- Email: [email protected]
On the Use of Stability Techniques to Determine Neural Network Models Suitable for Differential Geometric Based Nonlinear Controller Design
Copyright © 19961FAC 13th Triennial World Congress, San FrancIsco. USA
7a-ll 2
ON THE USE OF STABILITY TECHNIQUES TO DETERMINE NEURAL NETWORK MODELS SUITABLE FOR DIFFERENTIAL GEOMETRIC BASED NONLINEAR CONTROLLER DESIGN
C. J. Chessari· . G. W. Barton# and J. A. Romagnoli#
*Tech Control Ply Limited, 2 Lincoln Street Lane Cove,
N....·W 2066 Australia
#Departmenf oj'Che1llical E.ngineering, The Universi~v ()j'Sydney, NSJf' 2006. tustra/ia
Abstract: This paper outlines a strategy for developing neural network models suitable for nonlinear controller design. The strategy first employs a multiple neural network approach to the dynamic modelling of (minimum phase) affine systems. The resultant dynamic model is then used for controller design within a differential geometric framework. The focus of the work is to predict a priori whether the nonlinear controller will be stable. Stability checks are used in order to vet candidate models and to influence the neural network training procedure itself. Keywords: Neural networks: Identification algorithms: Nonlinear control systems: Stability tests: Differential Geometry.
1. INTRODUCTION In recent years powerful nonlinear controller design techniques based on differential geometry have been developed (lsidori. 1989: Kravaris and Chung, 1987). Unlike local Iinearisation techniques, these methods provide exact linearisation of a nonlinear process model that is independent of the operating point. Linear controllers can then be designed for the equivalent 'linear' system. The methods involve developing nonlinear coordinate transformations and implementing a feedback law and are based on the assumption that a good nonlinear model exists over the operating region of the plant.
accurate nonlinear representation of complex processes makes them an attractive modelling approach to be used as the first step in the development of a geometric nonlinear controller.
2. GEOMETRIC NONLINEAR CONTROL Only affine continuous single-input single-output (SISO) systems are considered in this paper. The nonlinear systems considered have the following form. i = lex) + g(x)u
y
Chemical processes are very often difficult to model due to their nonlinear behaviour. strong variable interactions. and the complexity of the underlying physical and chemical phenomena. The ability of neural networks to provide an
(1)
= hex)
where u is the manipulated input. y is the output, x represents a vector of states and fix) and g(x) are vector functions.
6161
In nonlinear control theory. linearisation consists of finding a state feedback control law. 11
= p(x) + q(x)v
(6)
(2)
where p(x) and q(x) are algebraic functions of the state variables with q(x )* 1I. and v is an external reference input, such that the subsequent closed-loop system is exactly input-outpullinear:
';,,-1 = ';n
';n = <1>(';) + G(';)u
~ = [lex)
+ p(x)g(x)] + [q(x)g(x)]v y=h(x)
where
The main impetus for feedback linearisation is that once the system is transformed. linear control loops can be designed around it. In the nonlinear case. state feedback changes the behaviour of the system by shaping the closedloop response. There are two alternative concepts of feedback Iinearisation: one makes the closed-loop state equations linear while the other makes the input-output behaviour of the closed-loop system linear. This work is concerned with the latter. specifically with the minimal order linearisation approach of Kravaris and Chung (1987) expressed as Theorem I. However. prior to stating this theorem the concepts of Lie derivative. relative order. coordinate transformation. and zero dynamics need to be considered. Lie derivative : Given a Cl scalar field hex) and a vector field .((x). the Lie derivative of hex) with respect to .((x) is defined by. n oh Lrh(x)= (dh(x).f(x»)= ~.f;(x) ox; (x)
Y
(3)
(4)
where <... > denotes the inner product. Relative order : For nonlinear non linear systems of the form given in Equation I the relative order is defined (Hirschorn. 1979) as the smallest integer r for which.
Fn-r (.;'), <1>( ~), and
G(~) are
all
scalar fields on ~ltn. with G(~)*O. The terms are respectively defined as Fj (~)
= Ll~i (x)
for i in [1. n-rJ.
<1>(~) = Lrrh(x). and G(~) = LgLilh(x). Zero dynamics : A nonlinear system of the form given in Equation I does not have a transfer function and consequently cannot have zeros that are a set of number derived from the roots of a polynomial. However, a nonlinear system does have an inverse and this inverse defines a dynamic system. From such considerations. the concept of zero dynamics of a nonlinear system is the dynamics of a minimal order realisation of its inverse. Byrnes and Isidori (1985) suggested that a nonlinear system be called minimum phase if its zero dynamics is asymptotically stable around the origin. otherwise it is called nonminimum phase.
With the relevant concepts described. it is now possible to return to the original description of the feedback theorem of Kravaris and Chung (1987) and its implications. Theorem 1 : Consider a non linear system of the form given by Equation I of relative order r. The state feedback of the form given by Equation 2 that makes the inputoutput behaviour of the closed-loop system linear and of minimal order is:
(5) 11
Co-ordinate tran~fim"ation : In order to obtain a minimal order inverse from which inverse dynamics can be elucidated, a co-ordinate transforntation to the ByrnesIsclori (1986) normal form is required. For a nonlinear dynamic system of relative order r this transformation is given by.
I~i (~), .. "
= ;n-r+1
(v-l:il(x) - P1Ej"l(xh. ·-Pr_,Ljh(x)- PrI(X») (7)
(Lg I!7' hex») where PI' ... , Pr are constant scalar parameters and the input-output behaviour of the closed-loop system is:
dry . dry dry dry --. + Pl--. +···+fir-I--. + fir - - = v £I,' cif' tlr' dt r
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(8)
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6164
. [~ll
6. STABILITY ISSUES PERTAINING TO NEURAL NETWORK MODELS Of critical importance is the requirement that any identified neural network model must be minimum phase. It is well known that a neural network error surface may have many acceptable minima (Hecht-Nielsen. 1989). In the present case, neural network training is not specifically encouraged to develop a model which is minimum phase even if the system is minimum phase. It is quite conceivable that the converged neural network provides a nonminimum phase representation of a minimum phase process. The final consideration in this work is concerned with this point that is, to establish conditions ensuring that a neural network model is indeed minimum phase. [n essence, this is a basic robustness bound for our model as the nonlinear controller must be developed from a minimum phase model in this controller design framework.
~ = ~:
=
(8t/J) ac x
(13)
where S are the transformed states and in particular the zero dynamics state is denoted by 11. It is clear from Equation 15 that the third differential equation is of interest as this describes the zero dynamics. In isolation. this equation is.
iJ
= (
~; ) . (f (x) + g (x)u)
(14)
which can be expressed in Lie derivative terminology as. (15)
The evaluation is begun by considering the development of the zero dynamics for the CSTR system. The first task is to construct a co-ordinate transformation, ;=~(x), from which a normal form of the model can be generated. As the CSTR system has three states, it is three-dimensional. A partial co-ordinate transformation is given by the following:
However. as the additional function is constructed such that Lg 1]( x) = 0 the zero dynamics is given by:
Additionally, a third function (say 11) is required to complete the transformation such that its Jacobian is nonsingular and the filOction 11 satisfies the partial differential equation given by L g 1]( x) = 0 . that is.
the./{x) function. Consequently. local asymptotic stability is satisfied if the partial derivative of the first row element of the.f(x) function is negative.
where gl. g2 and g3 are the components of the g vector field (that is, 0, 1 and O. respectively). By putting these values into Equation 12. it is clear that a 11 function should be chosen such that its derivative with respect to the second state is zero. A suitable (simple) candidate is choosing 11 = CA' With this choice. the nonsingularity of the Jacobian of the transformation is inferred by showing that the rank of this matrix is three. This was done using MATLAB (1991 ). The transformed equations of the original system are developed by taking the derivative of the transformation S = 4>(x) as follows.
r, = L/1](x)
(16)
Local asymptotic stability of Equation 16 (as required by Theorem 2) can be obtained by linear stability methods. As 11 = ~4' the derivative of the zero dynamics state with respect to the original states is [ I 0 0 ) and thus the Lie derivative term Lf1{x) is equal to the first row element of
The last point to be considered is that the requirement that the neural network model be minimum phase is a necessary (but not sufficient) condition to ensure acceptable controller behaviour. The neural network model must also adequately represent both /(x) and g(x). a condition that is automatically imposed by requiring a low RMS error. If both these conditions are satisfied. then it would appear that the neural network model will be sufficient for nonlinear controller synthesis. This point is illustrated with respect to the CSTR example. As stated previously. the neural network models which can be developed are not unique. Figure 5 shows the range of maximum derivative values that can be obtained from a number of CSTR models at various RMS fitting errors using different random initial starting weights. This figure suggests that models having the same overall RMS fitting error can have differing characteristics.
6165
6166
7a-ll 2
ON THE USE OF STABILITY TECHNIQUES TO DETERMINE NEURAL NETWORK MODELS SUITABLE FOR DIFFERENTIAL GEOMETRIC BASED NONLINEAR CONTROLLER DESIGN
C. J. Chessari· . G. W. Barton# and J. A. Romagnoli#
*Tech Control Ply Limited, 2 Lincoln Street Lane Cove,
N....·W 2066 Australia
#Departmenf oj'Che1llical E.ngineering, The Universi~v ()j'Sydney, NSJf' 2006. tustra/ia
Abstract: This paper outlines a strategy for developing neural network models suitable for nonlinear controller design. The strategy first employs a multiple neural network approach to the dynamic modelling of (minimum phase) affine systems. The resultant dynamic model is then used for controller design within a differential geometric framework. The focus of the work is to predict a priori whether the nonlinear controller will be stable. Stability checks are used in order to vet candidate models and to influence the neural network training procedure itself. Keywords: Neural networks: Identification algorithms: Nonlinear control systems: Stability tests: Differential Geometry.
1. INTRODUCTION In recent years powerful nonlinear controller design techniques based on differential geometry have been developed (lsidori. 1989: Kravaris and Chung, 1987). Unlike local Iinearisation techniques, these methods provide exact linearisation of a nonlinear process model that is independent of the operating point. Linear controllers can then be designed for the equivalent 'linear' system. The methods involve developing nonlinear coordinate transformations and implementing a feedback law and are based on the assumption that a good nonlinear model exists over the operating region of the plant.
accurate nonlinear representation of complex processes makes them an attractive modelling approach to be used as the first step in the development of a geometric nonlinear controller.
2. GEOMETRIC NONLINEAR CONTROL Only affine continuous single-input single-output (SISO) systems are considered in this paper. The nonlinear systems considered have the following form. i = lex) + g(x)u
y
Chemical processes are very often difficult to model due to their nonlinear behaviour. strong variable interactions. and the complexity of the underlying physical and chemical phenomena. The ability of neural networks to provide an
(1)
= hex)
where u is the manipulated input. y is the output, x represents a vector of states and fix) and g(x) are vector functions.
6161
In nonlinear control theory. linearisation consists of finding a state feedback control law. 11
= p(x) + q(x)v
(6)
(2)
where p(x) and q(x) are algebraic functions of the state variables with q(x )* 1I. and v is an external reference input, such that the subsequent closed-loop system is exactly input-outpullinear:
';,,-1 = ';n
';n = <1>(';) + G(';)u
~ = [lex)
+ p(x)g(x)] + [q(x)g(x)]v y=h(x)
where
The main impetus for feedback linearisation is that once the system is transformed. linear control loops can be designed around it. In the nonlinear case. state feedback changes the behaviour of the system by shaping the closedloop response. There are two alternative concepts of feedback Iinearisation: one makes the closed-loop state equations linear while the other makes the input-output behaviour of the closed-loop system linear. This work is concerned with the latter. specifically with the minimal order linearisation approach of Kravaris and Chung (1987) expressed as Theorem I. However. prior to stating this theorem the concepts of Lie derivative. relative order. coordinate transformation. and zero dynamics need to be considered. Lie derivative : Given a Cl scalar field hex) and a vector field .((x). the Lie derivative of hex) with respect to .((x) is defined by. n oh Lrh(x)= (dh(x).f(x»)= ~.f;(x) ox; (x)
Y
(3)
(4)
where <... > denotes the inner product. Relative order : For nonlinear non linear systems of the form given in Equation I the relative order is defined (Hirschorn. 1979) as the smallest integer r for which.
Fn-r (.;'), <1>( ~), and
G(~) are
all
scalar fields on ~ltn. with G(~)*O. The terms are respectively defined as Fj (~)
= Ll~i (x)
for i in [1. n-rJ.
<1>(~) = Lrrh(x). and G(~) = LgLilh(x). Zero dynamics : A nonlinear system of the form given in Equation I does not have a transfer function and consequently cannot have zeros that are a set of number derived from the roots of a polynomial. However, a nonlinear system does have an inverse and this inverse defines a dynamic system. From such considerations. the concept of zero dynamics of a nonlinear system is the dynamics of a minimal order realisation of its inverse. Byrnes and Isidori (1985) suggested that a nonlinear system be called minimum phase if its zero dynamics is asymptotically stable around the origin. otherwise it is called nonminimum phase.
With the relevant concepts described. it is now possible to return to the original description of the feedback theorem of Kravaris and Chung (1987) and its implications. Theorem 1 : Consider a non linear system of the form given by Equation I of relative order r. The state feedback of the form given by Equation 2 that makes the inputoutput behaviour of the closed-loop system linear and of minimal order is:
(5) 11
Co-ordinate tran~fim"ation : In order to obtain a minimal order inverse from which inverse dynamics can be elucidated, a co-ordinate transforntation to the ByrnesIsclori (1986) normal form is required. For a nonlinear dynamic system of relative order r this transformation is given by.
I~i (~), .. "
= ;n-r+1
(v-l:il(x) - P1Ej"l(xh. ·-Pr_,Ljh(x)- PrI(X») (7)
(Lg I!7' hex») where PI' ... , Pr are constant scalar parameters and the input-output behaviour of the closed-loop system is:
dry . dry dry dry --. + Pl--. +···+fir-I--. + fir - - = v £I,' cif' tlr' dt r
6162
(8)
6163
6164
. [~ll
6. STABILITY ISSUES PERTAINING TO NEURAL NETWORK MODELS Of critical importance is the requirement that any identified neural network model must be minimum phase. It is well known that a neural network error surface may have many acceptable minima (Hecht-Nielsen. 1989). In the present case, neural network training is not specifically encouraged to develop a model which is minimum phase even if the system is minimum phase. It is quite conceivable that the converged neural network provides a nonminimum phase representation of a minimum phase process. The final consideration in this work is concerned with this point that is, to establish conditions ensuring that a neural network model is indeed minimum phase. [n essence, this is a basic robustness bound for our model as the nonlinear controller must be developed from a minimum phase model in this controller design framework.
~ = ~:
=
(8t/J) ac x
(13)
where S are the transformed states and in particular the zero dynamics state is denoted by 11. It is clear from Equation 15 that the third differential equation is of interest as this describes the zero dynamics. In isolation. this equation is.
iJ
= (
~; ) . (f (x) + g (x)u)
(14)
which can be expressed in Lie derivative terminology as. (15)
The evaluation is begun by considering the development of the zero dynamics for the CSTR system. The first task is to construct a co-ordinate transformation, ;=~(x), from which a normal form of the model can be generated. As the CSTR system has three states, it is three-dimensional. A partial co-ordinate transformation is given by the following:
However. as the additional function is constructed such that Lg 1]( x) = 0 the zero dynamics is given by:
Additionally, a third function (say 11) is required to complete the transformation such that its Jacobian is nonsingular and the filOction 11 satisfies the partial differential equation given by L g 1]( x) = 0 . that is.
the./{x) function. Consequently. local asymptotic stability is satisfied if the partial derivative of the first row element of the.f(x) function is negative.
where gl. g2 and g3 are the components of the g vector field (that is, 0, 1 and O. respectively). By putting these values into Equation 12. it is clear that a 11 function should be chosen such that its derivative with respect to the second state is zero. A suitable (simple) candidate is choosing 11 = CA' With this choice. the nonsingularity of the Jacobian of the transformation is inferred by showing that the rank of this matrix is three. This was done using MATLAB (1991 ). The transformed equations of the original system are developed by taking the derivative of the transformation S = 4>(x) as follows.
r, = L/1](x)
(16)
Local asymptotic stability of Equation 16 (as required by Theorem 2) can be obtained by linear stability methods. As 11 = ~4' the derivative of the zero dynamics state with respect to the original states is [ I 0 0 ) and thus the Lie derivative term Lf1{x) is equal to the first row element of
The last point to be considered is that the requirement that the neural network model be minimum phase is a necessary (but not sufficient) condition to ensure acceptable controller behaviour. The neural network model must also adequately represent both /(x) and g(x). a condition that is automatically imposed by requiring a low RMS error. If both these conditions are satisfied. then it would appear that the neural network model will be sufficient for nonlinear controller synthesis. This point is illustrated with respect to the CSTR example. As stated previously. the neural network models which can be developed are not unique. Figure 5 shows the range of maximum derivative values that can be obtained from a number of CSTR models at various RMS fitting errors using different random initial starting weights. This figure suggests that models having the same overall RMS fitting error can have differing characteristics.
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