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Steady State Optimal Test Signal Design for Constrained Multivariable Systems
Copyright © IFAC System Identification Santa Barbara, California, USA, 2000
STEADY STATE OPTIMAL TEST SIGNAL DESIGN FOR CONSTRAINED MULTIVARIABLE SYSTEM Qiang Zhan and Christos Georgakis
Chemical Process Modeling and Control Center and Dept. of Chemical Engineering, Lehigh University, Bethlehem, PA 18015
Abstract: A new input signal design method is proposed for constrained multivariable systems that aims to maximize the signal to noise ratio while also meeting certain input and output constraints. The proposed methodology utilizes an approximate steady state model that is assumed to be known a-priori. The steady state operability concepts are used to incorporate the steady state input and output constraints into the signal design. The input test signal design problem is formulated as a nonlinear optimization problem and it is shown that all the existing methods can be unified under this proposed design framework . It will be proven that the design method is D-optimal from a steady state perspective. For ill-conditioned systems , more energy will be automatically directed in the weak directions . An example will be given to demonstrate the superiority of the method. Copyright @2000 IFAC Keywords: Identification, Input Signal Design, Multivariable Control
1. INTRODUCTION
the wide popularity of multivariable constrained control algorithms. However , this one-dimensional testing approach makes the identification experiments last weeks or even months. Furthermore, the data collected frequently contains poor information for the weak directions of the MIMO system.
Interaction between methods of identification and control is a recent topic of interest. It is well accepted that the control objective should be taken into account at the identification step to yield a "controlrelevant mode!." Once the data has been collected, one can try to minimize bias and covariance errors through the proper selection of the data prefilter and noise mode!. However, the information collected in a badly designed test cannot be improved. In the case where the signal to noise ratio is not sufficient, the identified model will not be very accurate. If a systematic design is used, as proposed in the present paper, then the input signal can be selected to have the maximum magnitude and the most appropriate directionality while also meeting steady state input and output constraints.
To increase the efficiency of the test and to shorten its length several inputs should be moved simultaneously. Furthermore, it has been shown (Andersen and Kummel, 1992a), (Andersen and Kummel, 1992b) that multi variable perturbation gives improved information on gain directionality. The use of a multivariable PRBS has also been reported for a real industrial Model Predictive Control project (Zhu , 1998). Such a multivariable signal has also been used effectively for the Tennessee Eastman process (Zhan and Georgakis , 1998). Koung and MacGregor (Koung and Macgregor , 1993) , (Koung and Macgregor, 1994) studied the open-loop excitation for MIMO system by using Singular Value
As pointed out in (Zhu, 1998), the industrial practice for process identification has remained focused on tests that change only one input at a time despite
121 3
where also here ai is the amplitude of the corresponding input signal. Since all the inputs are injected simultaneously, the test signal will be considerably shorter if the length of each sf data row remains the same. A discussion on the appropriate values of ai so that both input and output constraints are observed will be offered in a subsequent section.
consider constraints on the inputs and outputs explicitly and for this it will be referred as the Constrained Transformed Multivariable Input Signal (CTMIS) . Based on the above discussion, we can now postulate the design problem for an mxn system as the following optimization problem: maxabs(det(T)),
For the ORMIS method, the input data set can be expressed in the rotated input variables ~ as:
tij ER,
i,j = 1, · · · ,n (8)
tij
subject to: (9)
(5) Yl ~ GTs ~ Yh
(10)
Because we assume the existence of only a steady state model, the transient values will not be guaranteed to satisfy these constraints but this is the best we can do with the available preliminary process model.
where k is the single magnitude variable to be selected and ai is the singular values of the steady state gain matrix. The resulting signal in the original input variables can be expressed as :
Here s is a vector with all the possible values the PRBS can attain. For example , a 2 input system s can take the values of
(6)
In order to unify the MIS and ORMIS methods and to motivate our more general approach, we note that the signal sequences utilized in the original input variables u is a linear transformation of the vector of independent PRBS or other type of signals used .
For the case of a 3 input system, the possible values for S are eight.
3.3 Geometric Interpretation
The geometrical meaning the determinant of the linear transformation plays is to represent the volume change of a set or a region under a linear transformation. So the volume of the image T(Region) has a
(7)
For the MIS method, T is a diagonal matrix while for the ORMIS method, T is the orthogonal matrix defined in Equation (6) .
8
x-region
3.2 Proposed Approach
y=Tx ..
/l C/
Fig. 2. Linear Transformation of the Region
We can now easily generalize and claim that we need to select the matrix T that. will maximize the information contained in the input/output data. In fact, the objective used in this paper is to find the T with the maximal absolute value for its determinant. under the input/output constraints. It will be argued later that the transformation T with the largest determinant leads to the D-optimal design (Ljung, 1987).
simple formula: vol(y - region) = Idet(T)lvol(x - region) (12)
For the 2x2 system, it can be shown that all the methods are trying to maximize the area of different geometric shapes inside the int.ersection of the AIS and the DIS:
Our proposed methodology generalizes previous ones in that it considers any type of transformation T rather than only orthogonal ones as in the ORMIS method. Furthermore, the proposed methodology will
• The SIS method maximizes a rhombus wit.h diagonals parallel to the coordinate axes . • The MIS method maximizes a rectangular with sides parallel to the coordinate axes .
1215
equivalent with the maximization of det(TTT) which qualifies our design a D-optimal design (Ljung, 1987) . The D-optimal design has advantage over other design criteria because it is scaling free and the optimization problem is numerically easier.
5. IMPACT OF INPUT DESIGN ON CONTROLLER PERFORMANCE In this section, we provide a very approximate analysis of the effect the input signal design has on the control performance. By use of the SVD of the linear transformation T, we can write the parameter covariance matrix as follows :
uncorrelated with each other. We can then express ~GG - l as : ~GG - l
= EG - 1 = EVGI:;;IU6
(28)
For the ill-conditioned system, some very small singular values of G exist. Since the E matrix is assumed to have no directionality, the uncertainty directions corresponding to the big singular values in G - 1 will be greatly exaggerated. Now let's take a look at the ORMIS method proposed by Koung and MacGregor (Koung and Macgregor, 1994) . The linear transformation matrix T their method uses is simply: (29)
(22) If we express the parametric uncertainties in an explicit way. One direct realization of the identification error ~G is:
(23)
with the overall uncertainty matrix as:
['"
e2!
E=
eml
where tics:
eij
,On]
e!2
e2n
e22
. e m 2 .. '.
. e~n
(24)
where k is a constant. By noting that I:T = kI: and UT = VG, Equation 27 can be written as
c/,
(30)
This shows that the ill-conditioning of G can he compensated for the unconstrained system. The new CTMIS method does not have a closedloop form for the dependence of T on G due to the nonlinear constrained optimization used for its calculation . Because it is a generalization of the ORMIS method and all other prior methods , it numerically finds the optimal steady state signal within the input and output constraints which compensates the illconditioning effect of the system, leading to a Doptimal design.
is the stochastic variable with characteris6. EXAMPLES i:f.p
or
and
k :f. j
(25)
The following 2 by 2 high-purity distillation column model (Koung and Macgregor, 1993) is used here to illustrate the proposed design method .
(26)
G( ) =
From this equation, it can be clearly seen how the design of the transformation matrix will affect. the identification. The characteristics of the uncertainty matrix E is determined by the length of the testing data N and the noise energy Ai, not much can be done to improve it for fixed length of data.
Uj ] [ U2
E
[
11 ] [9] [-11 ] [-9] 9
'
11
'
-9
'
-11
(32)
For our approach, we assume that the constraints on both outputs are ± 1. Assume that the model uncertainty of the nominal model is 10%. In order to compare the results better, we use the exact steady state model as the nominal model.
If we first assume that the T matrix is identity, ~G
(31)
By assuming the hard limit of 11 on both inputs, Koung and Macgregor (Koung and Macgregor, 1994) designed the following input signal without explicitly considering t.he output constraints:
It has been shown that small value of 11 GG -1 - I 11 are desired for robust multivariable model based COlltrol (Koung and Macgregor , 1994) , (Lee et al., 1998) . Then
the elements in the error term
0.4 [0.505 -0.495] 1 _ 0.6z -1 0.495 -0.505
z
are completely
12 17
STEADY STATE OPTIMAL TEST SIGNAL DESIGN FOR CONSTRAINED MULTIVARIABLE SYSTEM Qiang Zhan and Christos Georgakis
Chemical Process Modeling and Control Center and Dept. of Chemical Engineering, Lehigh University, Bethlehem, PA 18015
Abstract: A new input signal design method is proposed for constrained multivariable systems that aims to maximize the signal to noise ratio while also meeting certain input and output constraints. The proposed methodology utilizes an approximate steady state model that is assumed to be known a-priori. The steady state operability concepts are used to incorporate the steady state input and output constraints into the signal design. The input test signal design problem is formulated as a nonlinear optimization problem and it is shown that all the existing methods can be unified under this proposed design framework . It will be proven that the design method is D-optimal from a steady state perspective. For ill-conditioned systems , more energy will be automatically directed in the weak directions . An example will be given to demonstrate the superiority of the method. Copyright @2000 IFAC Keywords: Identification, Input Signal Design, Multivariable Control
1. INTRODUCTION
the wide popularity of multivariable constrained control algorithms. However , this one-dimensional testing approach makes the identification experiments last weeks or even months. Furthermore, the data collected frequently contains poor information for the weak directions of the MIMO system.
Interaction between methods of identification and control is a recent topic of interest. It is well accepted that the control objective should be taken into account at the identification step to yield a "controlrelevant mode!." Once the data has been collected, one can try to minimize bias and covariance errors through the proper selection of the data prefilter and noise mode!. However, the information collected in a badly designed test cannot be improved. In the case where the signal to noise ratio is not sufficient, the identified model will not be very accurate. If a systematic design is used, as proposed in the present paper, then the input signal can be selected to have the maximum magnitude and the most appropriate directionality while also meeting steady state input and output constraints.
To increase the efficiency of the test and to shorten its length several inputs should be moved simultaneously. Furthermore, it has been shown (Andersen and Kummel, 1992a), (Andersen and Kummel, 1992b) that multi variable perturbation gives improved information on gain directionality. The use of a multivariable PRBS has also been reported for a real industrial Model Predictive Control project (Zhu , 1998). Such a multivariable signal has also been used effectively for the Tennessee Eastman process (Zhan and Georgakis , 1998). Koung and MacGregor (Koung and Macgregor , 1993) , (Koung and Macgregor, 1994) studied the open-loop excitation for MIMO system by using Singular Value
As pointed out in (Zhu, 1998), the industrial practice for process identification has remained focused on tests that change only one input at a time despite
121 3
where also here ai is the amplitude of the corresponding input signal. Since all the inputs are injected simultaneously, the test signal will be considerably shorter if the length of each sf data row remains the same. A discussion on the appropriate values of ai so that both input and output constraints are observed will be offered in a subsequent section.
consider constraints on the inputs and outputs explicitly and for this it will be referred as the Constrained Transformed Multivariable Input Signal (CTMIS) . Based on the above discussion, we can now postulate the design problem for an mxn system as the following optimization problem: maxabs(det(T)),
For the ORMIS method, the input data set can be expressed in the rotated input variables ~ as:
tij ER,
i,j = 1, · · · ,n (8)
tij
subject to: (9)
(5) Yl ~ GTs ~ Yh
(10)
Because we assume the existence of only a steady state model, the transient values will not be guaranteed to satisfy these constraints but this is the best we can do with the available preliminary process model.
where k is the single magnitude variable to be selected and ai is the singular values of the steady state gain matrix. The resulting signal in the original input variables can be expressed as :
Here s is a vector with all the possible values the PRBS can attain. For example , a 2 input system s can take the values of
(6)
In order to unify the MIS and ORMIS methods and to motivate our more general approach, we note that the signal sequences utilized in the original input variables u is a linear transformation of the vector of independent PRBS or other type of signals used .
For the case of a 3 input system, the possible values for S are eight.
3.3 Geometric Interpretation
The geometrical meaning the determinant of the linear transformation plays is to represent the volume change of a set or a region under a linear transformation. So the volume of the image T(Region) has a
(7)
For the MIS method, T is a diagonal matrix while for the ORMIS method, T is the orthogonal matrix defined in Equation (6) .
8
x-region
3.2 Proposed Approach
y=Tx ..
/l C/
Fig. 2. Linear Transformation of the Region
We can now easily generalize and claim that we need to select the matrix T that. will maximize the information contained in the input/output data. In fact, the objective used in this paper is to find the T with the maximal absolute value for its determinant. under the input/output constraints. It will be argued later that the transformation T with the largest determinant leads to the D-optimal design (Ljung, 1987).
simple formula: vol(y - region) = Idet(T)lvol(x - region) (12)
For the 2x2 system, it can be shown that all the methods are trying to maximize the area of different geometric shapes inside the int.ersection of the AIS and the DIS:
Our proposed methodology generalizes previous ones in that it considers any type of transformation T rather than only orthogonal ones as in the ORMIS method. Furthermore, the proposed methodology will
• The SIS method maximizes a rhombus wit.h diagonals parallel to the coordinate axes . • The MIS method maximizes a rectangular with sides parallel to the coordinate axes .
1215
equivalent with the maximization of det(TTT) which qualifies our design a D-optimal design (Ljung, 1987) . The D-optimal design has advantage over other design criteria because it is scaling free and the optimization problem is numerically easier.
5. IMPACT OF INPUT DESIGN ON CONTROLLER PERFORMANCE In this section, we provide a very approximate analysis of the effect the input signal design has on the control performance. By use of the SVD of the linear transformation T, we can write the parameter covariance matrix as follows :
uncorrelated with each other. We can then express ~GG - l as : ~GG - l
= EG - 1 = EVGI:;;IU6
(28)
For the ill-conditioned system, some very small singular values of G exist. Since the E matrix is assumed to have no directionality, the uncertainty directions corresponding to the big singular values in G - 1 will be greatly exaggerated. Now let's take a look at the ORMIS method proposed by Koung and MacGregor (Koung and Macgregor, 1994) . The linear transformation matrix T their method uses is simply: (29)
(22) If we express the parametric uncertainties in an explicit way. One direct realization of the identification error ~G is:
(23)
with the overall uncertainty matrix as:
['"
e2!
E=
eml
where tics:
eij
,On]
e!2
e2n
e22
. e m 2 .. '.
. e~n
(24)
where k is a constant. By noting that I:T = kI: and UT = VG, Equation 27 can be written as
c/,
(30)
This shows that the ill-conditioning of G can he compensated for the unconstrained system. The new CTMIS method does not have a closedloop form for the dependence of T on G due to the nonlinear constrained optimization used for its calculation . Because it is a generalization of the ORMIS method and all other prior methods , it numerically finds the optimal steady state signal within the input and output constraints which compensates the illconditioning effect of the system, leading to a Doptimal design.
is the stochastic variable with characteris6. EXAMPLES i:f.p
or
and
k :f. j
(25)
The following 2 by 2 high-purity distillation column model (Koung and Macgregor, 1993) is used here to illustrate the proposed design method .
(26)
G( ) =
From this equation, it can be clearly seen how the design of the transformation matrix will affect. the identification. The characteristics of the uncertainty matrix E is determined by the length of the testing data N and the noise energy Ai, not much can be done to improve it for fixed length of data.
Uj ] [ U2
E
[
11 ] [9] [-11 ] [-9] 9
'
11
'
-9
'
-11
(32)
For our approach, we assume that the constraints on both outputs are ± 1. Assume that the model uncertainty of the nominal model is 10%. In order to compare the results better, we use the exact steady state model as the nominal model.
If we first assume that the T matrix is identity, ~G
(31)
By assuming the hard limit of 11 on both inputs, Koung and Macgregor (Koung and Macgregor, 1994) designed the following input signal without explicitly considering t.he output constraints:
It has been shown that small value of 11 GG -1 - I 11 are desired for robust multivariable model based COlltrol (Koung and Macgregor , 1994) , (Lee et al., 1998) . Then
the elements in the error term
0.4 [0.505 -0.495] 1 _ 0.6z -1 0.495 -0.505
z
are completely
12 17