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[10] F. Deza, E. Busvelle, and J. P. Gauthier, “Exponentially converging observers for distillation columns and internal stability of the dynamic output feedback,” Chemical Eng. Sci., vol. 47, pp. 3935–3941, 1992. [11] G. Ciccarella, M. D. Mora, and A. Germani, “A Luenberger-like observer for nonlinear systems,” Int. J. Contr., vol. 57, pp. 537–556, 1993. [12] M. Farza, K. Busawon, and M. Hammouri, “Simple nonlinear observer for on-line estimation of kinetic rates in bioreactors,” Automatica, vol. 34, pp. 301–318, 1998. [13] J. P. Gauthier and I. A. K. Kupka, “Observability for systems with more outputs than inputs and asymptotic observers,” Mathematische Zeitschrift, vol. 223, pp. 47–78, 1996. [14] H. Hammouri, K. Busawon, and M. Farza, “Nonlinear observer for local uniform observable systems,” Ph.D. dissertation, Universite Claude Bernard—Lyon I, France, 1996. K. Busawon: Sur les observateurs des systemes nonlineares et le principe de séparation, No. 03396. [15] H. Nijmeijer, “Observability of a class of nonlinear systems: A geometric approach,” Ricerche di Automatica, vol. 12, pp. 1–19, 1981. [16] J. Levine and R. Marino, “Nonlinear system immersion, observers and finite-dimensional filters,” Syst. Contr. Lett., vol. 7, pp. 133–142, 1986. [17] A. J. Krener, “Nonlinear stability and detectability,” in Proc. Int. Symp. Mathematical Theory of Networks Systems, Regensburg, Germany, 1993, pp. 231–250. [18] X.-H. Xia and M. Zeitz, “On nonlinear continuous observers,” Int. J. Contr., vol. 66, pp. 943–954, 1997. [19] H. Nijmeijer and A. J. van der Schaft, Nonlinear Dynamical Control Systems. New York: Springer-Verlag, 1990. [20] G. B. Price, Multivariable Analysis. New York: Springer-Verlag, 1984. [21] E. D. Sontag, “Feedback stabilization using two-hidden layer nets,” IEEE Trans. Neural Nets, vol. 3, pp. 981–990, 1991.
linear AutoRegressive Moving Average (ARMA) and AutoRegressive Moving Average with Exogenous Inputs (ARMAX) structures with time-varying coefficients. In [2], the time variations are modeled by preselected basis functions. Here, in contrast, we drop the assumption that these time variations evolve according to some predetermined functional structure; the ARMA(X)2 coefficients are assumed to follow a random walk, and are estimated by a linear time-varying Kalman filter, as reported in our preliminary work [1]. The use of random walk to describe unknown time variations is not new: it can be found in econometrics [3], statistics [4], and earlier in optimal estimation [5], [6]. The goal of this note is not to estimate models of time-varying plants: it is to find input-output models for time-invariant nonlinear plants. The time-varying ARMA(X) structure is used as the approximating model, the random walk describes the unknown time-variations of the model parameters, and the Kalman filter estimates these parameters. In this note, time-varying ARMA and ARMAX models are jointly denoted by TVARMA(X). II. THE CONVENTIONAL ARMA AND ARMAX MODELS A linear, time-invariant (LTI), discrete-time state-space system (with output yk and input uk ) can be converted to an ARMA model n yk
=
=1
a(i)yk
n
0i +
=1
i
Input–Output Modeling of Nonlinear Systems with Time-Varying Linear Models
n yk
=
a(i)yk
=1
0i +
i
Index Terms—ARMAX model, input-output modeling, nonlinear systems, on-line, random walk Kalman filter, time domain.
n
I. INTRODUCTION
1Among
others, neural network methods fall under this category.
n
=1
b(i)uk
0i +
i
=
=1
a(i)yk
0i +
i
Manuscript received November 10, 1998; revised March 9, 1999, June 29, 1999, and December 20, 1999. Recommended by Associate Editor, J. C. Spall. This work was supported in part by National Science Foundation under Grant ECS-9753084. The author is with the Department of EECE, University of Louisiana at Lafayette, Lafayette, LA 70504 USA (e-mail:
[email protected]). Publisher Item Identifier S 0018-9286(00)06099-2.
(1)
n
=1
c(i)ek
0i + ek :
(2)
i
The coefficients a(i); b(i); and c(i) are can be computed from state-space or transfer function models. If the system model is not known, then the ARMAX coefficients can be estimated from measured values of yk ; uk ; and ek . However, the true observation noise ek cannot be measured; so we replace them with the residuals obtained from previous estimates. This modified version still represents the original system: we simply estimate different coefficients fd(i)g instead of fc(i)g. Thus, the ARMAX model that can be implemented in real time is yk
This note presents a technique for input-output (I/O) modeling of deterministic and stochastic nonlinear systems. For nonlinear plants, usually a specific structure for the unknown nonlinear function1 is assumed; recently, an alternative method proposed in [1] and [2] uses
0i
where a(i) and b(i) are constant coefficients [9]. Sometimes (1) is used to represent stochastic processes where uk is noise. Here, as in typical control system literature, uk is a deterministic input [7]–[9]. If the measurements are corrupted by noise, that is, xk+1 = F xk + Guk , and yk = H xk + ek , then (1) cannot represent the process correctly. The proper I/O relationship is then given by the ARMAX model:
Fahmida N. Chowdhury
Abstract—Time-varying ARMA (AutoRegressive Moving Average) and ARMAX (AutoRegressive Moving Average with Exogenous Inputs) models are proposed for input-output modeling of nonlinear deterministic and stochastic systems. The coefficients of these models are estimated by a Random Walk Kalman Filter (RWKF). This method requires no prior assumption on the nature of the model coefficients, and is suitable for real-time implementation since no off-line training is needed. A simulation example illustrates the method. Goodness of performance is judged by the quality of the residuals, histograms, autocorrelation functions and the KolmogorovSmirnoff test.
b(i)uk
i
n
=1
i
b(i)uk
0i +
n
=1
d(i)rk
0i + ek
(3)
i
where rk are residuals of the estimation process up to time k . III. INPUT–OUTPUT MODELING OF NONLINEAR SYSTEMS Generic models for nonlinear systems (NARMA and NARMAX—Nonlinear ARMA and ARMAX) have been proposed in [8]. The NARMAX model is yk
= f (a1 yk01 ; 1 1 1 ; an yk0n ; b1 uk01 ; 1 1 1 ; bn uk
0n ; c1 ek01 ; 1 1 1 ; cn ek0n ) + ek
(4)
where f ( ) is an unknown function. The corresponding NARMA model is obtained by deleting the terms containing ci . Artificial 2By
ARMA(X) we shall mean “ARMA or ARMAX.”
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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 2000
neural networks have been proposed and studied as efficient tools for estimating nonlinear systems (see, for example, [10]). NN-based NARMAX models are described in [11] and [12]. However, neural networks require lengthy training, convergence of the estimation process is sometimes uncertain, and choice of network architectures remains somewhat of an art. Under these circumstances, alternative methods are worth exploring. Comparative studies of NN-based NARMA(X) and Kalman filter based TVARMA(X) approaches by this author will be reported elsewhere; preliminary results of comparing NN-based NARMA and Kalman filter based TVARMA models are reported in [1]. A. Time-Varying ARMA and ARMAX Structures for Representing Nonlinear Plants In this note, rather than trying to estimate the unknown nonlinear function (as in the NARMA(X) model), we replace it with a family of piece-wise linear functions. The conjecture is that at each discrete time step, a linear system can match the I/O data of the nonlinear plant. This produces a time-varying family of ARMAX (TVARMAX) models: yk
=
n i=1
ak (i)yk0i +
n i=1
bk (i)uk0i +
n i=1
dk (i)rk0i + ek : (5)
Here, ak (i); bk (i), and dk (i) are the coefficients of the model at time-step k; rk = yk 0 y^k=k01 are the residuals of the estimation process from previous time-steps,3 and ek is the observation noise at time-step k . 1) Random Walk Kalman Filter (RWKF) for the Estimation Process: The TVARMAX structure is used here to illustrate the use of RWKF for estimating the model coefficients. The TVARMA case is obtained by deleting the d(i) and rk0i terms from the equations. Define the “state vector” as k
= [ak (1); 1 1 1 ; ak (n); bk (1); 1 1 1 ; bk (n); dk (1); 1 1 1 ; dk (n)]
T
= k + wk :
yk
=8
+ ek
(7)
0
Qk=k
= E [k 0 ^k=k ][k 0 ^k=k ]T
(10)
is updated (beginning with an initial guess value) at each time step according to Qk=k
=
I
0 Kk 8kT
Qk=k01 :
(11)
One-step-ahead projection of this covariance is given by Qk+1=k
= Qk=k + Rk :
(12)
The Kalman gain is computed as Kk
= Qk=k01 8k
Sk + 8Tk Qk=k01 8k
01
:
(13)
^k=k
= ^k=k01 + Kk rk :
(14)
Since the system matrix in our fictitious system is the Identity, the state estimate remains unchanged for the one-step-ahead projection:
= ^k=k :
(15)
(8)
(9)
In the above, wk (the random walk term, acting as process noise in the Kalman filter) and ek are zero-mean uncorrelated noise processes, with covariance matrices Rk = E [wk wkT ] (where E denotes expected value) and Sk = E [ek ekT ]. Equations (7) and (8) represent a discrete-time linear system with time-varying observation matrix 8k , identity system matrix, process
3The suffix k=k 1 indicates “at step k , with information available up to step k 1,” and the suffix k=k stands for “at step k updated with information available at step k .”
0
The covariance matrix of the parameter estimates
^k+1=k
where
8k = [yk01 ; 1 1 1 ; yk0n ; uk01 ; 1 1 1 ; uk0n ; rk01 ; 1 1 1 ; rk0n ]T :
1) The observation matrix 8k is time-varying, and includes past inputs, outputs and residuals [(9)]. 2) The “state vector” k [(6)] consists of the coefficients of the terms included in 8k . 3) The “system matrix” used in the algorithm is Identity. 4) The “process noise” is a design parameter that should be chosen depending on the severity of the nonlinearity.
The updated estimate of the state vector at step k is
The observed output of this fictitious state space model is given by T k k
noise wk , and observation noise ek . A Kalman filter can estimate k , generate the residual rk = yk 0 8Tk ^k=k01 , and update the model coefficients using the Kalman gain. The Kalman filter equations, customized to fit the TVARMAX model, are given below. Remark 1: The algorithm presented in this section is formally similar to the standard Kalman filter, but the interpretation and construction of some major terms are totally different:
(6)
where T indicates transpose, and k contains the TVARMAX coefficients to be estimated. The evolution of the state follows a random walk: k+1
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Initial guesses for the Kalman filter covariance matrices can be obtained by standard methods outlined by Box and Jenkins [13]. Remark 2: If a small amount of random noise is present in the measured outputs, the TVARMA model still provides very good I/O matching; however, the residuals are not Gaussian in such a case even if the observation noise is Gaussian. We demonstrate this through a simulation example later in the paper. Remark 3: This technique is not the same as extended Kalman filtering, although there is a similarity in the adaptive parameter estimation. Only the basic, linear Kalman filter is used here; there is no joint estimation of “states” and “parameters.” Moreover, system nonlinearities are assumed completely unknown. Remark 4: Accuracy of the model generated by the RWKF-based TVARMA(X) method has not been determined beyond computation of the mean-square-error during estimation process. Since the distribution of the residuals is non-Gaussian, the asymptotic distribution theory for Kalman filters developed by Spall [14], [15] should be applicable here.
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Fig. 1. The plant and the model estimator.
Fig. 3. Outputs of the plant and the TVARMA (RWKF) model.
Fig. 2 shows the histogram of the residuals. Despite the bell shape, the residuals fail the Kolmogorov-Smirnoff test [16] (for Gaussianness) at a 95% confidence level. The test is carried out on a normalized version of the residual; the maximum distance between the empirical and the normal distribution is 0.19, whereas a 95% confidence level requires a distance less than 0.11. However, the model output is very close to the plant output: Fig. 3 shows that the plant and estimated outputs practically overlap, and the mean-square error is in the order of 04 10 . V. CONCLUSION
Fig. 2. Histogram of residuals.
IV. EXAMPLE OF IMPLEMENTATION The proposed technique has been tested in many different examples, always with excellent results.4 Here, only one result is presented, with general observations. The number of past time events used in the model must be preset (that is, model order is assumed to be known). The covariance of the random walk term (Rk ) is a design parameter: in general, to track fast-changing ARMA coefficients, use a higher covariance for the random walk term. This would indicate a highly nonlinear system being approximated; conversely, if a system were known to be linear, Rk could be set to zero, resulting in an ARMAX model estimation. The overall effect of Rk is similar to that of a forgetting factor in the standard Kalman filter. The nonlinear plant used in the simulation example is shown in Fig. 1. The nonlinearity is f (x) = sin(x2 ). The input u is band-limited Gaussian white noise. A small Gaussian observation noise (mean zero, variance Sk = 0:01) is added to the output. Using the I/O data, the Kalman filter estimates a second-order TVARMA model and generates a sequence of estimated outputs. The initial guess value for k is zero; Rk is 0.01. The TVARMA model should perform very well in this case, but the distribution of the residuals would not be Gaussian.5 Due to space restrictions, the residuals and their autocorrelation functions are not shown in this paper.
The proposed technique is suitable for generating on-line I/O models for unknown nonlinear plants with or without observation noise. The main attractions are its accuracy and practicality, and its potential for real-time implementation. Its advantages over other existing methods are that there is no need for a separate off-line training phase6 and no need to assume a set of pre-determined basis functions for describing the time-variations7 of the model coefficients. The novelty of this paper is in combining existing tools (time-varying ARMA(X) models, representing nonlinearities with piecewise linear functions, and using RWKF) in a specific way to achieve real-time accurate models of nonlinear systems. This approach can be utilized in adaptive controller design for unknown dynamic systems. ACKNOWLEDGMENT The author is grateful to the anonymous reviewers and to the Associate Editor for their constructive and thorough reviews and many helpful suggestions. REFERENCES [1] A. Mu’awin and F. N. Chowdhury, “Extrapolative models of dynamic systems: Neural networks vs. Kalman filters,” in Proc. 29th Southeastern Symp. Syst. Theory (SSST’97): IEEE Comp. Soc. Press, Mar. 1997, pp. 315–319. [2] R. B. Mrad, S. D. Fassois, and J. A. Levitt, “A polynomial algebraic method for nonstationary TARMA signal analysis,” Signal Process., pt. I, vol. 65, pp. 1–19, 1998. and Part II, pp. 21–38.
4Details
are available from the author upon request. example illustrates that even with small observation noise, using the ARMA instead of the ARMAX formulation results in nonnormality of the residuals. 5The
6As 7As
required in the neural network approach. required in another, currently proposed approach [2].
IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 7, JULY 2000
[3] G. Kitagawa and W. Gersch, Smoothness Priors: Analysis of Time Series. New York: Springer, 1996, pp. 147–178. Lecture Notes in Statistics. [4] B. Abraham and J. Ledolter, Statistical Methods for Forecasting, 1983, pp. 364–368. Wiley Series in Prob. and Math. Statistics. [5] F. C. Schweppe, Uncertain Dynamic Systems. Englewood Cliffs, NJ: Prentice-Hall, 1973, pp. 432–433. [6] A. Gelb, Ed., Applied Optimal Estimation. Cambridge, MA: MIT Press, 1974, pp. 348–349. [7] E. Mosca, Optimal, Predictive, and Adaptive Control, T. Kailath, Ed. Englewood Cliffs, NJ: Prentice Hall Information and System Sciences Series, 1995. [8] I. J. Leontaritis and S. A. Billings, “Input-output parametric models for nonlinear systems,” Int. J. Contr., vol. 41, no. 2, pp. 329–344, 1985. [9] G. Goodwin and K. Sin, Adaptive Filtering Prediction and Control, T. Kailath, Ed. Englewood Cliffs, NJ: Prentice Hall, 1984, pp. 262–263. [10] P. J. Werbos, T. McAvoy, and T. Su, “Neural networks, system identification, and control in the chemical process industries,” in Handbook of Intelligent Control, D. White and D. Sofge, Eds. New York: Van Nostrand Reinhold, 1992, pp. 283–356. [11] F. N. Chowdhury, E. Lobo, X. Pei, and R. Thangavelu, “A neurostatistical method for fault detection in stochastic systems,” in Proc. IEEE Conf. Contr. Applica., 1998, pp. 283–287. [12] P. M. Norgaard. (1998) Toolkit for Use with MATLAB. [Online]. Available: http://kalman.iau.dtu.dk/research/control/nnsysid.html [13] G. E. P. Box and G. M. Jenkins, Time Series Analysis: Forecasting and Control. San Francisco, CA: Holden-Day, 1976. [14] J. C. Spall, “Asymptotic distribution theory for the Kalman filter state estimator,” Commun. Statist.-Theor. Meth., vol. 13, no. 16, pp. 1981–2003, 1984. , “The Kantorovich inequality for error analysis of the Kalman [15] filter with unknown noise distributions,” Automatica, vol. 31, no. 10, pp. 1513–1517, 1995. [16] A. Papoulis, Probability and Statistics. Englewood Cliffs, NJ: Prentice Hall, 1990, pp. 339–340.
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Robustification of Backstepping Against Input Unmodeled Dynamics Murat Arcak, Maria Seron, Julio Braslavsky, and Petar Kokotovic´ Abstract—We present two redesigns to robustify backstepping control laws against dynamic uncertainties at the input of the plant. In the first redesign we construct a passivating control law from a control Lyapunov function (CLF) obtained by backstepping. The second redesign is a simplification of the first, useful for high-order systems. We also compare the stability margins of the two main versions of backstepping: cancellation -backstepping. and Index Terms—Backstepping, robust stabilization, unmodeled dynamics.
I. INTRODUCTION In this paper we redesign backstepping schemes such as those in [1] and [2] to robustify them against input unmodeled dynamics. We consider systems in the form
_ = F (X ) + G(X )x1 x_ 1 = f1 (X; x1 ) + g1 (X; x1 )x2 x_ 2 = f2 (X; x1 ; x2 ) + g2 (X; x1 ; x2 )x3 X
(1)
.. .. . =.
_ = fn (X; x) + gn (X; x)v _ = q (; u) v = p(; u) (2) where jgi (X; 1 1 1 ; xi )j g0 > 0, 8 (X; 1 1 1 ; xi ) 2 r+i , i = 1; 1 1 1 ; n. The -subsystem (2) with input u 2 and output v 2 represents unmodeled dynamics, that is, (1) with v = u is the nominal system. When u = 0, the system (1), (2) has an equilibrium at zero, xn
whose stability properties are to be analyzed. For the X -subsystem, with x1 viewed as a virtual control input, a control Lyapunov function (CLF) V0 (X ) and a control law 30 (X ), 30 (0) = 0, are known such that, for all X 6= 0,
0 (F + G30 ) = 0U (X ) < 0: ( ) := @V 0 @X
LF +G3 V0 X
(3)
With the knowledge of V0 (X ) and 30 (X ), backstepping can be applied to guarantee global asymptotic stability (GAS) for the nominal system. However, a GAS control law for the nominal system does not guarantee GAS in the presence of unmodeled dynamics. To robustify backstepping designs, we propose two redesign methods: passivation and truncated passivation. In the first redesign we use the results of [3] and ensure GAS via the passivity properties of the closed-loop system. In the second redesign, we passivate the X -subsystem and proceed with Manuscript received November 25, 1998; revised September 29, 1999. Recommended by Associate Editor, J. Si. This work was supported in part by the National Science Foundation under Grant ECS-9812346 and the Air Force Office of Scientific Research under Grant F49620-95-1-0409. M. Arcak and P. Kokotovic´ are with the Center for Control Engineering and Computation, University of California, Santa Barbara, CA 93106-9560 USA (e-mail:
[email protected];
[email protected]). M. Seron is with the Departamento de Electronica, Universidad Nacional de Rosario, Riobamba 245 bis, 2000 Rosario, Argentina (e-mail:
[email protected]). J. Braslavsky is with the Departamento de Ciencia y Tecnologia, Universidad Nacional de Quilmes, Roque Saenz Peña 180, 1876 Bernal, Argentina (e-mail:
[email protected]). Publisher Item Identifier S 0018-9286(00)06073-6. 0018–9286/00$10.00 © 2000 IEEE
