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Robust stability of feedback linearised systems modelled with neural networks: dealing with uncertainty
Engineering Applications of Artificial Intelligence 13 (2000) 659–670
Robust stability of feedback linearised systems modelled with neural networks: dealing with uncertainty Miguel Ayala Bottoa,*, Bart Wamsb, Ton van den Boomb, Jose´ Sa´ da Costaa a
Department of Mechanical Engineering, Technical University of Lisbon, Instituto Superior Te´cnico, GCAR, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal b Department of Electrical Engineering, Faculty of Information Technology and Systems, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Abstract This paper presents a systematic procedure to analyse the stability robustness to modelling errors when a neural network model is integrated in an approximate feedback linearisation control scheme. The propagation through the control loop of the structured uncertainty from the neural network model parameters enables the construction of a polytopic uncertainty description for the overall linear closed-loop system. By using computationally efficient algorithms the solution of a set of linear matrix inequalities provides a Lyapunov function for the uncertain system, therefore proving robust stability of the overall control system. A nonlinear multivariable water vessel system is chosen as the case study for the application of this control strategy. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Feedback linearisation; Neural networks; Stability robustness; Linear matrix inequalities; Lyapunov stability
1. Introduction Over the last decade neural network models have become favorite candidates in the field of nonlinear systems identification due to their excellency in approximating multivariable nonlinear mappings. Their capability of generalising from fresh data as they keep learning during operation (Sjo¨berg, 1995a) is one of their main features which makes them attractive models for nonlinear control applications, as has been widely reported in the literature (Narendra and Parthasarathy, 1990; Hunt and Sbarbaro, 1991; Hunt et al., 1992; Suykens et al., 1996). However, for most of the in-line control implementations a relatively big sampling interval is needed for the neural network parameters adaptation, while it is not clear how the stability robustness of the resulting time-variant control loop can be analysed. Recent publications (Tanaka, 1996; Kuntanapreeda and Fullmer, 1996) suggest that related subjects are a current research field. *Corresponding author. Tel.: +351-218-419-028; fax: +351-218498-097. E-mail addresses: [email protected] (M.A. Botto), [email protected] (T. van den Boom).
This paper assumes that only structured uncertainty is present in the neural network model, i.e., the deviation of the estimated neural network parameters from the real ones is the main relevant cause for model-plant mismatch. Tools for selecting the model structure, and checking whether the process is in the model set, can be found in Sjo¨berg (1995a, 1995b), Billings and Voon (1986) and Ljung (1987). Since finding the best parameters for the neural network model (training) is essentially equivalent to a nonlinear least-squares estimation problem, it can be stated that the real parameters are within an interval from the estimated ones, provided certain conditions are met (Seber and Wild, 1989). In this case, quantitative measures for the uncertainty of the neural network parameters can be found by employing statistical properties of the nonlinear least-squares estimation, which enables a complete uncertainty description of the neural network model (Wams et al., 1998). Further, the knowledge of how such structured uncertainty description propagates through a given control loop will provide the means to formally establish sufficient conditions for robust stability of the overall closed-loop system with respect to plant uncertainty. One possible way to accomplish this goal is to integrate the parametric uncertain neural
0952-1976/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 2 - 1 9 7 6 ( 0 0 ) 0 0 0 5 0 - 6
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network model in a feedback linearisation control loop. This control strategy has been recognized as a powerful technique to tackle nonlinear control problems since it provides, under some mild assumptions, an exact linearisation of the process over the complete operating range (Slotine and Li, 1991; Nijmeijer and van der Schaft, 1990). In this paper, an approximate feedback linearisation is implemented through the computation of the first-order Taylor series expansion of the neural network dynamic model, around a state trajectory, resulting in an affine in the input system (te Braake et al., 1997). As long as the resulting Jacobian matrix is non-singular around the state trajectory, a simple linear numerical problem will provide the solution for the desired feedback linearizing control law (te Braake et al., 1999). It is shown in this paper that, under some mild assumptions, the parametric uncertainty of the resulting linear closed-loop system can be related with the neural network model uncertain parameters. Then, knowledge concerning their corresponding bounds along the operating trajectory provides the means for constructing a polytopic uncertainty description of the overall linear closed-loop system. The closed-loop system is proven robustly stable if there exists a Lyapunov function for the polytopic uncertainty description. Due to the fact that the uncertainty description is polytopic, finding that Lyapunov function can be recast as a linear matrix inequality (LMI) problem. LMI problems can be solved with computationally efficient algorithms, one of the main reasons why they became so popular in systems and control engineering applications (Kothare, 1997; Kothare et al., 1996; Boyd et al., 1994, van den Boom, 1997). It is stressed that in the proposed control strategy LMIs are employed off-line, and only once, to analyse the robustness properties of the closed-loop system rather than for control law synthesis as in Kiriakidis (1998). In the later case, the nonlinear system under consideration is captured in a set of linear models forming a polytopic description which serves as a basis for the control design. This control design involves solving a LMI online at each sampling instant, which turns it into a very complex task. This paper is divided into 5 sections. In Section 2 the principle of approximate input–output feedback linearisation is presented considering first the nominal case in Section 2.1, while the uncertain case which takes structured uncertainty into account is described in Section 2.2. Section 3 presents the stability robustness analysis procedure for a given linear uncertainty description, while Section 3.1 particularises it for the previously obtained closed-loop system. In Section 4 the application results of this control strategy to a nonlinear multivariable water vessel system are presented. Finally, in Section 5 some conclusions are drawn.
2. Approximate input–output feedback linearisation Input–output feedback linearisation (IOFL) aims at finding a static state feedback control law such that the resulting closed-loop system has a desired linear input– output behaviour. For a given nonlinear process, this procedure can be represented as in Fig. 1. The linear dynamics between y and the new external signal v are obtained by means of a proper design of the feedback linearising control law, C, which describes a nonlinear and state-dependent relation between the process input u and the external input v. This relation is given by u ¼ Cðx; xl ; vÞ;
ð1Þ
where x and xl are, respectively, the state vector of the process and the state vector of the desired resulting linear system. The development presented throughout this paper considers stable and square multiple-input multiple-output processes, with well-defined relative degree for all outputs, and which input–output behaviour can be modelled through single-layer feedforward neural networks. The next two subsections will explore the application of the input–output feedback linearisation principle when such models are exact (Section 2.1) and when modelling errors are present (Section 2.2). 2.1. The nominal case Consider a nonlinear square input–output dynamical system, with p inputs and p outputs, with unitary relative degree for all outputs, which can be assumed without loss of generality. The dynamic model of the system will be based on a particular (nonlinear autoregressive with exogeneous input) NARX description which in this case corresponds to a one hidden layer feedforward neural network, with hyperbolic tangent activation functions at the hidden neurons, and linear output functions at the output neurons (Narendra and Parthasarathy, 1990). Such models can be generically described using the following matrix notation: ykþ1 ¼ W tanhðVxk þ Guk þ bÞ þ d;
ð2Þ
where the activation function is applied elementwise to a matrix or vector, and ykþ1 ¼ ½y1kþ1 ; . . . ; y pkþ1 T ;
Fig. 1. Principle of input–output feedback linearisation.
ð3Þ
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xk ¼ ½ y1k ; . . . ; y1kÿn1 ; . . . ; y pkÿnp ; u1kÿ1 ; . . . ; u1kÿm1 ; . . . u pkÿmp T ; ð4Þ uk ¼ ½u1k ; . . . u pk T ;
p 1 1 ~k ¼½yk ; . . . ; ykÿn1 ÿ1 ; . . . ; ykÿnp ÿ1 ; x
Du1kÿ1 ; . . . ; Du1kÿm1 ; . . . Dupkÿmp T
ð5Þ
with ni and mj being the maximum delay of the ith output and jth input, respectively, present in the regression vector xk , while W 2 Rpnh , V 2 Rnh nx , G 2 Rnh p , b 2 Rnh 1 and d 2 Rp1 , represent the nominal weights of the network, with nh and nx being the number of neurons in the hidden layer and the length of the regression vector xk , respectively. The benefits from using the dynamic model description (2) are well expressed in the Cybenko’s (1989) theorem where it is stated that the use of a finite superposition of continuous sigmoidal functions can uniformly approximate any continuous function within any desired accuracy. The next step towards a straightforward design of the feedback linearising control law follows closely the ideas presented in te Braake et al. (1999), where an approximate feedback linearisation is achieved by first performing the Taylor expansion of process model (2) around the previous operating point, ðxkÿ1 ; ukÿ1 Þ, at each sampling instant. As the higher order terms of this Taylor expansion are assumed to be negligible, the following affine in the input model will be considered as an accurate representation of the nonlinear process dynamics: ykþ1 ¼ f ðxkÿ1 ; ukÿ1 Þ þ Fðxkÿ1 ; ukÿ1 ÞDxk þ Eðxkÿ1 ; ukÿ1 ÞDuk ;
ð6Þ
where f ðxkÿ1 ; ukÿ1 Þ represents the neural network model output vector computed at the operating point, Fðxkÿ1 ; ukÿ1 Þ is the matrix containing the partial derivatives of the neural network model (2) with respect to each element of the state vector xk , while Eðxkÿ1 ; ukÿ1 Þ contains the partial derivatives with respect to uk , both evaluated at the operating point, respectively given by f ðxkÿ1 ; ukÿ1 Þ ¼ W tanhðVxkÿ1 þ Gukÿ1 þ bÞ þ d;
ð7Þ
Fðxkÿ1 ; ukÿ1 Þ ¼ WGkÿ1 V;
ð8Þ
Eðxkÿ1 ; ukÿ1 Þ ¼ WGkÿ1 G;
ð9Þ
with Gkÿ1 ¼ I ÿ diag½tanh2 ðVxkÿ1 þ Gukÿ1 þ bÞ ÿ1
~k given by with the new state vector x
ð10Þ
ÿ1
ð12Þ
~ kÿ1 ; ukÿ1 Þ in (11) can be generically conMatrix Fðx structed from Fðxkÿ1 ; ukÿ1 Þ with the following procedure: ~ kÿ1 ; ukÿ1 Þ ¼ S þ Fðxkÿ1 ; ukÿ1 ÞN ð13Þ Fðx with S being an auxiliary matrix defined by S ¼ T sn1 snp , where each sni corresponds to a line vector defined as ð14Þ sni ¼ ½0 |fflffl{zfflffl0} 1 0|fflffl{zfflffl0} ai
bi
with ai and bi calculated by iÿ1 X nj þ 2ði ÿ 1Þ for i ¼ 2; . . . p with a1 ¼ 0 ai ¼ j¼1
ð15Þ bi ¼
p X
nj þ
j¼i
p X
mj þ 2ðp ÿ iÞ þ 1
for i ¼ 1; . . . p
j¼1
ð16Þ while matrix N in (13) 2 0 ... Nn1 6 0 N n2 0 6 6 .. N ¼ 6 ... . 6 4 0 ... ... 0 ... ...
is generically defined as 3 ... 0 ... 07 7 .. 7 .7 7 Nn 0 5
ð17Þ
p
0
I
with I being a square identity matrix with appropriate dimensions, and Nni a ½ðni þ 1Þ ðni þ 2Þ matrix defined by 3 2 1 ÿ1 0 0 ... 0 6 0 1 ÿ1 0 . . . 0 7 7 6 6 .. 7 .. .. ð18Þ Nni ¼ 6 ... 7: . . . 7 6 4 0 . . . . . . 1 ÿ1 0 5 0
...
...
0
1
ÿ1
Then, the rearranged linearised neural network model (11) can be exactly feedback linearised by means of the following control law: ~ kÿ1 ; ukÿ1 Þ~ Duk ¼Eðxkÿ1 ; ukÿ1 Þÿ1 ðÿFðx xk ð19Þ l þ CA~ xk þ CBvk Þ
while D ¼ 1 ÿ q , with q being the delay operator, thus Dxk ¼ xk ÿ xkÿ1 and Duk ¼ uk ÿ ukÿ1 . For reasons which will become clear on later developments, Eq. (6) is rearranged as
provided that matrix EðÞ is invertible for all admissible pairs ðxkÿ1 ; ukÿ1 Þ. The resulting linear closed-loop system will be described by
~ kÿ1 ; ukÿ1 Þ~ xk þ Eðxkÿ1 ; ukÿ1 ÞDuk ykþ1 ¼ Fðx
xlk þ CBvk ykþ1 ¼ CA~
ð11Þ
ð20Þ
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which can be represented in the following linear state-space form: ~lkþ1 ¼ A~ xlk þ Bvk ; x xlk ; yk ¼ C~
ð21Þ
~lk being the linear state vector, vk the newly created with x input signal, while A, B and C are appropriate choices for the linear state-space matrices. The control objective is to make the output of the nonlinear system follow the desired reference trajectory defined in signal v, by first canceling the system nonlinear dynamics over the full operating range, while simultaneously imposing a desired linear dynamic behaviour through a proper choice of the linear state-space matrices. The main reason to adopt the approximate feedback linearisation strategy here described is motivated from the observation that the use of an affine in the input model (6) enables a straightforward implementation of the resulting analytical feedback control solution (19). Notice that, if the original non-affine process model (2) was used instead, the solution for the feedback linearisation control action found through expression (1) would result in a very complex numerical optimisation procedure to be solved at each sampling instant. This would involve a time-consuming algorithm, for which there is no solution guaranteed.
ðxkÿ1 ; ukÿ1 Þ, is an accurate model representation of the nonlinear process dynamics, so the following neural network model description will be used in further developments (letting fall argument ðxkÿ1 ; ukÿ1 Þ for a clear notation): ^ k þ EDu ^ k ð23Þ ykþ1 ¼ f^ þ FDx with ^f ¼ W ^ ^ kÿ1 þ ^bÞ þ d; ^ tanhðVx ^ kÿ1 þ Gu
ð24Þ
^ ^ ¼W ^G ^ kÿ1 V; F
ð25Þ
^ ^ ¼W ^G ^ kÿ1 G; E
ð26Þ
where ^ kÿ1 þ ^bÞ: ^ kÿ1 þ Gu ^ kÿ1 ¼ I ÿ diag½tanh2 ðVx G
ð27Þ
A description is pursued in which the nominal part of (23) is described by (6), while the uncertain part is expressed in terms of dW, dV, dG, db and dd. Such description can be obtained by performing a first-order Taylor series expansion of (23) around the nominal weights of the neural network model, yielding ykþ1 ¼ f þ ðF þ dFÞDxk þ ðE þ dEÞDuk ; where 0
2.2. The uncertain case
dF ¼ dWGkÿ1 V þ WGkÿ1 dV þ WGkÿ1 FV kÿ1 V; 0
Despite feedforward neural networks have been generally accepted as suitable models for capturing the behaviour of highly nonlinear dynamical systems, it is also accepted that such models should not be considered exact. Therefore, model uncertainty should also be accounted for during the neural networks identification stage. This paper assumes that model uncertainty is due to the uncertainty on the weights of the neural network model. In fact, once the structure of the network (number of neurons, composition of the regression vector) is chosen, training the neural network can be regarded as a nonlinear least-squares estimation of its weights, by minimizing the sum of squared errors between the training data and the outputs of the neural network model. Provided that certain conditions are met, confidence regions for the estimated parameters can be obtained using statistical properties of a least-squares estimation (Seber and Wild, 1989) (see the appendix). It is assumed that the nonlinear process dynamics is described by the following neural network model: ^ þ d; ^ ^ k þ bÞ ^ tanhðVx ^ k þ Gu ð22Þ ykþ1 ¼ W ^ ¼ G þ dG; ^b ¼ ^ ¼ W þ dW, V ^ ¼ V þ dV, G where W b þ db and d^ ¼ d þ dd, with dW, dV, dG, db and dd representing the uncertainty in the neural network weights. Like in the nominal case, it is assumed that the first order Taylor expansion of (22), around
ð28Þ
dE ¼ dWGkÿ1 G þ WGkÿ1 dG þ WGkÿ1 FG kÿ1 G;
ð29Þ ð30Þ
with Gkÿ1 as defined in (10), and 0
Gkÿ1 ¼ diag½ÿ2 tanhðVxkÿ1 þ Gukÿ1 þ bÞ;
ð31Þ
FV kÿ1 ¼ diag½dVxkÿ1 þ db;
ð32Þ
FG kÿ1 ¼ diag½dGukÿ1 þ db;
ð33Þ
while f , F and E in (28) are defined by Eqs. (7)–(9), respectively. Finally, Eq. (28) can be rearranged in accordance with the procedure previously presented, leading to the following result: ~ þ dFÞ~ ~ xk þ ðE þ dEÞDuk ð34Þ ykþ1 ¼ ðF ~ ~k as defined in (12), F with the new state vector x ~ taken from Eq. (13), while dF ¼ dFN, with N taken from (17). The application of the feedback linearising control law (19) to the re-arranged model with uncertainties (34), results in a closed-loop system described by the following input–output behaviour: ~ ÿ dEE ÿ1 FÞ~ ~ xk xl þ CBvk þ ðdF ykþ1 ¼ CA~ k
þ dEE ÿ1 CA~ xlk þ dEE ÿ1 CBvk ;
ð35Þ
In order to proceed with the design of the input– output feedback linearisation scheme for this uncertain case, the following two assumptions must be made for
M.A. Botto et al. / Engineering Applications of Artificial Intelligence 13 (2000) 659–670
the characterisation of the desired linear state-space system: Assumption 1. The linear state vector must be such that ~k , which means a particular state-space descrip~lk ¼ x x tion for the linear system is due. Assumption 2. The output vector of the linear system xlk , where the observation should be given by, yk ¼ C~ T matrix C ¼ ½sn1 snp , with each sni defined as in (14). This means that the linear system output is yk ¼ ½y1k ypk T . As a natural consequence, the following will hold: C C T ¼ I, with I being the ½ p p identity matrix. With these assumptions, expression (35) can be rewritten as ~ ÿ dEE ÿ1 F ~ þ dEE ÿ1 CAÞ~ xl ykþ1 ¼ ðCA þ dF k
þ ðCB þ dEE ÿ1 CBÞvk
ð36Þ
which corresponds to the following state-space description for the uncertain linear closed-loop system: ~lkþ1 ¼ ðA þ dAÞ~ xlk þ ðB þ dBÞvk ; x yk ¼ C~ xlk ;
ð37Þ
where terms dA and dB correspond to state-dependent matrices that express the overall linear closed-loop uncertainty as a function of the weights of the neural network model, and are respectively given by ~ ÿ dEE ÿ1 F ~ þ dEE ÿ1 CAÞ; ð38Þ dA ¼ CT ðdF dB ¼ CT dEE ÿ1 CB:
ð39Þ
The knowledge concerning their quantitative measures and respective bounds will provide the basic ingredients for a complete stability robustness analysis of the closed-loop system.
3. Stability robustness analysis The development made in the previous section, taking into account model uncertainty, resulted in the uncertain linear closed-loop description given by (37). Assume that ½A þ dAðÞ B þ dBðÞ 2 Coð½A1 B1 ; ½A2 B2 ; . . . ; ½AL BL Þ
ð40Þ
where Co stands for a convex hull, i.e., for some l1 ; l2 ; . . . ; lL 0 summing to one: ½A þ dAðÞ
B þ dBðÞ ¼
L X
li ½Ai Bi :
ð41Þ
i¼1
Then, if Bi , with i ¼ 1; . . . ; L, is bounded, for showing closed-loop stability of the uncertain system it is
663
sufficient to find a positive-definite matrix P, such that ATi PAi ÿ P P
5 0 > 0
8i ¼ 1; . . . ; L:
ð42Þ
This result represents sufficient conditions for Lyapunov stability for system (37). Moreover, expression (42) can be rewritten as a linear matrix inequality (LMI) by using Qÿ1 ¼ P: ATi Qÿ1 Ai ÿ Qÿ1 Qÿ1
5 >
0 0
8i ¼ 1; . . . ; L:
ð43Þ
Then, by post- and pre-multiplying (43) by Q, with Q ¼ QT , leads to Q ÿ ðAi QÞT Qÿ1 Ai Q > Q >
0 0
8i ¼ 1; . . . ; L
ð44Þ
from which, by using Schur’s complement condition, the following LMI can be constructed: Q ðAi QÞT > 0 8i ¼ 1; . . . ; L: ð45Þ Ai Q Q Solving a set of LMIs can be easily done with use of computationally efficient algorithms, which turn them into a practical tool for control engineering purposes (Boyd et al., 1994). Therefore, the key to the solution of the stability robustness problem is to build the necessary LMIs which correspond to the closed-loop uncertainty description given by (37). 3.1. Uncertainty bounds The remaining task to complete the stability robustness analysis of the closed-loop system is to bound the intervals within which the entries of matrices A and B are uncertain, such that a polytopic uncertainty description can be constructed. Considering the involved expressions, finding an analytic expression for the bounds of (38) and (39) can introduce a great deal of conservatism. The following analytic expressions can be derived for the maximum absolute values for each of the uncertain terms: ~ max þ abs C T dEmax E ÿ1 F ~ max dAmax abs C T dF ð46Þ þ abs C T dEmax E ÿ1 CA ; dBmax abs C T dEmax E ÿ1 CB ;
ð47Þ
~ max ¼ S þ Fmax N, and ~ max ¼ dFmax N, F where dF dFmax ¼ dWb Gmax absðVÞ þ absðWÞGmax dVb 0
þ absðWÞGmax FV max absðVÞ;
ð48Þ
dEmax ¼ dWb Gmax absðGÞ þ absðWÞGmax dGb 0
þ absðWÞGmax FG max absðGÞ; Fmax ¼ absðWÞGmax absðVÞ;
ð49Þ ð50Þ
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M.A. Botto et al. / Engineering Applications of Artificial Intelligence 13 (2000) 659–670
while 2
Gmax ¼ absðI ÿ diag½tanh ðVxkÿ1 þ Gukÿ1 þ bÞÞ; 0
ð51Þ
Gmax ¼ absðdiag½ÿ2 tanhðVxkÿ1 þ Gukÿ1 þ bÞÞ;
ð52Þ
FV max ¼ diag½dVb absðxkÿ1 Þ þ dbb ;
ð53Þ
FG max ¼ diag½dGb absðukÿ1 Þ þ dbb ;
ð54Þ
where dWb , dVb , dGb and dbb are exactly known and correspond to the uncertainty associated with each
neural network weight, i.e., the maximum uncertainty bound for each neural network nominal weight (see Appendix). For a given operating point, ðxkÿ1 ; ukÿ1 Þ, the bounds for dA and dB can be directly found through (46) and (47), respectively. However, as can be seen from Eqs. (51)–(54), these bounds will be state dependent, and so a numerical search must be performed in order to assure that the maximum value for each dA and dB over all admissible operating points is found. The proposed strategy which assures that only admissible ðxk ; uk Þ pairs are considered, is to design a reference trajectory vk , as used in the feedback linearising control law (19), and feedback linearise the process model. Then, as the model output travels along this trajectory, bounds on the uncertainty entries of matrices A and B can be collected, for each encountered operating point, according to the expressions (46) and (47), respectively. Finally, the maximum bound encountered along that trajectory can be used to construct the polytopic description which enables stability robustness analysis. One of the advantages of this approach is that uncertainty is considered only in relevant regions of the space spanned by the ðxk ; uk Þ pair during operation. Moreover, while this procedure is being performed, it can be automatically checked whether matrix Eðxkÿ1 ; ukÿ1 Þ, as defined in (9), is invertible over all encountered operating points, a basic requirement for the success of the proposed control scheme. 4. Case study
Fig. 2. Two-by-two water vessel system.
The control strategy and the stability robustness analysis procedure described in the previous sections
Fig. 3. Applied inflows to generate training and validation data (dashed line: u1, solid line: u2).
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M.A. Botto et al. / Engineering Applications of Artificial Intelligence 13 (2000) 659–670
Fig. 4. Neural network model validation (dashed line: target outputs, solid line: neural network outputs). (a) Water level h1. (b) Water level h2.
are applied to the water vessel system shown in Fig. 2. This is a two inputs–two outputs nonlinear dynamical system whose inputs correspond to the water inflows at the two upper vessels, u1 and u2, while the outputs are the water levels at the two lower vessels, h1 and h2, respectively. As seen from Fig. 2, the water outflows of the two upper vessels are partly directed to the underneath vessel and to the other lower vessel in a 75% by 25% ratio, respectively. Furthermore, the highly nonlinear dynamic behaviour of the system is reinforced by the shape of the two lower vessels. A
detailed description of the system dynamics is given below: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh1 1 f 21 ¼ d21 2 h1 g; dt ¼ A21ðh1Þð f 11 ÿ f 21Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh2 1 f 22 ¼ d22 2 h2 g; dt ¼ A22ðh2Þð f 12 ÿ f 22Þ; f 11 ¼ a1 f 1 þ ð1 ÿ a2 Þ f 2; f 12 ¼ ð1 ÿ a1 Þ f 1 þ a2 f 2; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 1 ¼ d11 2 h11 g; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 ¼ d12 2 h12 g;
dh11 dt dh12 dt
1 ¼ A11 ðu1 ÿ f 1Þ; 1 ¼ A12 ðu2 ÿ f 2Þ; ð55Þ
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where the following parameters were used: A11 ¼ A12 ¼ p m2 , A21ðh1Þ ¼ p ð1 þ 0:3 sinð24 h1ÞÞ2 m2 ; d11 ¼ p A22ðh2Þ ¼ p ð1 þ 0:3 sinð24 h2ÞÞ2 m2 , 0:352 m2 ; d12 ¼ p 0:22 m2, d21 ¼ d22 ¼ p 0:2672 m2, a1 ¼ a2 ¼ 0:75 and g ¼10 m sÿ2 is the gravitational acceleration. The main control objective is twofold: to design a controller which is robustly stable with respect to model-plant uncertainty, while simultaneously guaranteeing independent water level tracking at the two lower vessels. In accordance with the procedure presented in this paper, the following three steps will be performed: 1. A neural network is trained to model the nonlinear multivariable input–output dynamic behaviour of the system, based on a given set of data measurements. 2. A polytopic closed-loop uncertainty description is constructed, based on confidence intervals for each parameter of the neural network model. 3. Finally, the control design is proven robustly stable by finding a Lyapunov function for the polytopic system through the solution of a set of LMIs equations. Step 1: Modelling the system. The input–output dynamic behaviour of the water vessel system is modelled by a single-layer feedforward neural network with hyperbolic tangent activation function having the following structure:
h1kþ1 h2kþ1
1 1 hk uk ¼ W tanh V 2 þ G 2 þ b þ d; hk uk
where h1k and h2k are the water levels at time instant k at the lower left vessel and at the lower right vessel, respectively, u1k and u2k are the water inflows at time instant k at the upper left vessel and at the right upper vessel, respectively. This neural network model was trained using the Levenberg–Marquardt optimisation algorithm, based on noisy input–output data measurements taken from the process simulation with a sampling time of 1 s. Fig. 3 shows the applied water inflows used to generate the training and validation data. The best structure found after the learning phase was completed, consisted of a feedforward neural network with two hidden neurons, i.e., W 2 R22 , V 2 R22 , G 2 R22 , b 2 R21 and d 2 R21 in (56). The validation results for the nonlinear output error simulation shown in Fig. 4 reveal an accurate dynamic performance of the neural network model over the entire operating space. Therefore, reliable confidence regions for the estimated neural network parameters can be obtained by using the statistical properties of the nonlinear least-squares estimation. Step 2: Obtaining a polytopic closed-loop uncertainty description. In accordance with the approximate input– output feedback linearisation procedure, the following desired linear dynamics are imposed to the overall closed-loop system: 2
~lkþ1 x ð56Þ
1:7 6 1 6 ¼4 0 0
ÿ0:7225 0 0 0
0 0 1:7 1
3 0 7 l 0 7x ~ ÿ0:7225 5 k 0
ð57Þ
Fig. 5. Approximate feedback linearisation applied to the model (dashed line: external inputs, solid line: neural network model outputs).
M.A. Botto et al. / Engineering Applications of Artificial Intelligence 13 (2000) 659–670
667
Fig. 6. Closed-loop response of the water vessel system using the neural network model. (a) Reference following (dashed line: external inputs, solid line: system outputs). (b) Control signals (dashed line: inflow upper left vessel, solid line: inflow upper right vessel).
2
0:0225 6 0 þ6 4 0 0
h1k h2k
1 ¼ 0
0 0
0 1
3 0 1 0 7 7 vk2 ; 0:0225 5 vk 0 0 l ~ : x 0 k
If the design goal is satisfied, the dynamics of the closed-loop system will mimic the behaviour
of two stable and decoupled single-input singleoutput systems. Following the procedure presented in Section 3.1, the maximum uncertainty bounds in the parameters of matrices A and B in (57), determined along the reference trajectory shown in Fig. 5, are found to be 2 3 6:76 3:53 3:39 1:44 6 0 0 0 0 7 ÿ3 7 dAmax ¼ 6 4 3:75 1:60 6:47 3:37 5 10 ; 0 0 0 0
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2
dBmax
3 3:75 2:22 6 0 0 7 ÿ5 7 ¼6 4 2:45 3:58 5 10 : 0 0
ð58Þ
These quantitative measures are sufficient to proceed with the stability robustness analysis of the uncertain closed-loop system. Step 3: Proving robust stability. The final step of the controller design is to construct a polytopic description based on the uncertain linear closedloop system described by (57) and (58) and build the corresponding LMIs. Then, as described in Section 3, if a positive-definite matrix is found as the solution for this set of LMIs the closed-loop system is proven robustly stable. In this case, the set of LMIs was solved using the LMI-Toolbox (Gahinet et al., 1995), resulting in the following positive-definite matrix: 2 ÿ2:95 108 3:65 108 6 ÿ2:95 108 2:55 108 6 P ¼6 ÿ6 4 7:15 10 ÿ5:49 10ÿ6 ÿ5:66 10ÿ6 7:15 10ÿ6 ÿ5:49 10ÿ6 3:62 108 ÿ2:93 108
4:64 10ÿ6
3 ÿ5:66 10ÿ6 4:64 10ÿ6 7 7 8 7 ÿ2:93 10 5
ð59Þ
2:53 108
This solution proves that the closed-loop system is robustly stable with respect to parametric uncertainties. The overall closed-loop performance of the water vessel system under approximate feedback linearisation is shown in Fig. 6, where it is visible that the system outputs are independently controlled with a small control effort.
5. Concluding remarks Approximate feedback linearisation has several features which makes it an appealing control strategy for nonlinear systems. From an application point of view these are related to the control law being easily determined from the system model, while the transformation of the nonlinear open-loop system into a linear closed-loop system allowed for linear techniques to be used in the analysis of the closed-loop properties. The main contribution of this paper is that it provides a systematic procedure for stability robustness analysis to structured uncertainty for neural network models under approximate feedback linearisation. It was shown how the closed-loop uncertainty can be expressed in terms of the uncertainty in the weights of a single layer feedforward neural network. This result enabled the
construction of a polytopic uncertainty description for the linear closed-loop system. Based on this polytopic description, stability of the uncertain closed-loop can be verified by finding a proper Lyapunov function. Due to a special form of the closed-loop uncertainty description, finding the Lyapunov function could be rewritten as a LMI problem, and thus can be tackled with the use of computationally efficient algorithms. The significance of this design strategy in the scope of nonlinear robust control is that it allows the application of linear robust control analysis to nonlinear control problems. The strategy presented in this paper can be applied to a broad class of complex nonlinear multivariable dynamic systems, as long as they are square and follow some mild assumptions. Simulation experiments showed a successful application of this control design procedure to a nonlinear multivariable water vessel system.
Acknowledgements This research is partially supported by PRAXIS/P/ EME/12124/1998, by a Research Fellowship from the Technical University of Delft, and by the Dutch Technology Foundation (STW).
Appendix. Confidence intervals for estimated weights Suppose that n observations ðyi ; xi Þ; i ¼ 1; 2; . . . ; n, are given from a fixed-regressor nonlinear model with a known functional relationship f , according to y ¼ f ðx; y Þ þ e;
ðA:1Þ
where e is a normally distributed zero mean white noise with variance s2 , and y the true value of y, which is known to belong to Y. The least-squares estimate of y , denoted by ^y, is obtained through the minimization of the following sum square of errors: n X ½ yi ÿ f ðxi ; yÞ2 ¼ jjy ÿ f ðx; yÞjj2 ðA:2Þ SðyÞ ¼ i¼1
over y 2 Y. It should be noted that, unlike the linear least-squares case, SðyÞ may have several local minima in addition to the absolute minimum, y . Assuming e to be independent and identically distributed with variance s2 , it can be shown that under certain regularity conditions, ^y is a consistent estimate of y . Moreover, ^y is also proven to be asymptotically normally distributed as n 7! 1. To show this, consider the linear Taylor series expansion of f , in a small neighbourhood of y , given by @f ðx; yÞ ðy ÿ y Þ f ðx; yÞ f ðx; y Þ þ @y y ðA:3Þ ¼ f ðx; y Þ þ Fðy ÿ y Þ:
M.A. Botto et al. / Engineering Applications of Artificial Intelligence 13 (2000) 659–670
Hence 2
SðyÞ ¼ jjy ÿ f ðx; yÞjj jjy ÿ f ðx; y Þ ÿ Fðy ÿ y Þjj2 ¼ jje ÿ Fbjj2 ;
ðA:4Þ
where e ¼ y ÿ f ðx; y Þ, and b ¼ y ÿ y . From the properties of the linear model, SðyÞ is minimized when b is given by ^ ¼ ðF T FÞÿ1 F T e: ðA:5Þ b Furthermore, for large n, ^ y is almost certain to be within ^ and y ÿ y b, a small neighbourhood of y . Hence, ^ therefore ÿ ^y ÿ y F T F ÿ1 F T e: ðA:6Þ Then, according to Seber and Wild (1989) and Ljung (1987), the following holds: given E Nð0; s2 In Þ, with E normally distributed with zero mean and variance s2 In , together with the appropriate regularity conditions, it can be stated that for large n: ^y ÿ y Np ð0; s2 ðF T FÞÿ1 Þ: ðA:7Þ y more than Hence, that y deviates from ^ ffiffiffiffiffiffiffiprobability ffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffithe ÿ1 a s2 ðF T FÞ corresponds to the ð1 ÿ aÞ-level of the normal distribution, available in standard statistical tables.
References Billings, S., Voon, W., 1986. Correlation based model validity test for non-linear models. International Journal of Control 4 (1), 235–244. Boyd, S., El Gahoui, L., Feron, E., Balakrishnan, V., 1994. Linear Matrix Inequalities in System and Control, Studies in Applied Mathematics, Vol. 15. SIAM, Philadelphia, PA. Cybenko, G., 1989. Approximation by superpositions of a sigmoidal function. Mathematics of Control Signals and Systems 2, 303–314. Gahinet, P., Nemirovski, A., Laub, A., Chilali, M., 1995. Matlab User’s Guide, LMI Control Toolbox. The MathWorks, Inc. Hunt, K.J., Sbarbaro, D., 1991. Neural networks for nonlinear internal model control. IEE Proceedings } D 138 (5), 431–438. _ Hunt, K., Sbarbaro, D., Zbikowski, R., Gawthrop, P., 1992. Neural networks for control systems } a survey. Automatica 28 (6), 1083–1112. Kiriakidis, K., 1998. An LMI approach to the control of a class of nonlinear systems. Proceedings of the 37th IEEE Conference on Decision and Control, FL, USA, pp. 1470–1475. Kothare, M., 1997. Control of systems subject to constraints. Ph.D. Thesis, California Institute of Technology, CA, USA. Kothare, M., Balakrishnan, V., Morari, M., 1996. Robust constrained predictive control using linear matrix inequalities. Automatica 32 (10), 1361–1379. Kuntanapreeda, S., Fullmer, R.R., 1996. A training rule which guarantees finite-region stability for a class of closed-loop neuralnetwork control systems. IEEE Transactions on Neural Networks 7 (3), 745–751. Ljung, L., 1987. System Identification: Theory for the User. PrenticeHall, Englewood Cliffs, NJ.
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Narendra, K.S., Parthasarathy, K., 1990. Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks 1 (1), 4–27. Nijmeijer, H., van der Schaft, A. J., 1990. Nonlinear Dynamical Control Systems. Springer, New York. Seber, G., Wild, C., 1989. Nonlinear Regression. John Wiley, New York. Sjo¨berg, J., 1995a. Non-linear system identification with neural networks. Ph.D. Thesis, Linko¨ping University, Department of Electrical Engineering, Sweden. Sjo¨berg, J., 1995b. Nonlinear black-box modeling in system identification: a unified overview. Automatica 31 (12), 1691–1724. Slotine, J.-J. E., Li, W., 1991. Applied Nonlinear Control. PrenticeHall, NJ, USA. Suykens, J., Vandewalle, J., de Moor, B., 1996. Artificial Neural Networks for Modelling and Control of Non-Linear Systems. Kluwer Academic Publishers, Dordrecht. Tanaka, K., 1996. An approach to stability criteria of neural-network control systems. IEEE Transactions on Neural Networks 7 (3), 629–642. te Braake, H., Ayala Botto, M., van Can, H., Sa´ da Costa, J., Verbruggen, H., 1999. Linear predictive control based on approximate input-output feedback linearisation. IEE Proceedings } Control Theory and Applications 146 (4), 295–300. te Braake, H., van Can, H., Scherpen, J., Verbruggen, H., 1997. Control of nonlinear chemical processes using dynamic neural models and feedback linearization. Computers and Chemical Engineering 22, 1113–1127. van den Boom, T., 1997. Robust nonlinear predictive control using feedback linearization and linear matrix inequalities. American Control Conference, Albuquerque, NM, pp. 3068–3072. Wams, B., Ayala Botto, M., van den Boom, T., Sa´ da Costa, J., 1998. Training neural networks for robust control of nonlinear MIMO systems. Proceedings of the International Conference on Control’98, Swansea, UK, pp. 141–146.
Miguel Ayala Botto was born in Lisbon, Portugal in 1965. He graduated in Mechanical Engineering from the Technical University of Lisbon in 1989 and received his M.Sc. and Ph.D. degrees in Mechanical Engineering from the same university in 1992 and 1996, respectively. He is currently Assistant Professor at the Control, Automation and Robotics Group of the Department of Mechanical Engineering at Instituto Superior Te´cnico, Technical University of Lisbon. From September 1999 he held a six-month sabbatical leave position at the Control Laboratory of the Department of Information Technology and Systems at the Delft University of Technology, the Netherlands. His current research interests include nonlinear control systems, neural networks modelling and identification for robust control.
Bart Wams is currently a Ph.D. student at the Control Laboratory of the Department of Information Technology and Systems at the Delft University of Technology, the Netherlands. His research topic is the incorporation of neural network models into nonlinear robust modelbased control. His research is supported by the Dutch Technology Foundation (STW), under project number AIF44.3595. Before he joined the Control Laboratory in 1996, he received his M.Sc. degree in Electrical Engineering at the same university. His M.Sc. thesis focused on control configurations for time-varying nonlinear processes, with guaranteed closed-loop stability.
Dr.ir. T.J.J. van den Boom is an Associate Professor at the Control Laboratory of the Department of Information Technology and
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Systems at the Delft University of Technology, the Netherlands. He is also a member of CIDIC, the European consortium on ‘‘Computer Integrated Design of Controlled Industrial Systems’’. In 1996 he held a six-month sabbatical leave position at the Swiss Federal Institute of Technology (ETH) in Zu¨rich. Special fields of interest: model predictive control, robust control, nonlinear control, anti-windup bumpless-transfer control schemes, system identification for robust control, application of advanced control in mechatronics and process industry.
Jose´ Sa´ da Costa received his Diploma in Mechanical Engineering from the Technical University of Lisbon in 1974 and his M.Sc. and Ph.D. in Automatic Control from the University of Manchester Institute of Science and Technology in 1979 and 1982, respectively. He is currently an Associate Professor in Control at the Department of Mechanical Engineering at Instituto Superior Te´cnico, Technical University of Lisbon. His current research interests include system modelling and identification, adaptive and intelligent control and robotics.
Robust stability of feedback linearised systems modelled with neural networks: dealing with uncertainty Miguel Ayala Bottoa,*, Bart Wamsb, Ton van den Boomb, Jose´ Sa´ da Costaa a
Department of Mechanical Engineering, Technical University of Lisbon, Instituto Superior Te´cnico, GCAR, Avenida Rovisco Pais, 1049-001 Lisbon, Portugal b Department of Electrical Engineering, Faculty of Information Technology and Systems, Delft University of Technology, P.O. Box 5031, 2600 GA Delft, The Netherlands
Abstract This paper presents a systematic procedure to analyse the stability robustness to modelling errors when a neural network model is integrated in an approximate feedback linearisation control scheme. The propagation through the control loop of the structured uncertainty from the neural network model parameters enables the construction of a polytopic uncertainty description for the overall linear closed-loop system. By using computationally efficient algorithms the solution of a set of linear matrix inequalities provides a Lyapunov function for the uncertain system, therefore proving robust stability of the overall control system. A nonlinear multivariable water vessel system is chosen as the case study for the application of this control strategy. # 2000 Elsevier Science Ltd. All rights reserved. Keywords: Feedback linearisation; Neural networks; Stability robustness; Linear matrix inequalities; Lyapunov stability
1. Introduction Over the last decade neural network models have become favorite candidates in the field of nonlinear systems identification due to their excellency in approximating multivariable nonlinear mappings. Their capability of generalising from fresh data as they keep learning during operation (Sjo¨berg, 1995a) is one of their main features which makes them attractive models for nonlinear control applications, as has been widely reported in the literature (Narendra and Parthasarathy, 1990; Hunt and Sbarbaro, 1991; Hunt et al., 1992; Suykens et al., 1996). However, for most of the in-line control implementations a relatively big sampling interval is needed for the neural network parameters adaptation, while it is not clear how the stability robustness of the resulting time-variant control loop can be analysed. Recent publications (Tanaka, 1996; Kuntanapreeda and Fullmer, 1996) suggest that related subjects are a current research field. *Corresponding author. Tel.: +351-218-419-028; fax: +351-218498-097. E-mail addresses: [email protected] (M.A. Botto), [email protected] (T. van den Boom).
This paper assumes that only structured uncertainty is present in the neural network model, i.e., the deviation of the estimated neural network parameters from the real ones is the main relevant cause for model-plant mismatch. Tools for selecting the model structure, and checking whether the process is in the model set, can be found in Sjo¨berg (1995a, 1995b), Billings and Voon (1986) and Ljung (1987). Since finding the best parameters for the neural network model (training) is essentially equivalent to a nonlinear least-squares estimation problem, it can be stated that the real parameters are within an interval from the estimated ones, provided certain conditions are met (Seber and Wild, 1989). In this case, quantitative measures for the uncertainty of the neural network parameters can be found by employing statistical properties of the nonlinear least-squares estimation, which enables a complete uncertainty description of the neural network model (Wams et al., 1998). Further, the knowledge of how such structured uncertainty description propagates through a given control loop will provide the means to formally establish sufficient conditions for robust stability of the overall closed-loop system with respect to plant uncertainty. One possible way to accomplish this goal is to integrate the parametric uncertain neural
0952-1976/00/$ - see front matter # 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 5 2 - 1 9 7 6 ( 0 0 ) 0 0 0 5 0 - 6
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network model in a feedback linearisation control loop. This control strategy has been recognized as a powerful technique to tackle nonlinear control problems since it provides, under some mild assumptions, an exact linearisation of the process over the complete operating range (Slotine and Li, 1991; Nijmeijer and van der Schaft, 1990). In this paper, an approximate feedback linearisation is implemented through the computation of the first-order Taylor series expansion of the neural network dynamic model, around a state trajectory, resulting in an affine in the input system (te Braake et al., 1997). As long as the resulting Jacobian matrix is non-singular around the state trajectory, a simple linear numerical problem will provide the solution for the desired feedback linearizing control law (te Braake et al., 1999). It is shown in this paper that, under some mild assumptions, the parametric uncertainty of the resulting linear closed-loop system can be related with the neural network model uncertain parameters. Then, knowledge concerning their corresponding bounds along the operating trajectory provides the means for constructing a polytopic uncertainty description of the overall linear closed-loop system. The closed-loop system is proven robustly stable if there exists a Lyapunov function for the polytopic uncertainty description. Due to the fact that the uncertainty description is polytopic, finding that Lyapunov function can be recast as a linear matrix inequality (LMI) problem. LMI problems can be solved with computationally efficient algorithms, one of the main reasons why they became so popular in systems and control engineering applications (Kothare, 1997; Kothare et al., 1996; Boyd et al., 1994, van den Boom, 1997). It is stressed that in the proposed control strategy LMIs are employed off-line, and only once, to analyse the robustness properties of the closed-loop system rather than for control law synthesis as in Kiriakidis (1998). In the later case, the nonlinear system under consideration is captured in a set of linear models forming a polytopic description which serves as a basis for the control design. This control design involves solving a LMI online at each sampling instant, which turns it into a very complex task. This paper is divided into 5 sections. In Section 2 the principle of approximate input–output feedback linearisation is presented considering first the nominal case in Section 2.1, while the uncertain case which takes structured uncertainty into account is described in Section 2.2. Section 3 presents the stability robustness analysis procedure for a given linear uncertainty description, while Section 3.1 particularises it for the previously obtained closed-loop system. In Section 4 the application results of this control strategy to a nonlinear multivariable water vessel system are presented. Finally, in Section 5 some conclusions are drawn.
2. Approximate input–output feedback linearisation Input–output feedback linearisation (IOFL) aims at finding a static state feedback control law such that the resulting closed-loop system has a desired linear input– output behaviour. For a given nonlinear process, this procedure can be represented as in Fig. 1. The linear dynamics between y and the new external signal v are obtained by means of a proper design of the feedback linearising control law, C, which describes a nonlinear and state-dependent relation between the process input u and the external input v. This relation is given by u ¼ Cðx; xl ; vÞ;
ð1Þ
where x and xl are, respectively, the state vector of the process and the state vector of the desired resulting linear system. The development presented throughout this paper considers stable and square multiple-input multiple-output processes, with well-defined relative degree for all outputs, and which input–output behaviour can be modelled through single-layer feedforward neural networks. The next two subsections will explore the application of the input–output feedback linearisation principle when such models are exact (Section 2.1) and when modelling errors are present (Section 2.2). 2.1. The nominal case Consider a nonlinear square input–output dynamical system, with p inputs and p outputs, with unitary relative degree for all outputs, which can be assumed without loss of generality. The dynamic model of the system will be based on a particular (nonlinear autoregressive with exogeneous input) NARX description which in this case corresponds to a one hidden layer feedforward neural network, with hyperbolic tangent activation functions at the hidden neurons, and linear output functions at the output neurons (Narendra and Parthasarathy, 1990). Such models can be generically described using the following matrix notation: ykþ1 ¼ W tanhðVxk þ Guk þ bÞ þ d;
ð2Þ
where the activation function is applied elementwise to a matrix or vector, and ykþ1 ¼ ½y1kþ1 ; . . . ; y pkþ1 T ;
Fig. 1. Principle of input–output feedback linearisation.
ð3Þ
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xk ¼ ½ y1k ; . . . ; y1kÿn1 ; . . . ; y pkÿnp ; u1kÿ1 ; . . . ; u1kÿm1 ; . . . u pkÿmp T ; ð4Þ uk ¼ ½u1k ; . . . u pk T ;
p 1 1 ~k ¼½yk ; . . . ; ykÿn1 ÿ1 ; . . . ; ykÿnp ÿ1 ; x
Du1kÿ1 ; . . . ; Du1kÿm1 ; . . . Dupkÿmp T
ð5Þ
with ni and mj being the maximum delay of the ith output and jth input, respectively, present in the regression vector xk , while W 2 Rpnh , V 2 Rnh nx , G 2 Rnh p , b 2 Rnh 1 and d 2 Rp1 , represent the nominal weights of the network, with nh and nx being the number of neurons in the hidden layer and the length of the regression vector xk , respectively. The benefits from using the dynamic model description (2) are well expressed in the Cybenko’s (1989) theorem where it is stated that the use of a finite superposition of continuous sigmoidal functions can uniformly approximate any continuous function within any desired accuracy. The next step towards a straightforward design of the feedback linearising control law follows closely the ideas presented in te Braake et al. (1999), where an approximate feedback linearisation is achieved by first performing the Taylor expansion of process model (2) around the previous operating point, ðxkÿ1 ; ukÿ1 Þ, at each sampling instant. As the higher order terms of this Taylor expansion are assumed to be negligible, the following affine in the input model will be considered as an accurate representation of the nonlinear process dynamics: ykþ1 ¼ f ðxkÿ1 ; ukÿ1 Þ þ Fðxkÿ1 ; ukÿ1 ÞDxk þ Eðxkÿ1 ; ukÿ1 ÞDuk ;
ð6Þ
where f ðxkÿ1 ; ukÿ1 Þ represents the neural network model output vector computed at the operating point, Fðxkÿ1 ; ukÿ1 Þ is the matrix containing the partial derivatives of the neural network model (2) with respect to each element of the state vector xk , while Eðxkÿ1 ; ukÿ1 Þ contains the partial derivatives with respect to uk , both evaluated at the operating point, respectively given by f ðxkÿ1 ; ukÿ1 Þ ¼ W tanhðVxkÿ1 þ Gukÿ1 þ bÞ þ d;
ð7Þ
Fðxkÿ1 ; ukÿ1 Þ ¼ WGkÿ1 V;
ð8Þ
Eðxkÿ1 ; ukÿ1 Þ ¼ WGkÿ1 G;
ð9Þ
with Gkÿ1 ¼ I ÿ diag½tanh2 ðVxkÿ1 þ Gukÿ1 þ bÞ ÿ1
~k given by with the new state vector x
ð10Þ
ÿ1
ð12Þ
~ kÿ1 ; ukÿ1 Þ in (11) can be generically conMatrix Fðx structed from Fðxkÿ1 ; ukÿ1 Þ with the following procedure: ~ kÿ1 ; ukÿ1 Þ ¼ S þ Fðxkÿ1 ; ukÿ1 ÞN ð13Þ Fðx with S being an auxiliary matrix defined by S ¼ T sn1 snp , where each sni corresponds to a line vector defined as ð14Þ sni ¼ ½0 |fflffl{zfflffl0} 1 0|fflffl{zfflffl0} ai
bi
with ai and bi calculated by iÿ1 X nj þ 2ði ÿ 1Þ for i ¼ 2; . . . p with a1 ¼ 0 ai ¼ j¼1
ð15Þ bi ¼
p X
nj þ
j¼i
p X
mj þ 2ðp ÿ iÞ þ 1
for i ¼ 1; . . . p
j¼1
ð16Þ while matrix N in (13) 2 0 ... Nn1 6 0 N n2 0 6 6 .. N ¼ 6 ... . 6 4 0 ... ... 0 ... ...
is generically defined as 3 ... 0 ... 07 7 .. 7 .7 7 Nn 0 5
ð17Þ
p
0
I
with I being a square identity matrix with appropriate dimensions, and Nni a ½ðni þ 1Þ ðni þ 2Þ matrix defined by 3 2 1 ÿ1 0 0 ... 0 6 0 1 ÿ1 0 . . . 0 7 7 6 6 .. 7 .. .. ð18Þ Nni ¼ 6 ... 7: . . . 7 6 4 0 . . . . . . 1 ÿ1 0 5 0
...
...
0
1
ÿ1
Then, the rearranged linearised neural network model (11) can be exactly feedback linearised by means of the following control law: ~ kÿ1 ; ukÿ1 Þ~ Duk ¼Eðxkÿ1 ; ukÿ1 Þÿ1 ðÿFðx xk ð19Þ l þ CA~ xk þ CBvk Þ
while D ¼ 1 ÿ q , with q being the delay operator, thus Dxk ¼ xk ÿ xkÿ1 and Duk ¼ uk ÿ ukÿ1 . For reasons which will become clear on later developments, Eq. (6) is rearranged as
provided that matrix EðÞ is invertible for all admissible pairs ðxkÿ1 ; ukÿ1 Þ. The resulting linear closed-loop system will be described by
~ kÿ1 ; ukÿ1 Þ~ xk þ Eðxkÿ1 ; ukÿ1 ÞDuk ykþ1 ¼ Fðx
xlk þ CBvk ykþ1 ¼ CA~
ð11Þ
ð20Þ
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which can be represented in the following linear state-space form: ~lkþ1 ¼ A~ xlk þ Bvk ; x xlk ; yk ¼ C~
ð21Þ
~lk being the linear state vector, vk the newly created with x input signal, while A, B and C are appropriate choices for the linear state-space matrices. The control objective is to make the output of the nonlinear system follow the desired reference trajectory defined in signal v, by first canceling the system nonlinear dynamics over the full operating range, while simultaneously imposing a desired linear dynamic behaviour through a proper choice of the linear state-space matrices. The main reason to adopt the approximate feedback linearisation strategy here described is motivated from the observation that the use of an affine in the input model (6) enables a straightforward implementation of the resulting analytical feedback control solution (19). Notice that, if the original non-affine process model (2) was used instead, the solution for the feedback linearisation control action found through expression (1) would result in a very complex numerical optimisation procedure to be solved at each sampling instant. This would involve a time-consuming algorithm, for which there is no solution guaranteed.
ðxkÿ1 ; ukÿ1 Þ, is an accurate model representation of the nonlinear process dynamics, so the following neural network model description will be used in further developments (letting fall argument ðxkÿ1 ; ukÿ1 Þ for a clear notation): ^ k þ EDu ^ k ð23Þ ykþ1 ¼ f^ þ FDx with ^f ¼ W ^ ^ kÿ1 þ ^bÞ þ d; ^ tanhðVx ^ kÿ1 þ Gu
ð24Þ
^ ^ ¼W ^G ^ kÿ1 V; F
ð25Þ
^ ^ ¼W ^G ^ kÿ1 G; E
ð26Þ
where ^ kÿ1 þ ^bÞ: ^ kÿ1 þ Gu ^ kÿ1 ¼ I ÿ diag½tanh2 ðVx G
ð27Þ
A description is pursued in which the nominal part of (23) is described by (6), while the uncertain part is expressed in terms of dW, dV, dG, db and dd. Such description can be obtained by performing a first-order Taylor series expansion of (23) around the nominal weights of the neural network model, yielding ykþ1 ¼ f þ ðF þ dFÞDxk þ ðE þ dEÞDuk ; where 0
2.2. The uncertain case
dF ¼ dWGkÿ1 V þ WGkÿ1 dV þ WGkÿ1 FV kÿ1 V; 0
Despite feedforward neural networks have been generally accepted as suitable models for capturing the behaviour of highly nonlinear dynamical systems, it is also accepted that such models should not be considered exact. Therefore, model uncertainty should also be accounted for during the neural networks identification stage. This paper assumes that model uncertainty is due to the uncertainty on the weights of the neural network model. In fact, once the structure of the network (number of neurons, composition of the regression vector) is chosen, training the neural network can be regarded as a nonlinear least-squares estimation of its weights, by minimizing the sum of squared errors between the training data and the outputs of the neural network model. Provided that certain conditions are met, confidence regions for the estimated parameters can be obtained using statistical properties of a least-squares estimation (Seber and Wild, 1989) (see the appendix). It is assumed that the nonlinear process dynamics is described by the following neural network model: ^ þ d; ^ ^ k þ bÞ ^ tanhðVx ^ k þ Gu ð22Þ ykþ1 ¼ W ^ ¼ G þ dG; ^b ¼ ^ ¼ W þ dW, V ^ ¼ V þ dV, G where W b þ db and d^ ¼ d þ dd, with dW, dV, dG, db and dd representing the uncertainty in the neural network weights. Like in the nominal case, it is assumed that the first order Taylor expansion of (22), around
ð28Þ
dE ¼ dWGkÿ1 G þ WGkÿ1 dG þ WGkÿ1 FG kÿ1 G;
ð29Þ ð30Þ
with Gkÿ1 as defined in (10), and 0
Gkÿ1 ¼ diag½ÿ2 tanhðVxkÿ1 þ Gukÿ1 þ bÞ;
ð31Þ
FV kÿ1 ¼ diag½dVxkÿ1 þ db;
ð32Þ
FG kÿ1 ¼ diag½dGukÿ1 þ db;
ð33Þ
while f , F and E in (28) are defined by Eqs. (7)–(9), respectively. Finally, Eq. (28) can be rearranged in accordance with the procedure previously presented, leading to the following result: ~ þ dFÞ~ ~ xk þ ðE þ dEÞDuk ð34Þ ykþ1 ¼ ðF ~ ~k as defined in (12), F with the new state vector x ~ taken from Eq. (13), while dF ¼ dFN, with N taken from (17). The application of the feedback linearising control law (19) to the re-arranged model with uncertainties (34), results in a closed-loop system described by the following input–output behaviour: ~ ÿ dEE ÿ1 FÞ~ ~ xk xl þ CBvk þ ðdF ykþ1 ¼ CA~ k
þ dEE ÿ1 CA~ xlk þ dEE ÿ1 CBvk ;
ð35Þ
In order to proceed with the design of the input– output feedback linearisation scheme for this uncertain case, the following two assumptions must be made for
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the characterisation of the desired linear state-space system: Assumption 1. The linear state vector must be such that ~k , which means a particular state-space descrip~lk ¼ x x tion for the linear system is due. Assumption 2. The output vector of the linear system xlk , where the observation should be given by, yk ¼ C~ T matrix C ¼ ½sn1 snp , with each sni defined as in (14). This means that the linear system output is yk ¼ ½y1k ypk T . As a natural consequence, the following will hold: C C T ¼ I, with I being the ½ p p identity matrix. With these assumptions, expression (35) can be rewritten as ~ ÿ dEE ÿ1 F ~ þ dEE ÿ1 CAÞ~ xl ykþ1 ¼ ðCA þ dF k
þ ðCB þ dEE ÿ1 CBÞvk
ð36Þ
which corresponds to the following state-space description for the uncertain linear closed-loop system: ~lkþ1 ¼ ðA þ dAÞ~ xlk þ ðB þ dBÞvk ; x yk ¼ C~ xlk ;
ð37Þ
where terms dA and dB correspond to state-dependent matrices that express the overall linear closed-loop uncertainty as a function of the weights of the neural network model, and are respectively given by ~ ÿ dEE ÿ1 F ~ þ dEE ÿ1 CAÞ; ð38Þ dA ¼ CT ðdF dB ¼ CT dEE ÿ1 CB:
ð39Þ
The knowledge concerning their quantitative measures and respective bounds will provide the basic ingredients for a complete stability robustness analysis of the closed-loop system.
3. Stability robustness analysis The development made in the previous section, taking into account model uncertainty, resulted in the uncertain linear closed-loop description given by (37). Assume that ½A þ dAðÞ B þ dBðÞ 2 Coð½A1 B1 ; ½A2 B2 ; . . . ; ½AL BL Þ
ð40Þ
where Co stands for a convex hull, i.e., for some l1 ; l2 ; . . . ; lL 0 summing to one: ½A þ dAðÞ
B þ dBðÞ ¼
L X
li ½Ai Bi :
ð41Þ
i¼1
Then, if Bi , with i ¼ 1; . . . ; L, is bounded, for showing closed-loop stability of the uncertain system it is
663
sufficient to find a positive-definite matrix P, such that ATi PAi ÿ P P
5 0 > 0
8i ¼ 1; . . . ; L:
ð42Þ
This result represents sufficient conditions for Lyapunov stability for system (37). Moreover, expression (42) can be rewritten as a linear matrix inequality (LMI) by using Qÿ1 ¼ P: ATi Qÿ1 Ai ÿ Qÿ1 Qÿ1
5 >
0 0
8i ¼ 1; . . . ; L:
ð43Þ
Then, by post- and pre-multiplying (43) by Q, with Q ¼ QT , leads to Q ÿ ðAi QÞT Qÿ1 Ai Q > Q >
0 0
8i ¼ 1; . . . ; L
ð44Þ
from which, by using Schur’s complement condition, the following LMI can be constructed: Q ðAi QÞT > 0 8i ¼ 1; . . . ; L: ð45Þ Ai Q Q Solving a set of LMIs can be easily done with use of computationally efficient algorithms, which turn them into a practical tool for control engineering purposes (Boyd et al., 1994). Therefore, the key to the solution of the stability robustness problem is to build the necessary LMIs which correspond to the closed-loop uncertainty description given by (37). 3.1. Uncertainty bounds The remaining task to complete the stability robustness analysis of the closed-loop system is to bound the intervals within which the entries of matrices A and B are uncertain, such that a polytopic uncertainty description can be constructed. Considering the involved expressions, finding an analytic expression for the bounds of (38) and (39) can introduce a great deal of conservatism. The following analytic expressions can be derived for the maximum absolute values for each of the uncertain terms: ~ max þ abs C T dEmax E ÿ1 F ~ max dAmax abs C T dF ð46Þ þ abs C T dEmax E ÿ1 CA ; dBmax abs C T dEmax E ÿ1 CB ;
ð47Þ
~ max ¼ S þ Fmax N, and ~ max ¼ dFmax N, F where dF dFmax ¼ dWb Gmax absðVÞ þ absðWÞGmax dVb 0
þ absðWÞGmax FV max absðVÞ;
ð48Þ
dEmax ¼ dWb Gmax absðGÞ þ absðWÞGmax dGb 0
þ absðWÞGmax FG max absðGÞ; Fmax ¼ absðWÞGmax absðVÞ;
ð49Þ ð50Þ
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while 2
Gmax ¼ absðI ÿ diag½tanh ðVxkÿ1 þ Gukÿ1 þ bÞÞ; 0
ð51Þ
Gmax ¼ absðdiag½ÿ2 tanhðVxkÿ1 þ Gukÿ1 þ bÞÞ;
ð52Þ
FV max ¼ diag½dVb absðxkÿ1 Þ þ dbb ;
ð53Þ
FG max ¼ diag½dGb absðukÿ1 Þ þ dbb ;
ð54Þ
where dWb , dVb , dGb and dbb are exactly known and correspond to the uncertainty associated with each
neural network weight, i.e., the maximum uncertainty bound for each neural network nominal weight (see Appendix). For a given operating point, ðxkÿ1 ; ukÿ1 Þ, the bounds for dA and dB can be directly found through (46) and (47), respectively. However, as can be seen from Eqs. (51)–(54), these bounds will be state dependent, and so a numerical search must be performed in order to assure that the maximum value for each dA and dB over all admissible operating points is found. The proposed strategy which assures that only admissible ðxk ; uk Þ pairs are considered, is to design a reference trajectory vk , as used in the feedback linearising control law (19), and feedback linearise the process model. Then, as the model output travels along this trajectory, bounds on the uncertainty entries of matrices A and B can be collected, for each encountered operating point, according to the expressions (46) and (47), respectively. Finally, the maximum bound encountered along that trajectory can be used to construct the polytopic description which enables stability robustness analysis. One of the advantages of this approach is that uncertainty is considered only in relevant regions of the space spanned by the ðxk ; uk Þ pair during operation. Moreover, while this procedure is being performed, it can be automatically checked whether matrix Eðxkÿ1 ; ukÿ1 Þ, as defined in (9), is invertible over all encountered operating points, a basic requirement for the success of the proposed control scheme. 4. Case study
Fig. 2. Two-by-two water vessel system.
The control strategy and the stability robustness analysis procedure described in the previous sections
Fig. 3. Applied inflows to generate training and validation data (dashed line: u1, solid line: u2).
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Fig. 4. Neural network model validation (dashed line: target outputs, solid line: neural network outputs). (a) Water level h1. (b) Water level h2.
are applied to the water vessel system shown in Fig. 2. This is a two inputs–two outputs nonlinear dynamical system whose inputs correspond to the water inflows at the two upper vessels, u1 and u2, while the outputs are the water levels at the two lower vessels, h1 and h2, respectively. As seen from Fig. 2, the water outflows of the two upper vessels are partly directed to the underneath vessel and to the other lower vessel in a 75% by 25% ratio, respectively. Furthermore, the highly nonlinear dynamic behaviour of the system is reinforced by the shape of the two lower vessels. A
detailed description of the system dynamics is given below: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh1 1 f 21 ¼ d21 2 h1 g; dt ¼ A21ðh1Þð f 11 ÿ f 21Þ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dh2 1 f 22 ¼ d22 2 h2 g; dt ¼ A22ðh2Þð f 12 ÿ f 22Þ; f 11 ¼ a1 f 1 þ ð1 ÿ a2 Þ f 2; f 12 ¼ ð1 ÿ a1 Þ f 1 þ a2 f 2; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 1 ¼ d11 2 h11 g; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi f 2 ¼ d12 2 h12 g;
dh11 dt dh12 dt
1 ¼ A11 ðu1 ÿ f 1Þ; 1 ¼ A12 ðu2 ÿ f 2Þ; ð55Þ
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where the following parameters were used: A11 ¼ A12 ¼ p m2 , A21ðh1Þ ¼ p ð1 þ 0:3 sinð24 h1ÞÞ2 m2 ; d11 ¼ p A22ðh2Þ ¼ p ð1 þ 0:3 sinð24 h2ÞÞ2 m2 , 0:352 m2 ; d12 ¼ p 0:22 m2, d21 ¼ d22 ¼ p 0:2672 m2, a1 ¼ a2 ¼ 0:75 and g ¼10 m sÿ2 is the gravitational acceleration. The main control objective is twofold: to design a controller which is robustly stable with respect to model-plant uncertainty, while simultaneously guaranteeing independent water level tracking at the two lower vessels. In accordance with the procedure presented in this paper, the following three steps will be performed: 1. A neural network is trained to model the nonlinear multivariable input–output dynamic behaviour of the system, based on a given set of data measurements. 2. A polytopic closed-loop uncertainty description is constructed, based on confidence intervals for each parameter of the neural network model. 3. Finally, the control design is proven robustly stable by finding a Lyapunov function for the polytopic system through the solution of a set of LMIs equations. Step 1: Modelling the system. The input–output dynamic behaviour of the water vessel system is modelled by a single-layer feedforward neural network with hyperbolic tangent activation function having the following structure:
h1kþ1 h2kþ1
1 1 hk uk ¼ W tanh V 2 þ G 2 þ b þ d; hk uk
where h1k and h2k are the water levels at time instant k at the lower left vessel and at the lower right vessel, respectively, u1k and u2k are the water inflows at time instant k at the upper left vessel and at the right upper vessel, respectively. This neural network model was trained using the Levenberg–Marquardt optimisation algorithm, based on noisy input–output data measurements taken from the process simulation with a sampling time of 1 s. Fig. 3 shows the applied water inflows used to generate the training and validation data. The best structure found after the learning phase was completed, consisted of a feedforward neural network with two hidden neurons, i.e., W 2 R22 , V 2 R22 , G 2 R22 , b 2 R21 and d 2 R21 in (56). The validation results for the nonlinear output error simulation shown in Fig. 4 reveal an accurate dynamic performance of the neural network model over the entire operating space. Therefore, reliable confidence regions for the estimated neural network parameters can be obtained by using the statistical properties of the nonlinear least-squares estimation. Step 2: Obtaining a polytopic closed-loop uncertainty description. In accordance with the approximate input– output feedback linearisation procedure, the following desired linear dynamics are imposed to the overall closed-loop system: 2
~lkþ1 x ð56Þ
1:7 6 1 6 ¼4 0 0
ÿ0:7225 0 0 0
0 0 1:7 1
3 0 7 l 0 7x ~ ÿ0:7225 5 k 0
ð57Þ
Fig. 5. Approximate feedback linearisation applied to the model (dashed line: external inputs, solid line: neural network model outputs).
M.A. Botto et al. / Engineering Applications of Artificial Intelligence 13 (2000) 659–670
667
Fig. 6. Closed-loop response of the water vessel system using the neural network model. (a) Reference following (dashed line: external inputs, solid line: system outputs). (b) Control signals (dashed line: inflow upper left vessel, solid line: inflow upper right vessel).
2
0:0225 6 0 þ6 4 0 0
h1k h2k
1 ¼ 0
0 0
0 1
3 0 1 0 7 7 vk2 ; 0:0225 5 vk 0 0 l ~ : x 0 k
If the design goal is satisfied, the dynamics of the closed-loop system will mimic the behaviour
of two stable and decoupled single-input singleoutput systems. Following the procedure presented in Section 3.1, the maximum uncertainty bounds in the parameters of matrices A and B in (57), determined along the reference trajectory shown in Fig. 5, are found to be 2 3 6:76 3:53 3:39 1:44 6 0 0 0 0 7 ÿ3 7 dAmax ¼ 6 4 3:75 1:60 6:47 3:37 5 10 ; 0 0 0 0
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2
dBmax
3 3:75 2:22 6 0 0 7 ÿ5 7 ¼6 4 2:45 3:58 5 10 : 0 0
ð58Þ
These quantitative measures are sufficient to proceed with the stability robustness analysis of the uncertain closed-loop system. Step 3: Proving robust stability. The final step of the controller design is to construct a polytopic description based on the uncertain linear closedloop system described by (57) and (58) and build the corresponding LMIs. Then, as described in Section 3, if a positive-definite matrix is found as the solution for this set of LMIs the closed-loop system is proven robustly stable. In this case, the set of LMIs was solved using the LMI-Toolbox (Gahinet et al., 1995), resulting in the following positive-definite matrix: 2 ÿ2:95 108 3:65 108 6 ÿ2:95 108 2:55 108 6 P ¼6 ÿ6 4 7:15 10 ÿ5:49 10ÿ6 ÿ5:66 10ÿ6 7:15 10ÿ6 ÿ5:49 10ÿ6 3:62 108 ÿ2:93 108
4:64 10ÿ6
3 ÿ5:66 10ÿ6 4:64 10ÿ6 7 7 8 7 ÿ2:93 10 5
ð59Þ
2:53 108
This solution proves that the closed-loop system is robustly stable with respect to parametric uncertainties. The overall closed-loop performance of the water vessel system under approximate feedback linearisation is shown in Fig. 6, where it is visible that the system outputs are independently controlled with a small control effort.
5. Concluding remarks Approximate feedback linearisation has several features which makes it an appealing control strategy for nonlinear systems. From an application point of view these are related to the control law being easily determined from the system model, while the transformation of the nonlinear open-loop system into a linear closed-loop system allowed for linear techniques to be used in the analysis of the closed-loop properties. The main contribution of this paper is that it provides a systematic procedure for stability robustness analysis to structured uncertainty for neural network models under approximate feedback linearisation. It was shown how the closed-loop uncertainty can be expressed in terms of the uncertainty in the weights of a single layer feedforward neural network. This result enabled the
construction of a polytopic uncertainty description for the linear closed-loop system. Based on this polytopic description, stability of the uncertain closed-loop can be verified by finding a proper Lyapunov function. Due to a special form of the closed-loop uncertainty description, finding the Lyapunov function could be rewritten as a LMI problem, and thus can be tackled with the use of computationally efficient algorithms. The significance of this design strategy in the scope of nonlinear robust control is that it allows the application of linear robust control analysis to nonlinear control problems. The strategy presented in this paper can be applied to a broad class of complex nonlinear multivariable dynamic systems, as long as they are square and follow some mild assumptions. Simulation experiments showed a successful application of this control design procedure to a nonlinear multivariable water vessel system.
Acknowledgements This research is partially supported by PRAXIS/P/ EME/12124/1998, by a Research Fellowship from the Technical University of Delft, and by the Dutch Technology Foundation (STW).
Appendix. Confidence intervals for estimated weights Suppose that n observations ðyi ; xi Þ; i ¼ 1; 2; . . . ; n, are given from a fixed-regressor nonlinear model with a known functional relationship f , according to y ¼ f ðx; y Þ þ e;
ðA:1Þ
where e is a normally distributed zero mean white noise with variance s2 , and y the true value of y, which is known to belong to Y. The least-squares estimate of y , denoted by ^y, is obtained through the minimization of the following sum square of errors: n X ½ yi ÿ f ðxi ; yÞ2 ¼ jjy ÿ f ðx; yÞjj2 ðA:2Þ SðyÞ ¼ i¼1
over y 2 Y. It should be noted that, unlike the linear least-squares case, SðyÞ may have several local minima in addition to the absolute minimum, y . Assuming e to be independent and identically distributed with variance s2 , it can be shown that under certain regularity conditions, ^y is a consistent estimate of y . Moreover, ^y is also proven to be asymptotically normally distributed as n 7! 1. To show this, consider the linear Taylor series expansion of f , in a small neighbourhood of y , given by @f ðx; yÞ ðy ÿ y Þ f ðx; yÞ f ðx; y Þ þ @y y ðA:3Þ ¼ f ðx; y Þ þ Fðy ÿ y Þ:
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Hence 2
SðyÞ ¼ jjy ÿ f ðx; yÞjj jjy ÿ f ðx; y Þ ÿ Fðy ÿ y Þjj2 ¼ jje ÿ Fbjj2 ;
ðA:4Þ
where e ¼ y ÿ f ðx; y Þ, and b ¼ y ÿ y . From the properties of the linear model, SðyÞ is minimized when b is given by ^ ¼ ðF T FÞÿ1 F T e: ðA:5Þ b Furthermore, for large n, ^ y is almost certain to be within ^ and y ÿ y b, a small neighbourhood of y . Hence, ^ therefore ÿ ^y ÿ y F T F ÿ1 F T e: ðA:6Þ Then, according to Seber and Wild (1989) and Ljung (1987), the following holds: given E Nð0; s2 In Þ, with E normally distributed with zero mean and variance s2 In , together with the appropriate regularity conditions, it can be stated that for large n: ^y ÿ y Np ð0; s2 ðF T FÞÿ1 Þ: ðA:7Þ y more than Hence, that y deviates from ^ ffiffiffiffiffiffiffiprobability ffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffithe ÿ1 a s2 ðF T FÞ corresponds to the ð1 ÿ aÞ-level of the normal distribution, available in standard statistical tables.
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Narendra, K.S., Parthasarathy, K., 1990. Identification and control of dynamical systems using neural networks. IEEE Transactions on Neural Networks 1 (1), 4–27. Nijmeijer, H., van der Schaft, A. J., 1990. Nonlinear Dynamical Control Systems. Springer, New York. Seber, G., Wild, C., 1989. Nonlinear Regression. John Wiley, New York. Sjo¨berg, J., 1995a. Non-linear system identification with neural networks. Ph.D. Thesis, Linko¨ping University, Department of Electrical Engineering, Sweden. Sjo¨berg, J., 1995b. Nonlinear black-box modeling in system identification: a unified overview. Automatica 31 (12), 1691–1724. Slotine, J.-J. E., Li, W., 1991. Applied Nonlinear Control. PrenticeHall, NJ, USA. Suykens, J., Vandewalle, J., de Moor, B., 1996. Artificial Neural Networks for Modelling and Control of Non-Linear Systems. Kluwer Academic Publishers, Dordrecht. Tanaka, K., 1996. An approach to stability criteria of neural-network control systems. IEEE Transactions on Neural Networks 7 (3), 629–642. te Braake, H., Ayala Botto, M., van Can, H., Sa´ da Costa, J., Verbruggen, H., 1999. Linear predictive control based on approximate input-output feedback linearisation. IEE Proceedings } Control Theory and Applications 146 (4), 295–300. te Braake, H., van Can, H., Scherpen, J., Verbruggen, H., 1997. Control of nonlinear chemical processes using dynamic neural models and feedback linearization. Computers and Chemical Engineering 22, 1113–1127. van den Boom, T., 1997. Robust nonlinear predictive control using feedback linearization and linear matrix inequalities. American Control Conference, Albuquerque, NM, pp. 3068–3072. Wams, B., Ayala Botto, M., van den Boom, T., Sa´ da Costa, J., 1998. Training neural networks for robust control of nonlinear MIMO systems. Proceedings of the International Conference on Control’98, Swansea, UK, pp. 141–146.
Miguel Ayala Botto was born in Lisbon, Portugal in 1965. He graduated in Mechanical Engineering from the Technical University of Lisbon in 1989 and received his M.Sc. and Ph.D. degrees in Mechanical Engineering from the same university in 1992 and 1996, respectively. He is currently Assistant Professor at the Control, Automation and Robotics Group of the Department of Mechanical Engineering at Instituto Superior Te´cnico, Technical University of Lisbon. From September 1999 he held a six-month sabbatical leave position at the Control Laboratory of the Department of Information Technology and Systems at the Delft University of Technology, the Netherlands. His current research interests include nonlinear control systems, neural networks modelling and identification for robust control.
Bart Wams is currently a Ph.D. student at the Control Laboratory of the Department of Information Technology and Systems at the Delft University of Technology, the Netherlands. His research topic is the incorporation of neural network models into nonlinear robust modelbased control. His research is supported by the Dutch Technology Foundation (STW), under project number AIF44.3595. Before he joined the Control Laboratory in 1996, he received his M.Sc. degree in Electrical Engineering at the same university. His M.Sc. thesis focused on control configurations for time-varying nonlinear processes, with guaranteed closed-loop stability.
Dr.ir. T.J.J. van den Boom is an Associate Professor at the Control Laboratory of the Department of Information Technology and
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Systems at the Delft University of Technology, the Netherlands. He is also a member of CIDIC, the European consortium on ‘‘Computer Integrated Design of Controlled Industrial Systems’’. In 1996 he held a six-month sabbatical leave position at the Swiss Federal Institute of Technology (ETH) in Zu¨rich. Special fields of interest: model predictive control, robust control, nonlinear control, anti-windup bumpless-transfer control schemes, system identification for robust control, application of advanced control in mechatronics and process industry.
Jose´ Sa´ da Costa received his Diploma in Mechanical Engineering from the Technical University of Lisbon in 1974 and his M.Sc. and Ph.D. in Automatic Control from the University of Manchester Institute of Science and Technology in 1979 and 1982, respectively. He is currently an Associate Professor in Control at the Department of Mechanical Engineering at Instituto Superior Te´cnico, Technical University of Lisbon. His current research interests include system modelling and identification, adaptive and intelligent control and robotics.