1

Controllability and observability covariance matrices for the analysis and order reduction of

stable nonlinear systems

Juergen Hahn1, Thomas F. Edgar1, and Wolfgang Marquardt2

1Department of Chemical Engineering

The University of Texas at Austin

Austin, TX 78712-1062 tfedgar@che.utexas.edu

Phone: +1 (512) 471 3080 Fax: +1 (512) 471 7060

2Lehrstuhl fr Prozesstechnik

RWTH Aachen

D - 52064 Aachen Germany

Abstract This paper presents a framework for nonlinear systems analysis that is based upon controllability and observability covariance matrices. These matrices are introduced in the paper and it is shown that gramians for linear systems form special cases of the covariance matrices. The covariance matrices can be transformed via a balancing-like transformation and nonlinearity measures are defined based upon these transformed covariance matrices. Subsequently, the covariance matrices are used for reduction of the nonlinear model. It is shown that the model reduction procedure reduces to balanced model truncation for linear systems for impulse inputs. Furthermore, it is also shown that several model reduction procedures that were developed by other researchers, and assumed to be independent from one another, are related. The findings are illustrated with an example.

Keywords: Nonlinear System, Covariance Matrix, Nonlinearity Measure, Model Reduction, Controllability & Observability Gramian

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1. Introduction Nonlinear model predictive control has become increasingly popular in the chemical process industry in the last few years [14]. This is due to the fact that a nonlinear model can provide a more accurate description of the process dynamics which results in better controller performance. However, nonlinear controllers have some drawbacks when compared to linear controllers due to the increased complexity introduced by the nonlinearity of the model. While a variety of methods for the analysis and design of linear controllers exists, relatively few methodologies are available for nonlinear systems. Additionally, the investigation of control-relevant properties of the nonlinear system can be extremely time-consuming, because no general framework has yet been developed. This paper presents a framework for the analysis and model reduction of nonlinear models for the purpose of control. The methodologies introduced within this framework are (1) simple to implement, thereby making it suitable for large-scale applications, (2) give information about control-relevant properties of a nonlinear model, and (3) reuse results obtained from earlier analysis steps. Specifically, the properties under investigation are the degree of nonlinearity that the model exhibits in the input-to-state and state-to-output behavior, the controllability and observability analysis of the model, and the essential number of states that contribute to the input-output behavior of the system. Based upon the results from this analysis, the model can be reduced to eliminate subspaces that do not contribute to the input-output behavior of the system.

The framework presented is based upon the computation of covariance matrices for the input-to-state (controllability) and state-to-output (observability) behavior of the nonlinear system. These covariance matrices have the advantage that they are relatively inexpensive to compute, easy to manipulate, and that the information contained can be used for several different tasks. First, the controllability and observability covariance matrices are computed for the nonlinear system. The controllability covariance matrix is computed from the system trajectories resulting from different excitations. The observability covariance matrix is computed from the behavior of the system outputs for different perturbations in the initial conditions of the system, while the system inputs are kept at their steady state values. Second, the covariance matrices are transformed by a balancing-like transformation such that each state of the transformed system is just as

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important for the input-to-state behavior as it is for the state-to-output behavior. Additionally, the states are ordered with decreasing importance for the input-output behavior of the system. Based upon changes in the transformed covariance matrices for different excitation or perturbation sizes, respectively, the degree of the nonlinearity of the process can be determined. In the third step the subsystem of states that is uncontrollable and unobservable can be eliminated from the model and additional states that contribute little to the input-output behavior of the system are truncated.

This procedure can result in faster computation times in dynamic simulations, but the main aim is to analyze the nonlinear system and its states for their relevance for controller design. An additional feature of the proposed method is that it reduces to well known linear techniques, when the model under investigation is linear.

The outline of the paper is as follows. A literature review is presented in the following subsection and section 2 introduces the concepts of controllability and observability covariance matrices. Nonlinear model reduction as well as nonlinearity quantification can be performed based upon the information contained in the covariance matrices. It is shown in section 3 that earlier methods based upon empirical gramians [1, 2] form a special case of methods based upon these covariance matrices and a framework for the analysis of nonlinear systems is introduced. Section 4 points out similarities between different model reduction procedures including the one presented in this paper. A case study to illustrate the use of this framework is presented in section 5 and properties of this framework are discussed in section 6.

1.1. Previous work on nonlinear model reduction Balanced model reduction for linear systems was first introduced by Moore [10] in order to eliminate states that are hard to observe or to control and therefore contribute little to the input-output behavior of a system. Scherpen [15] extended the balancing approach to nonlinear systems by introducing energy functions and investigating conditions that guarantee existence of a balanced realization. However, the procedures are only applicable to control-affine systems, present computational difficulties, and in general a closed form solution for the energy functions cannot be obtained. The only numerical implementation of Scherpens approach is given Newman and Krishnaprasad [11] who

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used a Monte-Carlo approach. They tested their algorithm on a pendulum with two states to compute approximations to the energy functions as well as a balancing transformation for the nonlinear system. However, after the coordinate transformation is applied, and even without reducing the model, the transformed system does not exhibit the same input-output behavior as the original system due to the approximations that were applied during the computational procedure.

Due to the problems encountered with nonlinear balancing procedures, several methods that perform a Galerkin projection, based upon a linear coordinate transformation, have been developed. Newman and Krishnaprasad [12] compared models describing chemical vapor deposition that were reduced by principal component analysis (PCA) and balancing, where the balancing transformation was found from the linearized system.

Pallaske [13] investigated a procedure where the linear transformation is found from a covariance matrix that is computed from data collected along system trajectories. These trajectories represent the system behavior under a constant input, but starting from different initial conditions. Lffler and Marquardt [8] extended this model reduction approach to models described by differential-algebraic equation systems. They investigated the case of trajectories generated by different initial conditions under constant inputs as well as the case where the trajectories start at the steady state operating point and are generated by step changes in the inputs to the system. Lee et al. [7] computed the linear coordinate transformation by balancing a system that was generated using subspace identification. This identified system was generated from data collected along system trajectories. Lall et al. [5, 6] introduced the concept of empirical gramians, which are an extension to the gramians for linear systems. These empirical gramians can

be computed for control-affine nonlinear systems. The computation procedure is based upon system trajectories that include changes in the system inputs as well as in the initial conditions. Based upon the empirical gramians a linear coordinate transformation can be computed and the model reduced via a Galerkin projection. Hahn and Edgar [1] showed that the procedure presented by Lall et al. [5, 6] is limited to control-affine systems and requires modifications when the steady state of the system is different from zero.

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Additionally, they investigated the extension of balanced residualization to nonlinear systems via the found coordinate transformation.

2. Covariance matrices for nonlinear system analysis Gramians and empirical gramians have been used for model reduction of linear and nonlinear systems as well as for nonlinearity quantification in recent work [1, 2]. However, the gramians

dteBBeW tATAtlinearCT

=

0,

(1)

dtCeCeW tATtAlinearOT

=

0,

(2)

can only be computed for linear systems. Empirical gramians [1, 2] are restricted to stable (in the sense of Lyapunov) nonlinear control-affine systems, so there is a need for a method that analyzes more general nonlinear systems. Furthermore, it is possible that the empirical controllability gramian that is computed by exciting the system with impulse inputs does not give sufficient information about the nonlinearity of the system for other types of inputs (for example step changes). Due to this, the concept of covariance matrices is introduced in this paper. These covariance matrices are not restricted to control-affine systems and can be computed for different input types as well. The information that is contained in these covariance matrices is similar to the information that can be obtained from empirical gramians and in fact the covariance matrices include empirical gramians as special cases. While these covariance matrices cannot represent the

whole global behavior of nonlinear systems, they can be used for the analysis and model reduction of nonlinear systems for a pre-specified operating region.

For linear models it is sufficient to use only impulse inputs for the excitation of the system to get information about the input-to-state behavior, since the controllability gramian contains complete information about the input-to-state behavior of a linear system. However, this is not the case for nonlinear systems, where a variety of different inputs needs to be considered for the excitation of the system. The controllability covariance matrices take this into account and rely on the input-to-state behavior as well as on the inputs used for their computation.

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2.1. Controllability covariance matrix For any stable nonlinear system

))(()())(),(()(

txhtytutxftx

=

=&

(3)

the sets

Tp = {T1, ,Tr ; Ti pp , TiTTi = I , i = 1, .., r} M = {c1, , cs ; ci , ci > 0 , i = 1, , s} Ep = {e1, , ep ; standard unit vectors in p} can be defined for the controllability covariance matrix with r, the number of matrices for perturbation directions, s, the number of different perturbation sizes for each direction, and p, the number of inputs to the system

Definition 1: Controllability covariance matrix Let Tp, Ep and M be given sets as described above. The controllability covariance matrix is defined by

= = =

=p

i

r

l

s

m

ilm

m

C dttrsc

W1 1 1 0

2 )(1

(4)

where ilm(t) nn is given by ilm(t) = (xilm(t)-xssilm)( xilm(t)-xssilm)T, and xilm(t) is the state of the nonlinear system corresponding to the input u(t) = cmTleiv(t)+uss(0).

The controllability covariance matrix is computed from data along selected system trajectories. The input to the system u(t) is defined as above, where cm describes the input size, TleI determines the input direction, v(t) denotes the nature of the input, and uss(0) refers to the input at the original steady state. The quantities xssilm(t) represent a desired system trajectory which is dependent upon v(t). If v(t) is piecewise constant as is the case for impulse and step inputs then xssilm(t) will also be piecewise constant. The nature of the input should be chosen in such a way that is consistent with typical input behavior of the plant. Based upon the nature of v(t), controllability covariance matrices can be put into one of the following categories.

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1. Impulse input: v(t) = (t) The desired state trajectories xssilm for this special case are constant over time and equal to the steady state value of the system states, xss. The controllability covariance matrix for impulse inputs for systems that are stable in the sense of Lyapunov is equal to the empirical controllability gramian as defined by Lall et al. [5, 6]. Additionally, if the system under investigation is linear, then the trajectories of the states are given by

( ) ilAtssilmAtssilm eBTexxextx ++= )0()( , (5) and it can be shown that the controllability covariance matrix reduces to the

controllability gramian for this special case. For a system at steady state before it is excited by impulse inputs, xilm(0) is equal to xss resulting in

( )( ) tATTlTiilAtmTilAtilAtmilm TeBTeeBTeceBTeeBTect 22)( == . (6) When this is substituted into the definition of the controllability covariance matrix it reduces to the definition the controllability gramian of a linear system.

linearCtATAt

r

l

s

m

p

i

tATTl

Tiil

Atm

m

C WdteBBedteBTeeBTecrsc

WTT

,01 1 1

02

21

===

= = =

(7)

Due to the fact that general nonlinear operations are not defined for impulses, the empirical controllability gramian can only be computed for control-affine systems. Since

the controllability covariance matrix computed from trajectories generated by impulse inputs for systems that are stable in the sense of Lyapunov is equal to the empirical

controllability gramian, which again reduces to the linear controllability gramian for linear systems, it is guaranteed that the covariance matrix is independent of the input size

cm.

2. Step input: v(t) = S(t) For this nature of input the system trajectory, xssilm(t), is given by the new steady state that the system will reach after the step input has been applied. For some models, step inputs will be able to capture more of the systems nonlinearities than impulse inputs. Also, step

inputs will generally cover a larger operating region for the same value of cm than

impulses will. Trajectories involving step inputs can be computed for any kind of stable

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nonlinear system from (4) and do not require Lyapunov stability. It is important to note that it is possible to compute controllability covariance matrices using step inputs for

systems that exhibit steady state multiplicity, even if the step input changes move the states from the region of attraction of one stable operating point to another stable

operating point outside of this region. However, special attention should be given to the interpretation of the results for cases where the system leaves the region of attraction of a

stable operating point. The proof that the controllability covariance matrix is independent of the input magnitude cm for linear systems is as follows.

The covariance matrix for a system described by equation (3) is ilm(t) = (xilm(t)-xssilm)( xilm(t)-xssilm)T (8)

For linear systems with step inputs of the form u(t) = cmTleiS(t)+uss the values of xilm(t) and their new steady state values are given by

( ) ++= t ilmtAssilmAtssilm deTBcexxextx 0 )()0()( , ssilmilmss xeTBcAx += 1 . (9) It is important to note that for this type of input the system trajectory, xssilm, is equal to a constant and is time-invariant. For a system at its steady state before it is excited by step

inputs, xilm(0) is equal to xss resulting in T

il

ttA

il

ttA

m

ilm eBTABeeBTABect

+

+=

1il

0

)(1il

0

)(2 deTdeT)( . (10)

The controllability covariance matrix that results from substituting (10) into (8) is independent of the step size cm to result in

( )( ) = =

==0

1

1 1 0

111 dtAeBBeAdteBTeAeBTeAr

W TtATAtp

i

r

l

Til

Atil

AtC

T .

(11)

3. A series of steps in v(t). The derived system trajectory xssilm produced by a series of step changes in the input will consist of a series of steps as well. This type of input includes steps as described above as

well as rectangular pulse inputs, which can be viewed as a series of two steps of equal size of different sign. The size of the steps as well as their occurrence can be varied. This

input type can be applied to the computation of the controllability covariance matrix for

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any stable nonlinear system and does not require Lyapunov stability. However, as for the case of a single step, special attention should be paid to the interpretation of the

information contained in covariance matrices computed for systems that leave the region of attraction of the original operating point. Also, it is important to keep in mind that even

these input sequences will have to be scaled by a factor of cm. The series of steps is given by

=

=

q

k

stepkk ttStv

1)()( , 11 k

(12)

where tkstep is the time when the k-th step change occurs, k is the size and direction of this step relative to the largest step in the sequence and q is the number of step changes in the sequence. Using this input, the state vector xilm(t) and its desired trajectory xssilm can be computed as

( ) =

++=t q

k

stepkkilm

tAss

ilmAtss

ilm dtSeTBcexxextx0

1

)( )()0()( (13)

ss

q

k

stepkkilm

ilmss xttSeTBcAx +=

=

1

1 )( (14)

for linear time invariant systems in deviation variables. It is now possible to show that the

controllability covariance matrix is independent of the input magnitude cm for linear systems under the assumption of xilm(0) equal to xss. In particular,

+=

=

=

q

k

stepkkil

t q

k

stepkkil

tAm

ilm ttSeBTAdtSeBTect1

1

01

)(2 )()()( ,

Tq

k

stepkkil

t q

k

stepkkil

tA ttSeBTAdtSeBTe

+

=

=

1

1

01

)( )()(

(15)

= =

=

=

+=

p

i

r

l

q

k

stepkkil

t q

k

stepkkil

tAC ttSeBTAdtSeBTe

rW

1 1 0 1

1

01

)( )()(1

dtttSeBTAdtSeBTeTq

k

stepkkil

t q

k

stepkkil

tA

+

=

=

1

1

01

)( )()( .

(16)

For q=1, t1step=0, and 1=1 this reduces to equations (10) and (11). The initial states of the system have been chosen equal to zero because only the influence of the input on the system should be reflected in the controllability covariance matrix.

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It is important to note that all of the chosen inputs will result in covariance

matrices that are independent of the input magnitude cm for a linear plant. However, this is usually not the case for nonlinear models and the changes in the covariance matrix for different input magnitudes provides a good indicator for the nonlinearity of a system.

2.2 Observability covariance matrix In order to define the observability covariance matrix the quantities

Tn = {T1, ,Tr ; Ti nn , TiTTi = I , i = 1, .., r} , M = {c1, , cs ; ci , ci > 0 , i = 1, , s} , and En = {e1, , en ; standard unit vectors in n} are required with r being number of matrices for perturbation directions, s the number of different perturbation sizes for each direction, and n the number of states of the full-order system.

Definition 2: Observability covariance matrix Let Tn, En and M be given sets as described above. The observability covariance matrix is defined by

= =

=r

l

s

m

Tl

lml

m

O dtTtTrsc

W1 1 0

2 )(1

(17)

where lm(t) nn is given by lmij(t) = (yilm(t)-yssilm)T( yjlm(t)-yssjlm), yilm(t) is the output of the system corresponding to the initial condition x(0) = cmTlei+xss, and yssilm is the steady state of the output that the system will reach after this perturbation.

The yilm(t) are the output measurements of the system and the yssilm represent the steady state of the output measurements for the initial condition x(0) = cmTlei+xss. If the initial condition x(0) is within the region of attraction of the operating point xss then the observability covariance matrix reduces to the empirical observability gramian. If the system under investigation is linear and stable, any initial condition will remain within the region of attraction of xss and the observability covariance matrix is identical to the

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empirical observability gramian, which again reduces to the observability gramian of the linear system as is shown below.

The trajectories for the outputs of a linear system subjected to the perturbed initial condition shown in definition 2 are given by

ssilmAtilm CxeTcCety +=)( , ssilmss Cxy = (18)

which results in

( ) ( )l

AtTtATlm

lm

jlAtTtAT

lTimjl

AtTil

Atm

lmij

TCeCeTct

eTCeCeTeceTCeeTCectT

T

2

22

)()(

=

==

(19)

When these equations are substituted in the definition of the observability covariance matrix, it reduces to the observability gramian of a linear system:

linearOAtTtA

r

l

s

m

lAtTtAT

lmm

O WdteCCedtTCeCeTcrsc

WTT

,

01 1 0

22

1===

= =

(20)

Thereby, the observability covariance matrix is independent of the perturbation magnitude cm for a linear system.

It is important to point out the differences in the computation procedures for the controllability and observability covariance matrices. The controllability covariance matrix is computed from the variance covariance matrices of the states for excitations in the inputs of the system. For each change in the input variables, a variance covariance matrix is computed, multiplied by a scaling factor, and added to the already computed variance covariance matrices. After each computation step, only the variance covariance

matrix needs to be stored and the data for the trajectories can be deleted. However, for the observability covariance matrix, data for trajectories corresponding to different initial conditions need to be stored, before a variance covariance matrix can be computed. Computation of the observability covariance matrix requires that each state of the system has been perturbed at least once before a covariance matrix can be computed. While it is sufficient for a linear system to perturb each state only once, this will lead to insufficient information for nonlinear systems and more complex perturbation patterns are required.

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3. Computing balancing-like coordinate transformations Each covariance matrix contains information about a specific behavior of the system. The controllability covariance matrices reflect the input-to-state behavior of the system for a particular type of excitation. It can be concluded from this information which states, or linear combinations of states, are affected the most by changes in the inputs to the system. Likewise, the observability covariance matrix reflects the state-to-output behavior of the system and it can be determined which states, or linear combinations of states, cause the largest changes in the output of the system. Each of the covariance

matrices can be diagonalized by a linear coordinate transformation. This can be done either for a covariance matrix by determining the eigenvectors and corresponding eigenvalues of the matrix [8, 13]. However, if only one of the covariance matrices is used, it is possible that a state that is very important for the input-to-state behavior of a system would be neglected, because this same state might be unimportant for the state-to-output behavior. This situation will lead to misleading conclusions for the input-output behavior of the system. In order to avoid this, the system can be transformed, such that the minimal part of both the controllability as well as the observability covariance matrix are equal and diagonal. This means that in case of a minimal linear system each state is just as important for the input-to-state behavior as it is for the state-to-output behavior. If the gramian matrices of a linear system are used as the covariance matrices, then the system is said to be in balanced form and the transformation is called a balancing transformation. For a nonlinear system the same procedure can be applied to the covariance matrices in order to determine the states that are important for the input-output behavior.

When computing the transformation for the covariance matrices of a system, it is

not important for the algorithm if the original model was linear or nonlinear, or even what type of input was used for the controllability covariance matrix. The problem can be formulated as finding an invertible state transformation that makes two symmetric positive semi-definite matrices diagonal and equal in the states that are both controllable and observable. The proof that such a transformation exists is given by Zhou and Doyle [16].

For a system of the form given in equation (3) a linear coordinate transformation

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Txx = (21) can be found that diagonalizes the covariance matrices and balances the states that are both observable and controllable as is shown in equations (22) and (23)

=

000000000000001

ITTW TC

(22)

=

00000000000000

)(3

1

11 TWT OT

(23)

where the 1 refers to the controllable and observable subspace. The other rows and columns refer to subspace that are either unobservable, uncontrollable or both unobservable and uncontrollable. The transformed system is then given by the set of equations (24).

))(())(()())(),(())(),(()(

1

1

txhtxThtytutxftutxTTftx

==

==

&

(24)

The advantage this formulation has over other balancing algorithms is that it can deal with systems that have rank-deficient covariance matrices, resulting from systems that are not completely observable or controllable. Thereby, no assumptions about observability and controllability of the system have to be made in order to apply this method. Furthermore, it is an approach that can be used for nonlinear systems and the properties of the procedure reduce to balancing of linear systems, if the system under investigation is linear and impulse inputs are used for the computation of the covariance matrices. Future research should address the effect that balancing of covariance matrices obtained from excitations/perturbations at different frequencies will have on model reduction.

3.1. Truncation of uncontrollable/unobservable subspaces and model reduction For the linear case the states corresponding to 1 make up the minimal realization of the system and therefore the nonminimal states can be truncated without modifying the input-output behavior of the system. While it is often true for nonlinear systems that there are

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more states corresponding to 1 than are required for a true minimal realization [3], the states not contributing

to either the controllability or the observability can still be truncated without changing the input-output behavior of the system. After truncation, the system is of the form

))(())(()()0()(

))(),(())(),(()(

1

,22

11

txhtxThty

xtx

tutxfPtutxTPTftxss

==

=

==

&

(25)

P is given by [I 0] and its rank is equal to the rank of 1 for the elimination of the unobservable as well as the uncontrollable states. The new system given by equation (25) can be further reduced by eliminating states that correspond to small values along the diagonal of 1. Since the states are arranged in order of decreasing importance for the input-output behavior of the system and the value along the diagonal is an indicator for the importance of a particular state, a difference of several orders of magnitudes of these values indicates that all following states can be truncated. The procedure to truncate additional states of the model is implemented the same way minimal realization is, except that for model reduction purposes the rank of P is chosen to be smaller than the rank of 1 and the size of P should be determined from the difference of the diagonal entries of 1. For a linear system this procedure reduces to the well-known method of balanced truncation, assuming that the gramians are used as covariance matrices. The model reduction procedure can theoretically be applied to any system for which covariance matrices can be computed, including system that are stable, but not in the sense of Lyapunov stability. However, the information that is contained in the covariance matrices will not necessarily reflect the global behavior of the system. Therefore, it is recommended that the proposed procedure for model reduction only be applied if the covariance matrices are computed from data collected along system trajectories that do not leave the region of attraction of the original operating point.

3.2. Nonlinearity measures One important property of a nonlinear system is the degree of nonlinearity that it exhibits in its input-output behavior. If the nonlinearity of the model is small over the region in which the process is operated then the system can be modeled as a linear model for

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control purposes. However, if the model exhibits a large degree of nonlinearity then the process performance can benefit when a nonlinear controller is designed.

The nonlinearity in the input-output behavior of a process can be described by nonlinearities in the input-to-state and in the state-to-output behavior for state space models. Nonlinearity in the input-to-state behavior is described by differences in the dynamics of states between the nonlinear and linear model caused by the input. This will be called input nonlinearity. State-to-output nonlinearity on the other hand will be called output nonlinearity and it describes deviations in the dynamics of the outputs of the nonlinear model from the linear system caused by perturbations in the states. The nonlinearity of the input-output behavior of the system is given by both measures. The procedure presented in this section is an extension to the one developed in [2], in that covariance matrices are used instead of empirical gramians for the nonlinearity quantification. Additionally, the algorithm presented in section 3 will be used in this article compared to the one developed in [2], because it is more general and is applicable for nonlinearity quantification and model reduction. Since the nonminimal part of a system does not contribute to the input-output behavior of the model it is assumed for the rest of this section that the nonminimal states were removed.

The computation of the two nonlinearity measures requires two covariance

matrices for each measure. One is the controllability/observability covariance matrix of the linear system and the other is the corresponding covariance matrix for the nonlinear system. Since the behavior of a smooth nonlinear system can be approximated by a linear system response close to a nominal operating point, the nonlinearity for an arbitrarily small operating region is always zero. After the computation of all four matrices, the

transformation that balances the matrices of the linear system is computed by the procedure shown in section 3. After the transformation and elimination of the nonminimal states, the linear covariance matrices are identical and equal to the diagonal matrix

TLINEARO

TTTLINEARCL PTWTPPTPTWW

1,

1,

)( == . (26) The transformed model can be easily interpreted as each state of the system being

just as important for the input-to-state behavior as for the state-to-output behavior. Comparing matrices of the balanced system has the advantage that it can determine if the

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input-output behavior of the nonlinear system deviates from its behavior for the linearized model. If a state is highly nonlinear in its input-to-state behavior, but does not affect the state-to-output behavior at all, then it will not influence the input-output behavior. This will be clear for a transformed system, but may not be the case if the system is left in its original form. The balancing-like transformation is determined by the procedure in section 3 for the linear minimal system, and then the same coordinate transformation is applied to the covariance matrices of the nonlinear system according to

TTNONLINEARCNC PTPTWW ,, = (27)

TNONLINEARO

TNO PTWTPW

1,

1,

)( = (28) Now the two nonlinearity measures as defined in [2] can be computed:

Input nonlinearity measure The input nonlinearity measure is defined by

=

= =

=1

1 1

1

1 1,

),(

),(),(n

iL

n

i

n

jNCL

C

iiW

jiWjiW

(29)

where the first entry of the ordered pair of matrices indicates the row and the second entry the column. Each ordered pair refers to a specific element of the matrix.

Output nonlinearity measure The output nonlinearity measure is defined by

=

= =

=1

1 1

1

1 1,

),(

),(),(n

iL

n

i

n

jNOL

O

iiW

jiWjiW

(30)

where the first entry of the ordered pair of matrices indicates the row and the second entry the column.

Both measures will be zero for linear systems since covariance matrices are independent of perturbation or excitation size as shown earlier. However, for large excitations or

17

perturbations and strongly nonlinear models the measures will approach high values, due to the fact that the states of the nonlinear model will have covariances different from zero for a nonlinear model that is simulated in a large operating region. Both measures can at least theoretically approach infinity for extreme nonlinearities, but a value of unity will already indicate a strongly nonlinear system. This is due to the fact that a nonlinearity measure of unity indicates that the differences in the elements of the covariance matrices between the linear and the nonlinear system are just as large as the variances of the balanced linear system.

4. Properties of the proposed methods Several properties of the method are presented in this section. Specifically, the similarities between previously proposed model reduction schemes [8, 13] and the procedure introduced in this paper are pointed out. It is also shown how the method can be used for the reduction of differential algebraic equation (DAE) systems.

4.1. Similarities between different methods All methods presented in section 1.1 have in common that they compute a linear coordinate transformation

Txx = (31) from some type of covariance matrix. The main differences are the computation procedures for the covariance matrices and whether one or two different covariance matrices are being used. It was shown in section 2 that the gramians used for balancing are a special type of covariance matrix. The similarities between the different procedures will be pointed out. In order to illustrate this the covariance matrix

( )( )

=

G

TTSSSS dtdGQxtxxtxQW

0

)()( (32)

as introduced by Pallaske [13] and later used by Lffler and Marquardt [8] is presented. The symbol G denotes a set of representative trajectories resulting from a variation of initial conditions and input signals. This covariance matrix W is then computed from data resulting from different state trajectories depending on the choice of G. Pallaske [13] computed the trajectories by keeping the inputs to the system at a constant value and the

18

trajectories started from initial conditions that were perturbed around the operating point. Lffler and Marquardt [8] used both the approach of perturbed initial conditions under constant inputs as well as the case where the system starts out at steady state and the trajectories are generated by changes in the inputs to the system. Under the assumption that G contains the trajectories generated by the inputs u(t) = cmTleiv(t)+uss(0) where the parameters are the same as for equation (2), the covariance matrix of the method of Lffler and Marquardt [8] reduces to

= = =

=

p

i

r

l

s

m

T dtQQW1 1 1 0

Tilmss

ilmilmss

ilm )x-(t) x)(x-(t)(x (33)

If Q is chosen to be equal to ( ) 212 mrsc multiplied by an identity matrix of dimension n by n, where n is the number of states, then W is equal to the controllability covariance matrix and the reduction procedure is identical to the method introduced in this work when only the controllability covariance matrix is used.

The coordinate transformation T is given by a matrix that contains the normalized eigenvectors of the covariance matrix W

11 )( = TTTW

=

0001

(34)

where is a diagonal matrix with nonnegative entries that are in decreasing order along the diagonal and T-1 is a nonsingular unitary matrix. The main difference between these methods and the balancing procedure is that in balancing a second covariance matrix has to be taken into account and the state transformation is required to diagonalize both covariance matrices as is shown in equations (22) and (23). In that sense the procedure by Lffler and Marquardt [8] can be viewed as a special case of the covariance-based balancing algorithm in that it only relies on the information contained in one of the covariance matrices.

4.2. Application to differential-algebraic equation systems The proposed algorithm is not only applicable to systems of ordinary differential equations, but can be used with only minor modifications for analysis of systems

19

described by differential-algebraic equation systems. This class of systems consists of a set of differential as well as algebraic equations, which may be given by

))(()())(),((0

))(),(()(1

txhtytutxg

tutxftx

=

=

=&

=

2

1

x

xx , 0)0( xx =

(35)

in its semi-explicit form. For the purpose of this paper the system given by (35) is assumed to have index 1, the algebraic equations g(x(t),u(t)) are locally solvable for x2 and the initial conditions satisfy the algebraic equations. Controllability covariance matrices can be computed for this type of system with the same algorithm that is used for a system of ODEs. However, computing observability covariance matrices for DAE systems only results in limited information. This is due to the fact that a DAE system of index 1 has only as many degrees of freedom for choosing a consistent set of initial conditions as it has differential equations. Therefore, after all of the differential variables have been perturbed independently from one another, no new information can be gained about the system from perturbing the algebraic equations. In fact, a perturbation of the algebraic variables will only lead to an initial condition that is (locally) a linear combination of the already computed case of perturbations of the differential variables. Due to this only the controllability covariance matrix can be taken into account for this type of analysis and model reduction of DAE systems. The model reduction part of the procedure is in fact equivalent to the one by Lffler and Marquardt [8] for the case where the trajectories are computed for changes in the inputs to the system.

The coordinate transformation T can be found by a singular value decomposition of the controllability covariance matrix as is shown in equation (34). The transformed system is given by the following set of equations:

))(())(()())(),(())(),((0

))(),(())(),(()(

1

1

11

txhtxThtytutxgtutxTg

tutxftutxTTftx

==

==

==

&

(36)

For the rest of this paper this special case will not be treated in any more detail. However, it is important to note that the same procedures that will be introduced for nonlinearity

20

quantification, elimination of uncontrollable and unobservable subspaces, as well as model reduction can be applied without further modification to DAE systems of index 1 of the form of equation (35).

5. Application of the proposed framework The proposed methods can be applied for the analysis and model reduction of nonlinear processes. This is illustrated in the following case study.

Case study: Analysis and reduction of two non-isothermal CSTRs in series Consider a system of two CSTRs in series with an irreversible reaction A B. Each

reactor is modeled by a mass, component and energy balance, resulting in six nonlinear differential equations. The volume and temperature of the second reactor can be measured. The manipulated variables are the valve position at the outlet of the second reactor, as well as the heat transfer rate to the first reactor. This results in a system with

six states, two inputs and two outputs, which makes this system controllable and observable. The example is based upon a model by Henson and Seborg [4], but in this case it includes volume balances for the two reactors, a controller for the flow rate leaving the second reactor, and the reactor temperature is controlled via the heat transfer rate. These modifications make the system control-affine so that comparisons between impulse and step inputs for the computation of the controllability covariance matrices can be made. The model exhibits steady state multiplicity with three steady states (two stable and one unstable). The original operating point of the system is chosen to be the stable steady state corresponding to high conversion of reactant A. Under the given operating conditions the volumes occupied by the liquid in the two reactors are equal to 200 l and 100 l, the temperatures are 446.5 K and 453.3 K, and the concentrations of component A are 0.0357 mol/l and 0.0018 mol/l, respectively.

For this model a nonlinear controller was found to offer much improved performance over a linear controller due to the nonlinearity of the model [4]. In fact for step changes of more than 8% in the valve position at the outlet of the second reactor, the system will leave the region of attraction of the operating point and move to another

21

steady state corresponding to significantly lower conversion and temperature. While the covariance matrices can be computed for the case where the system changes operating regions, one should be careful when analyzing the information contained in the covariance matrices. It is possible to use information gained from step inputs that exceed 8% for the purpose of quantifying the degree of nonlinearity of the system. However, when the covariance matrices are used for the reduction of the nonlinear model, then only data obtained from trajectories within the region of attraction of the operating point should be used. Also, the reduced model will only give a good approximation of the behavior of the full-order model within this operating region.

Figure 1 shows the output and input nonlinearity measures of the process for impulse and step changes. It can be seen from the graphs that the input nonlinearity measure is quite different for step and impulse inputs. This is due to the fact that this system can be perturbed by much larger impulse inputs and still return to its original steady state whereas the system will leave the region of attraction of the original operating point for much lower values of cm for step inputs.

It should be noted that while the observability covariance matrix is independent of the input type for the computation of the controllability covariance matrix this is not necessarily the case for the output nonlinearity measure. This is due to the fact that the output nonlinearity measure is computed from the deviations of the observability covariance matrix after a balancing transformation has been applied and this balancing transformation does depend upon both the observability and the controllability covariance matrices. In general the difference between balancing transformations that are computed for different input types will be of lesser importance, because the balancing transformation mainly determines which directions in state space contribute the most to the system dynamics and these directions will be very similar even for different input types. Because no difference could be seen for the output nonlinearity measure when comparing impulse and step inputs for this example, only one graph for the output nonlinearity is shown in Figure 1.

As can be seen in Figure 1, the input nonlinearity measure correctly predicts that the system will become highly nonlinear for step inputs larger than 7% and operating at a

22

significantly different steady state can be classified as an extreme nonlinearity. It is important to note that different results are obtained for different input magnitudes, as can be seen in Figure 1. Since both the input as well as the output nonlinearity measure reach high values for small values of cm the system can be classified as being strongly nonlinear in both input and output. This can be verified by comparisons between the performance of linear and nonlinear controllers for a variation of this process [4].

Based upon this analysis it can be concluded that in order to perform a reduction of this model the process should be operated in the region of attraction around the given operating point. If the model should be able to represent the dynamic behavior when moving from one operating point to another one that is vastly different, then the original model should not be reduced with the proposed procedure.

The reduction is performed by the method described in section 3 and its subsections. However, two different methods are used for the computation of the covariance matrices and the results are compared. The first method computes the covariance matrices of the model for input sizes cm of 0.3, corresponding to inputs that are equal to 30% of a unit impulse, and state perturbations of magnitude 0.03. The second method generates covariance matrices for step inputs of cm equal to 0.07 and state perturbations of magnitude 0.03. These perturbation/excitation magnitudes for each case were chosen so that a similar operating region is covered by each of them. In both cases the covariance matrices are transformed by the balancing-like transformation described in section 3. Based upon the transformed matrices it can be concluded that for both systems, a reduced model with four states will capture the majority of the system dynamics. The trajectories that are produced by a step change of 7 % in the valve position at the outlet of the second reactor are shown in Figures 2 and 3. The solid line is the behavior of the real system, whereas the dashed line represents the behavior by the system that was reduced using covariance matrices computed for step inputs and the dashed-dotted line corresponds to the system that was reduced using covariance matrices computed for impulse inputs. It can be concluded that it is important to consider the input type when computing the covariance matrices, because the reduced system that is based upon covariance matrices computed for step inputs performs better than the system based upon reduction via empirical gramians. This result was expected, because the system is

23

simulated using a step change in the input, and the covariance matrices that were computed using this type of input will give a better representation of the system behavior under these conditions.

6. Discussion The method of analyzing nonlinear systems via covariance matrices as opposed to empirical gramians or gramians of the linearized system has several advantages: 1) Unlike gramians for the linearized system, covariance matrices capture some of the

nonlinear behavior of the system. Furthermore, the content of the covariance matrices allows for the nonlinear behavior to be analyzed using the nonlinearity measures presented in section 3.2.

2) Covariance matrices have an advantage over empirical gramians in that they are applicable to a variety of input types. Impulses, steps, or series of steps can be used as inputs to the nonlinear system. For impulse inputs and under the assumption that the system is operated within the region of attraction of the operating point the covariance matrices reduce to the empirical gramians as one special case. However, other input types are allowed and this makes it possible to compute covariance matrices for non-control affine system, whereas empirical gramians are restricted to control affine systems.

3) Similarities between the methods introduced by Pallaske [13] and Lffler and Marquardt [8] and the reduction method proposed in this paper have been pointed out.

Covariance matrices contain empirical gramians, which can be seen as an extension of the theory for linear systems, because when the empirical gramians are computed for a linear system, they will reduce to the gramians of the linear system. Insofar, balanced truncation for linear systems forms a special case of model reduction via balancing of covariance matrices.

The model reduction method introduced by Pallaske [13] uses only one covariance matrix for model reduction. In his work the covariance matrix is computed by perturbing the states of the system, while keeping the inputs at their constant value. This method does not take into account the importance of states for the input-output behavior

24

of the system. Instead it reduces the fast modes of the system. Nevertheless, it has been shown that a variation of Pallaskes method, where a transformation is applied to the covariance matrix, is equivalent to balanced truncation [9]. Not surprisingly this transformation is equivalent to the transformation that balances the controllability and observability covariance matrices. However, in order to determine this balancing-like transformation, covariance matrices as presented in this paper have to be computed.

The method developed by Lffler and Marquardt [8] is motivated by Pallaskes [13] method for nonlinear model reduction. Lffler and Marquardt used only one covariance matrix for the reduction as well. However, in addition to computing the covariance matrix by state perturbations as Pallaske has done, they presented a procedure that computes the covariance matrix using step changes in the inputs of the system. It can

be shown that their method is equivalent to computing the controllability covariance matrix for step inputs for the system and using the information contained in this covariance matrix for nonlinear model reduction. Since only one covariance matrix is used, this method gives information about the input-to-state behavior of the system but not about the state-to-output behavior. If it is used for reduction of nonlinear ordinary differential equations systems this can be a drawback; however, Lffler and Marquardt applied their method to the reduction of differential algebraic equation systems, where no meaningful observability covariance matrix can be computed as has been shown in section 4.2. It has been pointed out that this method is another special case of the reduction via balancing of covariance matrices as introduced in this paper.

7. Conclusions This paper presents a new approach for the analysis and reduction of nonlinear systems. The proposed method is based upon determining the nonlinearity of the input-output behavior in a first step. If the system is found to behave nearly like a linear system then it can be modeled as such. If the nonlinearity of the system is too severe to make this assumption (C,O>>0), but not too severe for model reduction then model reduction with the presented method can be performed. The nonlinearity measures are based upon the comparison of controllability and observability covariance matrices based upon data collected within the operating region

25

of the process. Large deviations between the controllability covariance matrix for a linear and a nonlinear system indicate nonlinearity in the input-to-state behavior. Differences between the observability covariance matrices for linear and nonlinear systems can be found for models that exhibit nonlinear state-to-output behavior. Both measures must be taken into account to assess the input-output behavior of a system, as it is important for most controller design problems as well as model reduction.

The model reductions itself is performed by finding a balancing-like transformation that makes the covariance matrices equal as well as diagonal in the states corresponding to the subsystem that is both controllable and observable. This transformation is then applied to the system and states that correspond to low values on the diagonal of the balanced covariance matrix as well as states that are either unobservable or uncontrollable are truncated. This method of model analysis and reduction has been illustrated with an example from the literature.

As a last result it was shown that the proposed model reduction method includes the well known method of balanced truncation for linear systems as well as Lffler and Marquardts method [8] for the reduction of ODE or DAE systems.

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Reference

[1] Hahn, J.; Edgar, T.F. An improved method for nonlinear model reduction using balancing of empirical gramians. In press Comp. Chem. Eng., 2002.

[2] Hahn, J.; Edgar, T.F. A Gramian Based Approach to Nonlinearity Quantification and Model Classification. Indust. & Eng. Chem. Research 40, 2001, 5724-5731.

[3] Hahn, J.; Edgar, T.F. A balancing approach to minimal realization and model reduction of stable nonlinear systems. Indust. & Eng. Chem. Research 41, 2002, 2204-2212.

[4] Henson, M.A.; Seborg, D.E. (Editors) Nonlinear Process Control; Prentice Hall: Englewood Cliffs, NJ, 1997.

[5] Lall, S; Marsden, J.E.; Glavaski, S. Empirical model reduction of controlled nonlinear systems, 14th IFAC World Congress, Beijing, China, 1999.

[6] Lall, S; Marsden, J.E.; Glavaski, S. A subspace approach to balanced truncation for model reduction of nonlinear control systems. Submitted to Intern. J. on Robust and Nonlinear Control, 2000.

[7] Lee, K.S.; Eom, Y.; Chung, J.W.; Choi, J.; Yang, D. A control-relevant model reduction technique for nonlinear systems. Comp. Chem. Eng. 24, 2000, 309-315.

[8] Lffler, H.-P.; Marquardt, W. Order reduction of nonlinear differential-algebraic process models. Journal of Process Control, 1991, 32-40.

[9] Marquardt, W. Nonlinear Model Reduction for Optimization Based Control of Transient Chemical Processes. Proc. CPC VI, 2001, 30-60.

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[10] Moore, B.C. Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Automatic Control 26, No. 1, 1981, 17-32

[11] Newman, A.; Krishnaprasad, P.S. Computing balanced realizations for nonlinear systems. 14th Int. Symp. Math. Theory Networks and Systems, Perpignan, France, 2000.

[12] Newman, A.; Krishnaprasad, P.S. Nonlinear model reduction for RTCVD. IEEE Proc.32nd Conference on Information Sciences and Systems, Princeton, NJ, 1998.

[13] Pallaske, U. Ein Verfahren zur Ordnungsreduktion mathematischer Prozessmodelle. Chemie-Ingenieur-Technik, 1987, 604-605.

[14] Qin, S.J.; Badgwell, T.A. An Overview of Nonlinear Model Predictive Control Applications. In F. Allgwer and A. Zheng (Editors) Nonlinear Model Predictive Control. Birkhuser, 1999.

[15] Scherpen, J.M.A., Balancing for nonlinear systems. Systems and Control Letters 21, 1993, 143-153.

[16] Zhou, K.; Doyle, J.C. Essentials of Robust Control; Prentice Hall: Englewood Cliffs, NJ, 1998.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.350

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

cm

C

&

O

Input Nonlinearity Measure (Impulse)Output Nonlinearity MeasureInput Nonlinearity Measure (Step)

Figure 1: Output nonlinearity measure and input nonlinearity measures for step and impulse inputs

29

0 5 10 15 20 25

98

100

102

104

106

108

110

112

114

116

118

Time [min]

Volu

me

[l]

Volume vs Time

FullorderReducedorder (covariance matrix)Reducedorder (empirical gramian)

Figure 2: Volume of reactor 2 vs. time

30

0 5 10 15 20 25442

444

446

448

450

452

454

Time [min]

Tem

pera

ture

[K]

Temperature vs Time

FullorderReducedorder (covariance matrix)Reducedorder (empirical gramian)

Figure 3: Temperature of reactor 2 vs. time