the intersample output estimator in Figure 2. In order to feed back to the controller a fast rate signal yf (k), representing the output y(k), we use the slow sampled output y(iq) at every q period, giving y(0), y(q), and y(2q), etc., and use the estimated output yˆ(iq + j) to fill in the missing samples in y(k). Hence in Figure 2, yf (k) connects to y(iq) at times k = iq, and connects to yˆ(iq + j) at k = iq + j, j = 1, 2, . . . , (q − 1); thus the output of the switch, yf (k), is a fast rate signal. The operation of the periodic switch can be expressed in the following equation: y(iq), k = iq, yf (k) = (3) yˆ(iq + j), k = iq + j, j = 1, 2, . . . , (q − 1). To summarize, the dual-rate adaptive control scheme uses a fast single-rate controller and a periodic switch. It is conceptually simple, easy to implement in digital computers, and practical for industry. In connection with Figure 2, the objectives of this paper can be restated more clearly as follows: • Establish a dual-rate model which is suitable for identification using the given dual-rate data. • Propose an algorithm to estimate the intersample outputs {y(kq + j) : j = 1, 2, . . . , (q − 1)}. • Design an adaptive controller so that the output y(k) tracks the given desired output yr (k) by minimizing the tracking error criterion given by (4)

J[u(k)] = [yf (k + 1) − yr (k + 1)]2

for deterministic systems, or (5)

J[u(k)] = E{[yf (k + 1) − yr (k + 1)]2 |Fk−1 }

for stochastic systems, and study the properties of the closed-loop system. Here, {Fk } is the σ algebra sequence generated by the observations up to and including time k.

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3. Control algorithm description. Let us introduce some notation first. The symbol I stands for an identity matrix of appropriate dimensions. The superscript T denotes the matrix transpose. For a square matrix X, |X| = det[X] represents the determinant; λi [X] denotes the ith eigenvalue of X; and λmax [X] and λmin [X] represent the maximum and minimum eigenvalues of X, respectively. The norm of a matrix M is defined by M 2 = tr[M M T ]. The notation 1n denotes an n-dimensional column vector whose elements are all 1; f (k) = o(g(k)) represents f (k)/g(k) → 0 as k → ∞; for g(k) ≥ 0, we write f (k) = O(g(k)) or f (k) ∼ g(k) if there exists a positive constant δ1 such that |f (k)| ≤ δ1 g(k). The model in (1)–(2) is not suitable for dual-rate adaptive control because it would involve the unavailable outputs {y(kq + j) : j = 1, 2, . . . , (q − 1)}. To obtain a model that we can use directly on the dual-rate data, by a polynomial transformation technique, G(z) can be converted into a form so that the denominator is a polynomial in z −q instead of z −1 . For a general discussion, let the roots of A(z) be zi to get A(z) =

n

(1 − zi z −1 ).

i=1

Define φq (z) :=

n

(1 + zi z −1 + zi2 z −2 + · · · + ziq−1 z −q+1 ) =

i=1

n 1 − z q z −q i

i=1

1 − zi z −1

.

Multiplying the numerator and denominator of G(z) by φq (z) and using the polynomial transformation formula [27, 32], 1 − xq = (1 − x)(1 + x + x2 + · · · + xq−1 ), we get a new model: P (z) =

β(z) B(z)φq (z) =: , A(z)φq (z) α(z)

or (6)

α(z)y(k) = β(z)¯ u(k)

with α(z) = 1 + α1 z −q + α2 z −2q + · · · + αn z −qn , β(z) = β1 z −1 + β2 z −2 + · · · + βqn z −qn . Equation (6) has the advantage that the denominator is a polynomial of z −q ; arising from here is a recursive equation using only slowly sampled outputs. The control algorithm we propose later for dual-rate systems will be based on this model, which does not involve the unavailable intersample outputs. From the model in (6), we easily get the recursive equation: y(k) = − =−

n i=1 n i=1

αi y(k − iq) + αi y(k − iq) +

qn i=1 qn i=1

βi u ¯(k − i) βi

m j=1

cj γj (u(k − i)).

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Define the parameter vector θ and information vector ϕ(k) as ⎡ ⎤ ⎡ ⎤ α −y(k − q) ⎢ β ⎥ ⎢ −y(k − 2q) ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ c2 β ⎥ ⎢ ⎥ .. n0 θ=⎢ ⎥ ∈ R , ϕ(k − 1) = ⎢ ⎥ ∈ Rn0 , n0 := (mq + 1)n, . ⎢ .. ⎥ ⎢ ⎥ ⎣ . ⎦ ⎣ −y(k − nq) ⎦ ψ(k − 1) cm β ⎤ ⎤ ⎤ ⎡ ⎡ ⎡ β1 c2 α1 ⎢ β2 ⎥ ⎢ c3 ⎥ ⎢ α2 ⎥ ⎥ ⎥ ⎥ ⎢ ⎢ ⎢ α = ⎢ . ⎥ ∈ Rn , β = ⎢ . ⎥ ∈ Rqn , c = ⎢ . ⎥ ∈ Rm−1 , ⎣ .. ⎦ ⎣ .. ⎦ ⎣ .. ⎦ ⎡

αn

⎤

βqn

cm

ψ1 (k − 1) ⎢ ψ2 (k − 1) ⎥ ⎥ ⎢ ψ(k − 1) = ⎢ ⎥ ∈ Rmqn , .. ⎦ ⎣ . ψm (k − 1) ⎤ ⎡ γj (u(k − 1)) ⎢ γj (u(k − 2)) ⎥ ⎥ ⎢ ψj (k − 1) = ⎢ ⎥ ∈ Rqn , j = 1, 2, . . . , m. .. ⎦ ⎣ . γj (u(k − nq)) Then we have (7)

y(k) = ϕT (k − 1)θ.

Notice that if k is an integer multiple of q, then ϕ(k − 1) uses only available dual-rate data—only the past measurement outputs (slow rate) and inputs (fast rate). From here, we can see that the number of parameters will be greatly increased after the parameterization of nonlinear systems with dual-rate sampling; this will lead to increased computational complexity, which may be reduced by using the hierarchical identification scheme [13, 14, 15, 16, 17, 18]. Let yr (k) be a desired output signal; define the output tracking error ξ(k + 1) = y(k + 1) − yr (k + 1). In the deterministic case, if the control signal u(k) is chosen according to the equation yr (k + 1) = ϕT (k)θ obtained by minimizing the criterion function in (4), then the tracking error ξ(k + 1) approaches zero asymptotically. For stochastic systems, based on the model in (7), introducing a zero-mean white noise disturbance term v(k), we have (8)

y(k) = ϕT (k − 1)θ + v(k).

Let θˆ be the estimate of unknown parameter vector θ; then yˆ(k + 1) = ϕT (k)θˆ is the output prediction, which is computed by the intersample output estimator in Figure 2. According to the certainty equivalence principle [25], or minimizing the criterion function in (5), the control law takes the following form: (9)

ˆ yr (k + 1) = ϕT (k)θ.


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Replacing k in (8) with kq gives y(kq) = ϕT (kq − 1)θ + v(kq).

(10)

ˆ Then the recursive least squares algorithm may be used to produce the estimate θ(kq) of θ at current time kq, and the algorithm is as follows: ˆ ˆ θ(kq) = θ(kq − q) + P (kq)ϕ(kq − 1)e(kq), ˆ e(kq) = [y(kq) − ϕT (kq − 1)θ(kq − q)],

(11) (12) (13)

ˆ = θ(kq), ˆ θ(i) i = kq, kq + 1, . . . , kq + q − 1,

(14)

P −1 (kq) = P −1 (kq − q) + ϕ(kq − 1)ϕT (kq − 1), ˆ θ(kq) = [θˆ1 (kq), θˆ2 (kq), . . . , θˆn (kq)]T .

(15)

0

Based on (9), the control law is given by (16)

ˆ ϕT (kq + j)θ(kq) = yr (kq + 1 + j), j = 0, 1, . . . , q − 1.

To initialize the control algorithm in (11)–(16), we take P (0) = p0 I, with p0 normally ˆ a large positive number, e.g., p0 = 106 , and θ(0) = θˆ0 some small real vector, e.g., ˆ ˆ θ(0) = 1n0 /p0 . Notice that the parameter estimate θ is updated every q (fast) samples, namely, at the slow rate, as is the covariance matrix P ; in between the slow samples, we simply hold θˆ unchanged. Thus, every time θˆ is updated, we have q new input samples and one new output sample. Defining the gain vector L(kq) := P (kq)ϕ(kq − 1) and applying the matrix inversion lemma (A + BC)−1 = A−1 − A−1 B(I + CA−1 B)−1 CA−1 to (14), it is easy to get that the covariance matrix P can be updated as follows: P (kq) = [I − L(kq)ϕT (kq − 1)]P (kq − q), P (kq − q)ϕ(kq − 1) . L(kq) = T 1 + ϕ (kq − 1)P (kq − q)ϕ(kq − 1) Note c1 = 1; the estimates ˆ ˆ 2 (kq), . . . , α ˆ n (kq)]T , α(kq) = [ˆ α1 (kq), α ˆ β(kq) = [βˆ1 (kq), βˆ2 (kq), . . . , βˆqn (kq)]T ˆ of α and β can be read from the first n and second qn entries of θ(kq), respectively. Let ˆ(kq) = [ˆ c c2 (kq), cˆ3 (kq), . . . , cˆm (kq)]T be the estimate of c. Referring to the definition of θ, we get that the estimates of cj , j = 2, 3, . . . , m, may be computed by cˆj =

θˆn+(j−1)qn+i (kq) , j = 2, 3, . . . , m; i = 1, 2, . . . , qn. βî (kq)

2264


From here, we can see that there is considerable redundancy in determination of each coefficient cˆj in the nonlinear function f (·), since for each cj we have qn estimates cˆj for i = 1, 2, . . . , n. Because we do not need such n estimates for cˆj , one simple way is to take their average as the estimate of cj [12], i.e., qn 1 θˆn+(j−1)qn+i (kq) cˆj (kq) = , j = 2, 3, . . . , m. qn i=1 βî (kq)

Of course, the singular value decomposition technique or least squares optimization methods can also be used to find the estimate of c [3, 7]. The control signal u(kq + j) in (16) may be obtained by solving the following nonlinear equation: n

(17)

θî (kq)y(kq + j + 1 − iq) +

i=1

nq m

θˆn+(l−1)qn+i (kq)γl (u(kq + j + 1 − i))

l=1 i=1

= yr (kq + j + 1),

j = 0, 1, . . . , q − 1.

Here, a difficulty arises because over the interval [kq, kq + q), except for j = q − 1, the expression on the right-hand side of the above equation contains the future and past missing outputs y(kq + j + 1 − iq). So it looks impossible to compute the control law by (17) and to realize the algorithm in (11)–(16). Our solution is based on the adaptive control scheme stated in section 2—these unknown outputs y(kq + j) in (17) are replaced by their estimates yˆ(kq + j). Hence we have n

(18)

y (kq + j + 1 − iq) + θî (kq)ˆ

i=1

nq m

θˆn+(l−1)qn+i (kq)γl (u(kq + j + 1 − i))

l=1 i=1

= yr (kq + j + 1) or m

θˆn+(l−1)qn+1 (kq)γl (u(kq + j)) = yr (kq + j + 1) −

y (kq + j + 1 − iq) θî (kq)ˆ

i=1

l=1

−

(19)

n

nq m

θˆn+(l−1)qn+i (kq)γl (u(kq + j + 1 − i)),

j = 0, 1, . . . , q − 2.

l=1 i=2

In fact, it is only when j = q − 1 that the control term u(kq + j) does not involve the missing outputs and can be generated by m

θˆn+(l−1)qn+1 (kq)γl (u(kq + q − 1)) = yr (kq + q) −

y (kq + q − iq) θî (kq)ˆ

i=1

l=1

(20)

n

−

nq m

θˆn+(l−1)qn+i (kq)γl (u(kq + q − i)).

l=1 i=2

In our adaptive control algorithm in (11)–(16) (or (11)–(15), (19), and (20)), based on the parameter estimation, the control signal u(k) is computed by the model inverse of the Hammerstein model defined in (1) and (2) using past inputs u(k − j) for j = 1, 2, . . . , the current output y(k) (ˆ y (k)), past outputs y(k − j) (ˆ y (k − j)) for


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j = 1, 2, . . . , and desired output yr (k + 1) (for one-step-ahead self-tuning control) as k increases, and the input u(k) is made to drive the system output at time k + 1 to the target value yr (k + 1). In order to ensure that the input signal is bounded, this model inverse must be stable; thus the inverse of the nonlinearity f (·) must exist (i.e., the nonlinearity f (·) must be invertible) and G(z) is minimum phase, exactly as in the case of linear models. The control signal u(k) is also determined by using the following approach. Assuming b1 = 0 or β = 0, we compute the control signal u(k) by (21)

u(kq + j) = fˆ−1 (¯ u(kq + j)), i = 0, 1, 2, . . . , q − 1,

where fˆ−1 (·) is the inverse of the estimated nonlinear function fˆ(·) defined by (22)

u ¯(kq + j) = fˆ(u(kq + j)) = γ1 (u(kq + j)) + cˆ2 (kq)γ2 (u(kq + j)) + · · · + cˆm (kq)γm (u(kq + j)),

and the intermediate signal u ¯(k) is obtained from the inverse of the linear model, i.e., n 1 yr (kq + j + 1) + u ¯(kq + j) = α ˆ i (kq)ˆ y (kq + j + 1 − iq) βˆ1 (kq) i=1 nq − βî (kq)u(kq + j − i + 1) , j = 0, 1, . . . , q − 2, u ¯(kq + q − 1) =

1 βˆ1 (kq)

i=2

yr (kq + q) +

n

α ˆ i (kq)y(kq + q − iq)

i=1

−

nq

ˆ βi (kq)u(kq + j − i + 1) .

i=2

The above two control law equations contain only nq+n parameters (less than mqn+n parameters); however, this increased computation is still tolerable and affordable. Due to the nonlinear function f (·), the control signal u(k) in (22) is usually solved by numerical methods. The following assumption is required and is also reasonable: Given u ¯(k), there exists a u(k) satisfying (1). Note that we do not require that u(k) be uniquely determined by given u ¯(k), because all such u(k)’s produce the same u ¯(k), which does not affect our output tracking performance. 4. Output tracking performance. Let us first introduce some definitions and assumptions. The sequence {v(k), Fk } is assumed to be a martingale difference sequence defined on a probability space {Ω, F, P }, where {Fk } is the σ algebra sequence generated by the observations up to and including time k [25]. The noise sequence {v(k)} satisfies the following conditions: (A1) E[v(k)|Fk−1 ] = 0 a.s.; (A2) E[v 2 (k)|Fk−1 ] = σ 2 (k) ≤ σ ¯ 2 < ∞ a.s.; (A3) lim sup k→∞

k 1 2 v (i) ≤ σ ¯ 2 < ∞ a.s. k i=1

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That is, {v(k)} is an independent random noise sequence with zero mean and bounded time-varying variance. Define r(kq) := tr[P −1 (kq)], r(0) :=

n0 . p0

It follows easily that r(kq) = λ1 [P −1 (kq)] + λ2 [P −1 (kq)] + · · · + λn0 [P −1 (kq)] ≤ n0 λmax [P −1 (kq)], |P −1 (kq)| = λ1 [P −1 (kq)]λ2 [P −1 (kq)] · · · λn0 [P −1 (kq)] 0 ≤ λnmax [P −1 (kq)] ≤ rn0 (kq),

ln |P

(23)

−1

(kq)| = O(ln r(kq)).

In order to study the output tracking performance of the adaptive control algorithm proposed earlier, the following lemma is required. Lemma 1. For the algorithm in (11)–(16), the following inequality holds: ∞ ϕT (iq − 1)P (iq)ϕ(iq − 1) i=1

{ln r(iq)}c

< ∞ a.s. for any c > 1.

Proof. From the definition of P (kq), we have P −1 (iq − q) = P −1 (iq) − ϕ(iq − 1)ϕT (iq − 1) = P −1 (iq)[I − P (iq)ϕ(iq − 1)ϕT (iq − 1)]. Taking determinants on both sides gives |P −1 (iq − q)| = |P −1 (kq)||I − P (kq)ϕ(kq − 1)ϕT (kq − 1)| = |P −1 (iq)|[1 − ϕT (iq − 1)P (iq)ϕ(iq − 1)]. Thus, (24)

|P −1 (iq)| − |P −1 (iq − q)| . |P −1 (iq)|

ϕT (iq − 1)P (iq)ϕ(iq − 1) =

Noting that |P −1 (iq)| is a nondecreasing function of i, dividing both sides of (24) by {ln r(kq)}c gives ∞ ϕT (iq − 1)P (iq)ϕ(iq − 1) i=1

{ln r(iq)}c

=

∞ |P −1 (iq)| − |P −1 (iq − q)| i=1

≤ nc0 = nc0

∞ |P −1 (iq)| − |P −1 (iq − q)|

|P −1 (iq)|{ln |P −1 (iq)|}c i=1 ∞ |P −1 (iq)| i=1

≤ nc0

|P −1 (iq)|{ln r(iq)}c

|P −1 (iq−q)|

∞ i=1

|P −1 (iq)|

|P −1 (iq−q)|

dx |P −1 (iq)|{ln |P −1 (iq)|}c dx x(ln x)c


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|P −1 (∞)|

dx x(ln x)c |P −1 (0)| −1 c 1 −n0 |P (∞)| · = −1 c − 1 {ln x}c−1 |P (0)| 1 nc0 1 = − c − 1 {ln |P −1 (0)|}c−1 {ln |P −1 (∞)|}c−1 < ∞ a.s., c > 1. =

nc0

We shall prove the main results of this paper by formulating a martingale process as in [13, 14, 19, 21] and by using stochastic process theory and the martingale convergence theorem (Lemma D.5.3 in [25]). Theorem 2. For the system in (10), assume that (A1)–(A3) hold, B(z) is stable, and the reference input yr (k) is bounded in the sense of (A4) |yr (k)| < ∞. Then the adaptive control algorithm in (11)–(17) guarantees that the output tracking error at the output sampling instants has the property of minimum variance, i.e., k 1 [yr (iq) − y(iq) + v(iq)]2 = 0 a.s.; k→∞ k i=1

lim

lim sup k→∞

k 1 E{[yf (iq) − yr (iq)]2 |Fiq−1 } ≤ σ ¯ 2 < ∞ a.s. k i=1

Proof. Define the parameter estimation error vector as ˜ ˆ θ(kq) = θ(kq) − θ. Using (10) and (11), we have ˜ ˜ ˆ θ(kq) = θ(kq − q) + P (kq)ϕ(kq − 1)[ϕT (kq − 1)θ + v(kq) − ϕT (kq − 1)θ(kq − q)] ˜ (25) := θ(kq − q) + P (kq)ϕ(kq − 1)[−˜ y (kq) + v(kq)], where (26)

˜ ˆ y˜(kq) := ϕT (kq − 1)θ(kq − q) = ϕT (kq − 1)θ(kq − q) − ϕT (kq − 1)θ.

By using (10) and (16), it follows that y˜(kq) = yr (kq) − y(kq) + v(kq). Define a nonnegative definite function ˜ V (kq) = θ˜T (kq)P −1 (kq)θ(kq). Using (10), (25), and (26), we have ˜ V (kq) = θ˜T (kq − q)P −1 (kq)θ(kq − q) + 2θ˜T (kq − q)ϕ(kq − 1)[−˜ y (kq) + v(kq)] +ϕT (kq − 1)P (kq)ϕ(kq − 1)[−˜ y (kq) + v(kq)]2 ˜ = θ˜T (kq − q)[P −1 (kq − q) + ϕT (kq − 1)ϕ(kq − 1)]θ(kq − q) +2˜ y (kq)[−˜ y (kq) + v(kq)] + ϕT (kq − 1)P (kq)ϕ(kq − 1)[−˜ y (kq) + v(kq)]2 = V (kq − q) − [1 − ϕT (kq − 1)P (kq)ϕ(kq − 1)]˜ y 2 (kq) +ϕT (kq − 1)P (kq)ϕ(kq − 1)v 2 (kq) (27)

+2[1 − ϕT (kq − 1)P (kq)ϕ(kq − 1)]˜ y (kq)v(kq).

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Noting that y˜(kq), ϕT (kq − 1)P (kq)ϕ(kq − 1) are uncorrelated with v(kq) and are Fkq−1 -measurable, taking the conditional expectation on both sides of (27) with respect to Fkq−1 and using (A1)–(A2) give E[V (kq)|Fkq−1 ] ≤ V (kq − q) − [1 − ϕT (kq − 1)P (kq)ϕ(kq − 1)]˜ y 2 (kq) σ2 . +2ϕT (kq − 1)P (kq)ϕ(kq − 1)¯ Let W (kq) :=

V (kq) , c > 1. [ln r(kq)|]c

Noting that ln r(kq) is nondecreasing, we have V (kq − q) 1 − ϕT (kq − 1)P (kq)ϕ(kq − 1) 2 − y˜ (kq) [ln r(kq)]c [ln r(kq)]c 2ϕT (kq − 1)P (kq)ϕ(kq − 1) 2 σ ¯ + [ln r(kq)]c 1 − ϕT (kq − 1)P (kq)ϕ(kq − 1) 2 ≤ W (kq − q) − y˜ (kq) [ln r(kq)]c 2ϕT (kq − 1)P (kq)ϕ(kq − 1) 2 σ ¯ . + [ln r(kq)]c

E[W (kq)|Fkq−1 ] ≤

(28)

In terms of Lemma 1, we can see that the sum of the last right-hand term of (28) for k from k = 1 to k = ∞ is finite. Since 1 − ϕT (kq − 1)P (kq)ϕ(kq − 1) = [1 + ϕT (kq − 1)P (kq − q)ϕ(kq − 1)]−1 ≥ 0, applying the martingale convergence theorem (Lemma D.5.3 in [25]) to (28), we conclude that W (kq) converges a.s. to a finite random variable, say, W0 ; i.e., W (kq) =

V (kq) → W0 < ∞ a.s., or V (kq) = O([ln r(kq)]c ) a.s., [ln r(kq)]c

and also ∞ 1 − ϕT (kq − 1)P (kq)ϕ(kq − 1)

[ln r(kq)]c

k=1

y˜2 (kq) < ∞ a.s.

Due to ϕ (kq − 1)P (kq)ϕ(kq − 1) = o(1), we have T

∞

(29)

i=1

y˜2 (iq) < ∞ a.s. [ln r(iq)]c

As r(kq) → ∞, using the Kronecker lemma (Lemma D.5.5 in [25]) yields k 1 y˜2 (iq) = 0 a.s. k→∞ [ln r(kq)]c i=1

lim

Since [ln r(kq)]c = o(r(kq)), we have (30)

k k 1 2 y˜ (iq) = 0 a.s. k→∞ r(kq) k i=1

lim


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Since B(z) is stable, applying Lemma B.3.3 in [25] to (10) and using (A3) yields k k 1 2 c1 2 u (iq) ≤ y (iq) + c2 a.s., k i=1 k i=1

where ci represent finite positive constants. According to the definitions of r(kq) and ϕ(kq), it is not difficult to get k c3 2 r(kq) ≤ y (iq) + c4 k k i=1

=

k c3 [yr (iq) − y˜(iq) + v(iq)]2 + c4 k i=1

≤

k c5 2 y˜ (iq) + c6 a.s. k i=1

Thus, from (30) 1 k

0 = lim

k→∞

k

1 k

y˜2 (iq)

i=1 r(kq) k

≥ lim

k→∞

c5 k

k

y˜2 (iq)

i=1 k y˜2 (iq)

≥ 0 a.s., + c6

i=1

and hence k 1 2 lim y˜ (iq) = 0 a.s., k→∞ k i=1

or k 1 lim [yr (iq) − y(iq) + v(iq)]2 = 0 a.s. k→∞ k i=1

(31) Since

E{[yr (kq) − y(kq) + v(kq)]2 |Fkq−1 } = E[(yr (kq) − y(kq))2 + 2yr (kq)v(kq) − 2y(kq)v(kq) + v 2 (kq)|Fkq−1 ] = E[(yr (kq) − y(kq))2 |Fkq−1 ] + 0 − 2σ 2 (kq) + σ 2 (kq) = E[(yr (kq) − y(kq))2 |Fkq−1 ] − σ 2 (kq) a.s., and yf (kq) = y(kq) at the output sampling instants, we have lim sup k→∞

k k 1 1 E{[yf (iq) − yr (iq)]2 |Fiq−1 } = lim sup E{[y(iq) − yr (iq)]2 |Fiq−1 } k i=1 k→∞ k i=1 k 1 2 σ (iq) ≤ σ ¯ 2 a.s. = lim sup k→∞ k i=1

This proves Theorem 2. Since single-rate systems belong to a special class of dual-rate systems with q = 1, the results in Theorem 2 still hold for single-rate Hammerstein nonlinear systems, which, to the best of our knowledge, have not even been reported in the literature before.

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5. Global convergence. In this section, we analyze global stability properties of the closed-loop system under the adaptive control proposed earlier. From (3) and (10), we have (32)

yf (kq) = y(kq) = ϕT (kq − 1)θ + v(kq),

(33)

yf (kq + j) = yˆ(kq + j), j = 1, 2, . . . , q − 1.

From Figure 2 and (10), since v(kq) is a “white” noise, the best estimates of all missing output y(kq + j) are given by ˆ yˆ(kq + j + 1) = ϕˆT (kq + j)θ(kq), j = 0, 1, . . . , q − 2, with ⎡

−ˆ y (kq + j + 1 − q) −ˆ y (kq + j + 1 − 2q) .. .

⎢ ⎢ ⎢ ϕ(kq ˆ + j) = ⎢ ⎢ ⎣ −ˆ y (kq + j + 1 − nq) ψ(kq + j) ⎡

⎤

⎡ ψ1 (kq + j) ⎥ ⎥ ⎢ ψ2 (kq + j) ⎥ ⎢ ⎥ ∈ Rn0 , ψ(kq + j) = ⎢ .. ⎥ ⎣ . ⎦ ψm (kq + j)

γi (u(kq + j)) γi (u(kq + j − 1)) .. .

⎢ ⎢ ψi (kq + j) = ⎢ ⎣

⎤ ⎥ ⎥ ⎥ ∈ Rmqn , ⎦

⎤ ⎥ ⎥ ⎥ ∈ Rqn , i = 1, 2, . . . , m. ⎦

γi (u(kq + j + 1 − nq)) The missing output estimates yˆ(kq + j) can also be computed from the recursive equation: (34)

yˆ(kq + j + 1) = −

n

y (kq + j + 1 − iq) θî (kq)ˆ

i=1

+

nq m

θˆn+(l−1)qn+i (kq)γl (u(kq + j + 1 − i)),

l=1 i=1

j = 0, 1, . . . , q − 2. Comparing (18) with (34), we find that the missing intersample output estimates yˆ(kq + j), j = 1, 2, . . . , q − 1, equal the desired outputs yr (kq + j); so we have (35)

(36)

ˆ j = 1, 2, . . . , q − 1, yr (kq + j) = yˆ(kq + j) = ϕˆT (kq + j)θ(kq), ⎤ ⎡ −yr (kq + j + 1 − q) ⎢ −yr (kq + j + 1 − 2q) ⎥ ⎥ ⎢ ⎥ ⎢ .. ϕ(kq ˆ + j) = ⎢ ⎥. . ⎥ ⎢ ⎣ −yr (kq + j + 1 − nq) ⎦ ψ(kq + j)

It is easy to understand that the unknown intersample outputs y(kq + j) are replaced by the desired outputs yr (kq + j) because our goal is to make y(k) track yr (k). Hence, combining (20) with (35)–(36) generates the control signal sequence {u(kq+j),


2271

ˆ j = 0, 1, . . . , q − 1} based on the parameter estimates θ(kq) obtained. Thus, the following theorem is easily established. Theorem 3. Assume that the conditions of Theorem 2 hold, A(z) and B(z) both are stable, and f (·) is invertible. Then the adaptive control algorithm in (11)–(15), (19), and (20) ensures that the closed-loop system is stable and globally convergent with probability 1; moreover, • the input and output variables are uniformly bounded, i.e., lim sup k→∞

k 1 2 [u (i) + y 2 (i) + yf2 (i)] < ∞ a.s.; k i=1

• the average output tracking error is less than or equal to σ ¯ 2 /q, i.e., lim sup k→∞

k 1 σ ¯2 E{[yf (i) − yr (i)]2 |Fi−1 } ≤ a.s. k i=1 q

Proof. Since yr (k) is bounded, from Theorem 2 and condition (A3), it is easy to get that the outputs y(kq) at the output sampling instants are uniformly bounded, i.e., lim sup k→∞

k 1 2 y (iq) ≤ δy < ∞ a.s. k i=1

Also, the intersample output estimates yˆ(kq + j), j = 1, 2, . . . , (q − 1), satisfy yˆ(kq + j) = yr (kq + j), j = 1, 2, . . . , q − 1. So yf (kq + j) is bounded. According to (32) and (33), yf (k) is bounded. Since A(z) and B(z) are stable, so are α(z) and β(z), and u(k) is bounded in terms of Lemma B.3.3 in [25]. Hence we have k 1 2 lim sup u (i) < ∞ a.s., k→∞ k i=1

lim sup k→∞

k 1 2 lim sup y (i) < ∞ a.s., k→∞ k i=1

k 1 2 y (i) < ∞ a.s., k i=1 f

which means that all the input and output variables are uniformly bounded. Also, lim sup k→∞

k k 1 1 E{[yf (i) − yr (i)]2 |Fi−1 } = lim sup E{[y(iq) − yr (iq)]2 |Fiq−1 } k i=1 k→∞ kq i=1

+ lim sup k→∞

q−1 k 1 E{[ˆ y (iq + j) − yr (iq + j)]2 |Fiq+j−1 }. kq j=1 i=1

Since yf (i) = yˆ(i) = yr (i) at the missing output sampling instants, the last term on the right-hand side is zero, and the first term is no more than σ ¯ 2 /q from Theorem 2. This proves Theorem 3. Theorems 2 and 3 indicate that the proposed adaptive control scheme in the dualrate setting can achieve the property of minimum variance at the output sampling instants. Between the output sampling instants, we have yˆf (kq + j) = yˆ(kq + j) = yr (kq + j), j = 1, 2, . . . , q − 1. The control scheme here is easily extended to non-

2272


minimum phase cases by defining a new weighted criterion: J[u(k)] = E{[P (z)yf (k + d) − Q(z)yr (k + d)]2 + [R(z)u(k)]2 |Fk−1 }. Here, P (z), Q(z), and R(z) are polynomials in z −1 . If the single-rate system considered is an AR model with exogenous input (ARX model), then by the model transform, the noise model should be φq (z)v(k) instead of v(t). This leads to an ARMAX model. In this case, we may use the recursive extended least squares algorithm in [25] to estimate the parameters. The persistent excitation condition is required for the convergence of the parameter estimation. Like in linear single-rate cases [25], adaptive control algorithms do not guarantee the convergence of the parameter estimation to their true values. In order to avoid generating u(k) with too large magnitudes, for a given small positive ε, if |βˆ1 (kq)| < ε, we take βˆ1 (kq) = sgn[βˆ1 (kq)]ε, where the sign function is defined by 1 x ≥ 0, sgn(x) = 0 x < 0. 6. Example. In this section, we illustrate the results reported with simulation examples, including an experimental water level system. Example 1. For the Hammerstein nonlinear model shown in Figure 1, we select the dual-rate sampling ratio as q = 2, the fast-rate discrete-time (dynamic) model G(z) as a second-order one with G(z) =

B(z) 0.412z −1 + 0.309z −2 , = A(z) 1 − 1.60z −1 + 0.80z −2

and the static nonlinear function f (·) as u ¯(k) = f (u(k)) = c1 u(k) + c2 u2 (k) + c3 u3 (k) = u(k) + 0.5u2 (k) + 0.25u3 (k). We choose the noise sequence {v(k)} to be a white noise sequence with zero mean and variance σ ¯ 2 = 0.052 and the desired output to be yr (400i + j) = (−1)i+1 , i = 0, 1, 2, . . . ; j = 1, 2, . . . , 400. The adaptive control algorithm in section 3 is applied to this system. The system output y(k) and the desired output yr (k) are shown in Figure 3 with q = 2. Figure 4 with q = 1 is the simulated results of the ˚ Aström–Wittenmark self-tuning regulator (A–W STR) suitable for single-rate systems [2]. From Figures 3 and 4, we can see that the control algorithm proposed in this paper can achieve both less and more stationary average tracking errors than the A–W STR algorithm. This is due to yˆ(kq + j) − yr (kq + j) = 0, j = 1, 2, . . . , q − 1, between output sampling instants. Thus, the closed-loop tracking performance is satisfactory. This confirms the results reported earlier. Example 2. This example is a computer-controlled experimental water tank system at the University of Alberta, and is shown in Figure 5. In this system, the manipulated variable is the position of the inlet water valve, denoted u(t); the measured variable is the water level in the tank, denoted y(t); the inner variable, namely, the water flow rate u ¯(t) (the output of the water valve) is not measured. The map from u(t) to u ¯(t) is nonlinear and can be approximated by a third-order polynomial, u ¯(t) = c1 u(t) + c2 u2 (t) + c3 u3 (t)


2273

1.5 1

y(k), yr(k)

0.5 y(k) 0

−0.5 yr(k) −1 −1.5

200

400

600

800

1000

k

1200

1400

1600

1800

2000

1600

1800

2000

Fig. 3. y(k) and yr (k) versus k (q = 2).

1.5 1

r

y(k), y (k)

0.5 y(k)

0 −0.5 yr(k) −1 −1.5

200

400

600

800

1000

1200

1400

k Fig. 4. y(k) and yr (k) versus k (q = 1).

(keeping the first few terms in the Taylor series expansion). For our study, the input u(t) is updated every h = 40 s to get u(k), and the output y(t) is sampled every qh = 80 s (hence q = 2), yielding y(kq). Around an operating point, we use a random binary sequence generated by MATLAB as the input signal. By data correlation analysis on the collected input-output data, we determined that the model order associated with G(z) from u ¯(k) to y(k) is 2 with a delay of 2 samples, and hence G(z) has the form of G(z) =

b2 z −2 + b3 z −3 . 1 + a1 z −1 + a2 z −2

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Fig. 5. Experimental setup of a water tank.

16 15.5 15 y(k), yr(k)

14.5 y(k)

14

13.5

y (k) r

13 12.5 12

200

400

600

800

1000

1200

1400

1600

1800

2000

k Fig. 6. y(k) and yr (k) versus k (q = 2).

Using the polynomial transform, we estimated the new model from u ¯(k) to y(kq) as follows: P (z) =

0.92691z −2 + 0.83503z −3 − 0.65555z −4 − 0.66384z −5 β(z) = . α(z) 1 − 1.55795z −2 + 0.60728z −4

The parameters of P (z) were obtained by the dual-rate least squares algorithm in (11)–(15); see [20]. Similarly, we took {v(k)} to be a white noise sequence with zero mean and variance σ ¯ 2 = 0.052 and the desired output to be yr (400i + j) = 14 + (−1)i , i = 0, 1, 2, . . . ; j = 1, 2, . . . , 400. Here, yr (k) = 14 is the operating point. The adaptive control algorithm in section 3 was applied to this water tank system. The system output y(k) and the desired output yr (k) are shown in Figure 6. From Figure 6, we conclude that satisfactory tracking performance has been achieved, and this shows that the algorithm reported is effective.


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