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The State Space and Transfer Function Approaches in Practical Linear Multivariable Systems Design*
'IHE STATE SPACE AND TRANSFER FUNCTION APPROACHES
IN PRACTICAL LINEAR MULTIVARIABLE SYSW
Isaac Horowitz Professor Department of Applied Mathematics The Weizmann I n s t i t u t e of Science Rehovot, Israel and Department of Electrical Engineering University of Colorado Boulder, Colorado
U r i Shaked Department of Applied Mathematics The Weizmann I n s t i t u t e of Science Rehovot, Israel
ABSTRACT A great deal of attention has been paid recently t o the application of s t a t e space formalism t o linear m l t i v a r i a b l e time-invariant system design. I t i s shown that t h i s approach is inherently very limited for practical use and has led t o some naive results. The design based on s t a t e space formulation does not cope with the whole problem of practical design and i t s results are more easily derived by transform methods. There i s then much greater awareness of practical constraints and therefore it is easier t o avoid the unrealistic conclusions often reached by s t a t e space approach. Transfer function f o m l i s m i s then used t o present a procedure f o r s t a bilizing a miltivariable feedback system with significant plant uncertainty. The method i s t o shape the nominal loop functions and decrease t h e i r magnitudes with frequency as rapidly as possible such t h a t the system i s stable over the range of plant uncertainty. The procedure is t o handle them one by one such t h a t each stage i s basically a single loop shaping problem. A demonstrative example is given.
IrnOWCT ION Much emphasis has been placed i n the l a s t decade on the s t a t e space approach t o the design of linear nultivariable time-invariant systems. Starting from a s e t of differential equations which describes the plant, there have been published m y papers dealing with the following main topics: +(I-5) ( i ) Pole ~ssignrnent (6-11) ( i i ) Eigenvalue and Eigenvector Sensitivity (12-16) (iii) Decoupling or Non-interaction Theory Many of these papers used s t a t e space formalism t o develop useful concepts and system c r i t e r i a such as controllability, observability, and inverse system. However, when s t a t e space formulation i s retained for actual synthesis, the insight into the real problems of a practical system design i s l o s t .
DESIGN*
Practical systems usually suffer from a p a r t i a l ignorance of the real values of the plant parameters o r from slow parameter variations. To reduce the effect of these on the response, a feedback loop i s formed. However, i n practice the system i s always suffering from noise effects t h a t enter into the feedback loop through the sensors and transducers. The noise effects, which are proportional, a t h' h frequencies, t o the feedback compensator gain (lf7, force the designer t o reduce the feedback gain a t high fre, uencies, as f a s t . a s possible, taking into account h e system s t a b l l l t y . I t i s the intention of t h i s paper t o demonstrate that many of the design techniques based on s t a t e space formalism have failed t o deal with the real problems on practical system design. The papers published on these techniques e i t h e r used methods that are applicable only for ideal systems (with no ignorance and noise), or, even when aimed t o cope with changes of the plant parameters, they treated e i t h e r unrealistic d i f f e r e n t i a l changes or wrong sensitivity c r i t e r i a . STATE SPACE FEEDBACK A plant is described by the p a i r of vector equations
where u and y are m dimensional input and output of the plant, x i s the n dimensional s t a t e vector. A. B and C are constant matrices of proper dimensions. Two different categories of s t a t e feedback are generally used. (i)
State feedback of the type u=Kx+Hw
where w i s the m dimensional input t o the whole system and K, H are constant matrices of suitable dimensions. Using (3), it i s assumed that a l l s t a t e s are available f o r measurement, and these are
Superior numbers r e f e r t o similarly-numbered references a t the end of t h i s paper.
*
(3
This research was p a r t i a l l y supported by NASA under Grant NGR 06-003-083 and by the National Science Foundation under Research Grant GK33485.
used to obtain arbitrary pole placemzpt (l), desired eigensensit'vity to plant variation( or decoupling(12f. It is hardly conceivable that there is access to all states, but even if they are available it is impractical to spend much effort on so many transducers when it can be done with only a few at the real outputs of the system. Resistors and capacitors are much cheaper than transducers. The application of (3) to the plant leads to a very special type of two degrees of freedom structure where the matrix H acts as the prefilter to the system. As H is a constant matrix, it does not introduce new dynamics to the system and therefore the system is actually of the one degree of freedom type. For example, in the case of a single input single out u controllable system, one can show, following818)
of eqns. (1-2) with the feedback law (S), then
where T (s) , N(s) , D(s) and ki, same as in Theorem 1 and
Proof Assuming the system is already given in its companion form, where B~ is (0,O.. .l) , the last row of A is (-ao,-a,. .-an-1) and C=(cD,. .cn-,), then -
.
.
C I S -BKA --'BH=c[~,s.-.sn-l]
1.
T
THEOREM 1
i=l,n-1, are the
=
H
det [Is-A-
where T(s) is the system transfer function, N(s) and D(s) are the numerator and denominator of the plant transfer function, ki, i=l,n-1 are linear combinations of the elements of K and H is the scalar gain used in (3). It is therefore seen that in spite of having two degrees of freedom H and K, the burden of pole assignment, sensitivity or (in multivariable systems), decoupling should lie on K only. Moreover, there is no control over the zeros of T (s)
.
Methods that used this type of feedback for design did not take into account the noise effect. The design, which fixes K according to other criteria, may end, especially for the pole assignment problem, with a very high gain of one or more of the components of K This implies that some of the feedback loops may introduce into the system a fantastic amount of noise.
.
(ii) State feedback of the type
*ere ~ ( ~ ) = d ~ u / dB. t ~,,i-0,q-1 H and K iire constant hatrices of suitable dimensions. This feedback is a lied when some of the states In such a case (3) can no are inaccessible longer be used, as the ;election of ki is now constrained and therefore it is impossible to realize an arbitrary allocation for the poles.
(-?f.
The application of (5) is an improvement but basically the synthesis is still of the one degree of freedom type. THEOREM 2 Given the single input output controllable system
Theorem 2 shows that as V(s) is dictated by the pole allocation that is required, there is still no control over the system zeros. EIGENVALUES AND EIGENVECMR SENSITIVITY Much has been written about eigenvalue and eigenvector sensitivity. Expressions were derived using state space formalism for the differential eigenvalue sensitivity (which corresponds to the sensitivity of the closed loop poles due to differential changes in parameters of the system), and for differential eigenvector sensitivity (which corresponds to the sensitivity of the residues of the closed loop poles). These analytical expressions are based on the assumption that the eigenvalues are distinct (7-11). The attempts to use these expressions for sensitivity design (6,7, applied the state feedback (3) which, as mentioned, reduces design freedom. The designers examined the K row of eq. (3) and tried to find those ki which will fix relatively insensitive closed loop poles. There are several objections to design for eigensensitivity reduction in general and to the methods used to accomplish it in particular. (i) In practice, any linear time-invariantplant representation is an approximation of a nonlinear system and therefore inherently has a great deal of associated uncertainty. A design based on differential sensitivity of any kind (and not necessarily of eigenvalues and eigenvectors) is therefore impractical as it cannot guarantee a small
sensitivity of the system response for the practical significant uncertainty or changes of plant parameters. For example, if the m i n k of the sensitivity function is very narrow, this approach may lead to a design which is much inferior to one that takes the large scale changes of the parameters into account. This last has higher differential sensitivity but allocates the closed loop poles in such a region that the sensitivity to significant changes of parameters will be minimal. If one insists on dealing with differential sensithe transfer functivity, it is much easier to that for a given tion approach. It is known n input n output plant whose matrix of transfer functions is P(s)
(*3
where Po is the nominal P , L and T, are the nominal open-loop and closed-loop matrices of transfer functions and AT the transfer function matrix of the changes in the closed-loop transfer functions. Denoting
No example has been given in the literature where numeric specificath in the system output response have been satisfied using eigensensitivity. DECOUPLING OR NON-INI'ERAGTION THEORY Decoupling theory deals with the possibility of compensating a given multivariable system in such a way that each of the system outputs can be independently controlled by a corresponding system input. Many criteria, sufficient and necessary for the existence of a decoupling transformation in a multivariable system represented b tate space formalism have been published (12-Y5f algorithms were proposed for achieving it(
~7 .
Instead of dealing with combinations of high dimensional state space ma es and raising these to a relatively high power , one may easily use the transfer function approach.
If35
Definition(13) A system is decoupled if its matrix of transfer
it is easily seen that, if AP is'brnal1"enough (not necessarily differential)
functions is diagonal and nonsingular.
THEOREM 3
By means of (9) it is possible to choose the elements of Lo taking into account not only the permitted boundaries on AT but also stability and noise effects reduction.
Given an n input n output plant that is represented by a square n dimensional matrix of transfer functions P. The plant can be decoupled if and only if P is nonsingular. Proof
(fi) The above objection to differential sensitivity in general is even stronger in the case of eigensensitivity. Many different pole-zero configurations lead to very similar time responses. For example, two configurations of 3 poles and one zero leading to system transfer function of the type
are considered. The first one has ~=0.9,z=1.2 and p=l , where the second has p=l.Z, z=0.4 and p=0.5. The step responses of the two systems are given in Fig. 1. (iii) Even when it is agreed to design for eigensensitivity, the expression one gets for the partial derivatives of the system time response to a given input with respect to one of the plant parameters is a very complicated combination of all eigenvalues and eigenvector sensitivities, with weights that depend on the final choice of the system poles and zeros (8$11). Therefore it becomes essentially impossible to use in a design as it is not a riori known how to divide the burden among a t e eigenvalues and eigenvectors.
--?hi--.
(iv) It is assumed that all states may be sensed and that the eigenvalues are distinct.
Using the two degrees of freedom structure it is known that:
where T , M and G are n dimensional matrices of transfer functions of the whole system, the feedback compe s tor and the prefilter (respectively)?l7). If P is nonsingular, then for any selection of M for which [I+PM]-' is still defined,a matrix G can be found so that T d l be a diagonal nonsingular matrix. On the other hand, if P is singular, T must be singular too. One of the attractive features of decoupling theory is that at first sight it looks as if by decoupling an n input n output system, the whole design problem reduces to n simple single output problems. The truth is that when a complete knowledge and no variation of plant parameters are obtained, the general case is not much of a problem either. It is not even necessary to use feedback, the prefilter is designed to realize the desired system transfer function and the design is complete. has already been stated, it is impractical to deal with any aspect of system design without taking system sensitivity to uncertainty of parameters into account. This is even more important in the case of non-interacting control, as it can be shown that a system may be decoupled for nominal
As
values of its parameters, where a slight change of these may lead to a highly interacting system (especially in the case where P is nearly singular)
.
From the above it is seen that though attempts have been made to deal with some of the main problems in the design of control systems, the results are far from being satisfactory. The designer is delving into the mathematics of a model that describes a special limited aspect of the design only, without looking at the whole picture. The real problem is to cope simultaneously with realization of the system time response within prescribed boundaries in the time domain, in spite of significant ignorance of the plant parameters, system stability within prescribed margins, maximal reduction of noise effect at the plant input and attenuation of possible disturbances acting at different points of the system. This problem for multivariable systems is difficult enough even with the transfer function approach, where the designer deals with n dimensional matrices (when n is the number of inputs or outputs). If one tries to solve the same problem by the state space approach, it will require matrix calculations of many times higher dimensions. It is true that in the case of an ideal system, in which no ignorance of plant parameters exists, the system can be reduced to one of the canonical forms and then it is not much of a problem to handle the big matrices; however, when uncertainty or change in plant parameters is encountered, the transfonnations which lead to the nominal representation will no longer do so. The transfer function approach is far more efficient in dealing with the real problem. Recently a method has been proposed for ractical single input single output system desigu (18). The method uses templates that describe in the Nichols Chart the uncertainty of the plant transfer function. A minimal open loop gain is found, taking into account time domain boundaries, stability, noise effects and disturbance attenuation. In the multivariable case it has not been found possible (in contrast to the single input single output case) to satisfy specified tolerances to large parameter ignorance (in the low frequency range), and to achieve loop shaping for stability over the range of parameter variations (in the high frequency range), together. Therefore two distinct procedures are involved. The loop shaping for stability despite parameter variations is especially difficult. Consider an n input n output feedback system in which there is embedded a plant whose n x n matrix of transfer function P(s) has any number of parameters knmn only to lie within specified ranges which may be very large. Denoting the nominal plant, nominal open loop, nominal systems and the real system n x n matrices of transfer functions by Po, Lo, To and T (respectively), it is known that
and that the matrix Lo may be chosen diagonal
Bf '/f s. of design with no loss of
for all as generality
If the elements of P are stable, then the nominal loop functions ci mustbe shaped so that the determinant A = det [Lo+P,P-'1
has no right half plane zeros over the range of plant parameters. Until now the technique has been to divide the frequency spectrum into n-1 distinct regions, and it was then possible to deal in each region ge with two and rarely with three dominant II. 'tf 77l1y Nevertheless, shaping of the loops still iivolves tedious work because of the necessity of simultaneously shaping more than one loop function over the same frequency region.
.
A different method, in which there is a need to shape one loop function at a time only, is now proposed. The technique is first illustrated for n=2. The determinant becomes
where Aij
are the elements of the matrix Ah
A =
=
det A
and f
= -
X,,Rl+AX
Ll A l l ' The low frequency values of L-(jw) are chosen to obtain the benefits of feedback, i.e., acceptable sensitivity to plant parameter ignorance and/or disturbance attenuation. With no loss of generaliv, assume that this results in 1111 R, at the upper edge of the low frequency region. The first step is then to shape Rl(jw) so that 111 + A, is stable over the entire range of A,., values a d l that f, values are large enough in magnitude with maximal phase lag. In this way II,may obtain larger phase lag, and therefore decrease faster, without encircling or intersecting the f,. regions of the corresponding frequencies. The choice for i1 should therefore be such that it is not surrounding or cutting the -Al, regions, and that L,+hll values are as small as possible with minimal phase lag. +
I
In the second step'the f, regions are found and is shaped to "sneak" through these without en::rcling or intersecting them. In the general nth order case assume without loss for i=l,n-1 The of generality that leil
I
.
A second order case, d@jlj has already been solved
, is
by the previous method
taken.
The A elemen* are:A, ks/s+a, h =ks/s+b and ni=ck2s2/(s+a) (s+b) &e values k and c vary independently from 0.5 to 1 and those of a and b from 1 to 2 The low frequency requirements on g, and R, are as follows: the mgritudeof!?., must be 100 at 0 rps, 10 at 2 rps, while that of 11, must be 5 at 0 rps, and 1.5 at 1 rps.
.
OF
.
in this case 12 I a I , the regions of -A,, are first considere& This is done in Fig. 2 for 5 different frequencies. R, is then shaped so that will be stable over the entire range of kiues and that it will acquire as high phase lead as possible, especially in the high frequency region where k, is supposed to "sneak" through the resulting regions of f,= (-x,~R,+A~)/(R,+A,,)
As
.
R,
basically a sin e loop shaping problem. In each step, say the i , it is possible to increase the bandwidth of ti , to some extent, in order to economize on R+ which has higher bindwidth. This transparen=$ is useful to permit trade-off between the ti in proportion to the individual sensor noise strengths. The advantage of the present technique over the foner method is demonstrated in an example of a two input two output system.
! k .
Example
has been chosen to be
REFERENCES (1) Brockett, R. W., "Poles Zeroes and Feedback: State Space Interpretation", IEEE Trans. Automatic Control Vol. AC-10, pp. 129-135, April 1965. (2) Wonham, W. M., "On Pole Assignment in MultiInput Controllable Linear Systems", IEEE Trans. Automatic Control Vol. AC-12, pp. 660665, December 1967. (3) Pearson, J. B. and C. Y. Ding, "Compensator Design for Multivariable Linear Systems", IEEE Trans. Automatic Control Vol. AC-14, pp. 130134, April 1969.
and is shown in Fig. 2.
(4)
Next, the resulting boundaries of f, are found and shown in Fig. 3. The choice for L?, prevents the regions of f, for frequencies of 5 rps and more, from sliding toward the smaller phase lag region and so enables I R, I . to decrease by 10 db per octave, still guaranteeing large stability margins.
(5) Paul, C. A., "Pole Specifications in Decoupled System", Int. J. Control Vol. 15, No. 4, pp. 651-664, 1972.
Rl
, which
is also shown in Fig. 3 is
It is of interest to compare the present design which has been proved, by a computer solution, to be stable, with the former design. Fig. 4 shows the Bode plot of both designs for R, and demonstrates the superiority of the present method. S W Y AND a3NCLUSIONS The ability of state space formalism to cope with practical system design is considered. It is shown that although this formulation introduces useful analytic concepts, it fails to deal with the whole problem of system design. Concepts of design, such as pole assignment, decoupling and eigensensitivits are treated separately without taking other factors such as noise effects and significant parameters ignorance into account. It is shown that the state space formulation artificially increases the problem complexity by introducing many redundant variables that increase the dimension of the matrices without actually improving the design. It is also demonstrated that the inadequate results found by this formalism can be easily obtained using the transfer function approach. Confined to the latter, a new method for stability design, in the pres ence of significant plant parameter ignorance, is introduced. The method handles the nominal loop functions Ili one by one such that each step is
Howze, J. W. and J. B. Pearson, "Decoupling and Arbitrary Pole Placement in Linear Systems Using Output Feedback", IEEE Trans. Automatic Control Vol. AC-15, pp. 660-663,December 1970.
(6) Morgan, Jr., B. S., "The Synthesis of Linear Multlvariable Systems by State Variable Feedback," IEEE Trans. Automatic Control Vol. AC-9 pp. 405-411, October 1964. (7) Morgan, Jr., B. S., "Sensitivity Analysis and Synthesis of Multivariable Systems", IEEE Trans. Automatic Control Vol. AC-11, pp. 506512, July 1966. (8) Reddy, D. C., "Parameter Sensitivity of Linear Time Invariant, Dynamic Systems", Report R-315, Coordinated Science Laboratory, University of Illinois, Urbana, Illinois, September 1966.
(9) Mantey, P. E., "Eigenvalue Sensitivity and State-Variable Selection", IEEE Trans. Automatic Control, Vol. AC-13, pp. 263-209, June 1968. (10) Reddy, D. C., "Eigenfunction Sensitivity and Parameter Variation Problem", Int. J. Control, Vol. 9, No. 5, pp. 561-568, 1969. (11) Crossley, T. R. and B. Porter, "Eigenvalue and Eigenvector Sensitivities in Linear Systems Theory", Int. J. Control, Vol. 10, No. 2, pp. 163-170, 1969.
Fdb, Peter L. and W. A. hblovich, "Decoupling in the Design and Synthesis of Multivariable Control Systems", IEEE Trans. Automatic Control, Vol. AC-12, pp. 651-659, December 1967. Gilbert, E. 0. "The Decoupling of Multivariable Systems by State Feedback", SIAM J. Control, Vol. 7, pp. 50-64, 1969. McLane, P. J. and E. J. Davison, "Disturbance Localization and Decoupling in Stationary Linear Multivariable Systems", IEEE Trans. Automatic Control, Vol. AC-15, pp. 133-134, February 1970. brse, A. S. and W. M. Wonham, "Status of Noninteracting Control", IEEE Trans. Automatic Control, Vol. AC-16, pp. 568-581, December 1971. Gilbert, E. G. and J. R. Pionidmy, "A Computer Program for the Synthesis of Decoupled Multivariable Feedback Systems", IEEE Trans. Automatic Control, Vol. AC-14, pp. 652-659, December 1969. Horowitz, I., Synthesis of Feedback Systems, Academic Press, 1963, (31. 10.
FIG. 2:
Shaping
of
R,
for the stability
(112+A22)
Ding, C. Y., F. N. Brasch, Jr. and J. B. Pearson, "On Multivariable Systems", IEEE Trans. Automatic Control, Vol. AC-15, pp. 9697, February 1970. Sidi, M. and I. Horowitz, "Synthesis of Feedback Systems with Large Plant Ignorance for Prescribed Time-Domain Tolerances", to appear in Int. J. of Control.
Time in seconds
FIG. 1:
Two responses which are essentially similar
F I G . 3:
for the stability
1
30
-
former design
a 1
(d
z
-10-20
-
-30
-
FIG. 4 :
Bode p l o t o f t h e two d e s i g n s f o r R, -
IN PRACTICAL LINEAR MULTIVARIABLE SYSW
Isaac Horowitz Professor Department of Applied Mathematics The Weizmann I n s t i t u t e of Science Rehovot, Israel and Department of Electrical Engineering University of Colorado Boulder, Colorado
U r i Shaked Department of Applied Mathematics The Weizmann I n s t i t u t e of Science Rehovot, Israel
ABSTRACT A great deal of attention has been paid recently t o the application of s t a t e space formalism t o linear m l t i v a r i a b l e time-invariant system design. I t i s shown that t h i s approach is inherently very limited for practical use and has led t o some naive results. The design based on s t a t e space formulation does not cope with the whole problem of practical design and i t s results are more easily derived by transform methods. There i s then much greater awareness of practical constraints and therefore it is easier t o avoid the unrealistic conclusions often reached by s t a t e space approach. Transfer function f o m l i s m i s then used t o present a procedure f o r s t a bilizing a miltivariable feedback system with significant plant uncertainty. The method i s t o shape the nominal loop functions and decrease t h e i r magnitudes with frequency as rapidly as possible such t h a t the system i s stable over the range of plant uncertainty. The procedure is t o handle them one by one such t h a t each stage i s basically a single loop shaping problem. A demonstrative example is given.
IrnOWCT ION Much emphasis has been placed i n the l a s t decade on the s t a t e space approach t o the design of linear nultivariable time-invariant systems. Starting from a s e t of differential equations which describes the plant, there have been published m y papers dealing with the following main topics: +(I-5) ( i ) Pole ~ssignrnent (6-11) ( i i ) Eigenvalue and Eigenvector Sensitivity (12-16) (iii) Decoupling or Non-interaction Theory Many of these papers used s t a t e space formalism t o develop useful concepts and system c r i t e r i a such as controllability, observability, and inverse system. However, when s t a t e space formulation i s retained for actual synthesis, the insight into the real problems of a practical system design i s l o s t .
DESIGN*
Practical systems usually suffer from a p a r t i a l ignorance of the real values of the plant parameters o r from slow parameter variations. To reduce the effect of these on the response, a feedback loop i s formed. However, i n practice the system i s always suffering from noise effects t h a t enter into the feedback loop through the sensors and transducers. The noise effects, which are proportional, a t h' h frequencies, t o the feedback compensator gain (lf7, force the designer t o reduce the feedback gain a t high fre, uencies, as f a s t . a s possible, taking into account h e system s t a b l l l t y . I t i s the intention of t h i s paper t o demonstrate that many of the design techniques based on s t a t e space formalism have failed t o deal with the real problems on practical system design. The papers published on these techniques e i t h e r used methods that are applicable only for ideal systems (with no ignorance and noise), or, even when aimed t o cope with changes of the plant parameters, they treated e i t h e r unrealistic d i f f e r e n t i a l changes or wrong sensitivity c r i t e r i a . STATE SPACE FEEDBACK A plant is described by the p a i r of vector equations
where u and y are m dimensional input and output of the plant, x i s the n dimensional s t a t e vector. A. B and C are constant matrices of proper dimensions. Two different categories of s t a t e feedback are generally used. (i)
State feedback of the type u=Kx+Hw
where w i s the m dimensional input t o the whole system and K, H are constant matrices of suitable dimensions. Using (3), it i s assumed that a l l s t a t e s are available f o r measurement, and these are
Superior numbers r e f e r t o similarly-numbered references a t the end of t h i s paper.
*
(3
This research was p a r t i a l l y supported by NASA under Grant NGR 06-003-083 and by the National Science Foundation under Research Grant GK33485.
used to obtain arbitrary pole placemzpt (l), desired eigensensit'vity to plant variation( or decoupling(12f. It is hardly conceivable that there is access to all states, but even if they are available it is impractical to spend much effort on so many transducers when it can be done with only a few at the real outputs of the system. Resistors and capacitors are much cheaper than transducers. The application of (3) to the plant leads to a very special type of two degrees of freedom structure where the matrix H acts as the prefilter to the system. As H is a constant matrix, it does not introduce new dynamics to the system and therefore the system is actually of the one degree of freedom type. For example, in the case of a single input single out u controllable system, one can show, following818)
of eqns. (1-2) with the feedback law (S), then
where T (s) , N(s) , D(s) and ki, same as in Theorem 1 and
Proof Assuming the system is already given in its companion form, where B~ is (0,O.. .l) , the last row of A is (-ao,-a,. .-an-1) and C=(cD,. .cn-,), then -
.
.
C I S -BKA --'BH=c[~,s.-.sn-l]
1.
T
THEOREM 1
i=l,n-1, are the
=
H
det [Is-A-
where T(s) is the system transfer function, N(s) and D(s) are the numerator and denominator of the plant transfer function, ki, i=l,n-1 are linear combinations of the elements of K and H is the scalar gain used in (3). It is therefore seen that in spite of having two degrees of freedom H and K, the burden of pole assignment, sensitivity or (in multivariable systems), decoupling should lie on K only. Moreover, there is no control over the zeros of T (s)
.
Methods that used this type of feedback for design did not take into account the noise effect. The design, which fixes K according to other criteria, may end, especially for the pole assignment problem, with a very high gain of one or more of the components of K This implies that some of the feedback loops may introduce into the system a fantastic amount of noise.
.
(ii) State feedback of the type
*ere ~ ( ~ ) = d ~ u / dB. t ~,,i-0,q-1 H and K iire constant hatrices of suitable dimensions. This feedback is a lied when some of the states In such a case (3) can no are inaccessible longer be used, as the ;election of ki is now constrained and therefore it is impossible to realize an arbitrary allocation for the poles.
(-?f.
The application of (5) is an improvement but basically the synthesis is still of the one degree of freedom type. THEOREM 2 Given the single input output controllable system
Theorem 2 shows that as V(s) is dictated by the pole allocation that is required, there is still no control over the system zeros. EIGENVALUES AND EIGENVECMR SENSITIVITY Much has been written about eigenvalue and eigenvector sensitivity. Expressions were derived using state space formalism for the differential eigenvalue sensitivity (which corresponds to the sensitivity of the closed loop poles due to differential changes in parameters of the system), and for differential eigenvector sensitivity (which corresponds to the sensitivity of the residues of the closed loop poles). These analytical expressions are based on the assumption that the eigenvalues are distinct (7-11). The attempts to use these expressions for sensitivity design (6,7, applied the state feedback (3) which, as mentioned, reduces design freedom. The designers examined the K row of eq. (3) and tried to find those ki which will fix relatively insensitive closed loop poles. There are several objections to design for eigensensitivity reduction in general and to the methods used to accomplish it in particular. (i) In practice, any linear time-invariantplant representation is an approximation of a nonlinear system and therefore inherently has a great deal of associated uncertainty. A design based on differential sensitivity of any kind (and not necessarily of eigenvalues and eigenvectors) is therefore impractical as it cannot guarantee a small
sensitivity of the system response for the practical significant uncertainty or changes of plant parameters. For example, if the m i n k of the sensitivity function is very narrow, this approach may lead to a design which is much inferior to one that takes the large scale changes of the parameters into account. This last has higher differential sensitivity but allocates the closed loop poles in such a region that the sensitivity to significant changes of parameters will be minimal. If one insists on dealing with differential sensithe transfer functivity, it is much easier to that for a given tion approach. It is known n input n output plant whose matrix of transfer functions is P(s)
(*3
where Po is the nominal P , L and T, are the nominal open-loop and closed-loop matrices of transfer functions and AT the transfer function matrix of the changes in the closed-loop transfer functions. Denoting
No example has been given in the literature where numeric specificath in the system output response have been satisfied using eigensensitivity. DECOUPLING OR NON-INI'ERAGTION THEORY Decoupling theory deals with the possibility of compensating a given multivariable system in such a way that each of the system outputs can be independently controlled by a corresponding system input. Many criteria, sufficient and necessary for the existence of a decoupling transformation in a multivariable system represented b tate space formalism have been published (12-Y5f algorithms were proposed for achieving it(
~7 .
Instead of dealing with combinations of high dimensional state space ma es and raising these to a relatively high power , one may easily use the transfer function approach.
If35
Definition(13) A system is decoupled if its matrix of transfer
it is easily seen that, if AP is'brnal1"enough (not necessarily differential)
functions is diagonal and nonsingular.
THEOREM 3
By means of (9) it is possible to choose the elements of Lo taking into account not only the permitted boundaries on AT but also stability and noise effects reduction.
Given an n input n output plant that is represented by a square n dimensional matrix of transfer functions P. The plant can be decoupled if and only if P is nonsingular. Proof
(fi) The above objection to differential sensitivity in general is even stronger in the case of eigensensitivity. Many different pole-zero configurations lead to very similar time responses. For example, two configurations of 3 poles and one zero leading to system transfer function of the type
are considered. The first one has ~=0.9,z=1.2 and p=l , where the second has p=l.Z, z=0.4 and p=0.5. The step responses of the two systems are given in Fig. 1. (iii) Even when it is agreed to design for eigensensitivity, the expression one gets for the partial derivatives of the system time response to a given input with respect to one of the plant parameters is a very complicated combination of all eigenvalues and eigenvector sensitivities, with weights that depend on the final choice of the system poles and zeros (8$11). Therefore it becomes essentially impossible to use in a design as it is not a riori known how to divide the burden among a t e eigenvalues and eigenvectors.
--?hi--.
(iv) It is assumed that all states may be sensed and that the eigenvalues are distinct.
Using the two degrees of freedom structure it is known that:
where T , M and G are n dimensional matrices of transfer functions of the whole system, the feedback compe s tor and the prefilter (respectively)?l7). If P is nonsingular, then for any selection of M for which [I+PM]-' is still defined,a matrix G can be found so that T d l be a diagonal nonsingular matrix. On the other hand, if P is singular, T must be singular too. One of the attractive features of decoupling theory is that at first sight it looks as if by decoupling an n input n output system, the whole design problem reduces to n simple single output problems. The truth is that when a complete knowledge and no variation of plant parameters are obtained, the general case is not much of a problem either. It is not even necessary to use feedback, the prefilter is designed to realize the desired system transfer function and the design is complete. has already been stated, it is impractical to deal with any aspect of system design without taking system sensitivity to uncertainty of parameters into account. This is even more important in the case of non-interacting control, as it can be shown that a system may be decoupled for nominal
As
values of its parameters, where a slight change of these may lead to a highly interacting system (especially in the case where P is nearly singular)
.
From the above it is seen that though attempts have been made to deal with some of the main problems in the design of control systems, the results are far from being satisfactory. The designer is delving into the mathematics of a model that describes a special limited aspect of the design only, without looking at the whole picture. The real problem is to cope simultaneously with realization of the system time response within prescribed boundaries in the time domain, in spite of significant ignorance of the plant parameters, system stability within prescribed margins, maximal reduction of noise effect at the plant input and attenuation of possible disturbances acting at different points of the system. This problem for multivariable systems is difficult enough even with the transfer function approach, where the designer deals with n dimensional matrices (when n is the number of inputs or outputs). If one tries to solve the same problem by the state space approach, it will require matrix calculations of many times higher dimensions. It is true that in the case of an ideal system, in which no ignorance of plant parameters exists, the system can be reduced to one of the canonical forms and then it is not much of a problem to handle the big matrices; however, when uncertainty or change in plant parameters is encountered, the transfonnations which lead to the nominal representation will no longer do so. The transfer function approach is far more efficient in dealing with the real problem. Recently a method has been proposed for ractical single input single output system desigu (18). The method uses templates that describe in the Nichols Chart the uncertainty of the plant transfer function. A minimal open loop gain is found, taking into account time domain boundaries, stability, noise effects and disturbance attenuation. In the multivariable case it has not been found possible (in contrast to the single input single output case) to satisfy specified tolerances to large parameter ignorance (in the low frequency range), and to achieve loop shaping for stability over the range of parameter variations (in the high frequency range), together. Therefore two distinct procedures are involved. The loop shaping for stability despite parameter variations is especially difficult. Consider an n input n output feedback system in which there is embedded a plant whose n x n matrix of transfer function P(s) has any number of parameters knmn only to lie within specified ranges which may be very large. Denoting the nominal plant, nominal open loop, nominal systems and the real system n x n matrices of transfer functions by Po, Lo, To and T (respectively), it is known that
and that the matrix Lo may be chosen diagonal
Bf '/f s. of design with no loss of
for all as generality
If the elements of P are stable, then the nominal loop functions ci mustbe shaped so that the determinant A = det [Lo+P,P-'1
has no right half plane zeros over the range of plant parameters. Until now the technique has been to divide the frequency spectrum into n-1 distinct regions, and it was then possible to deal in each region ge with two and rarely with three dominant II. 'tf 77l1y Nevertheless, shaping of the loops still iivolves tedious work because of the necessity of simultaneously shaping more than one loop function over the same frequency region.
.
A different method, in which there is a need to shape one loop function at a time only, is now proposed. The technique is first illustrated for n=2. The determinant becomes
where Aij
are the elements of the matrix Ah
A =
=
det A
and f
= -
X,,Rl+AX
Ll A l l ' The low frequency values of L-(jw) are chosen to obtain the benefits of feedback, i.e., acceptable sensitivity to plant parameter ignorance and/or disturbance attenuation. With no loss of generaliv, assume that this results in 1111 R, at the upper edge of the low frequency region. The first step is then to shape Rl(jw) so that 111 + A, is stable over the entire range of A,., values a d l that f, values are large enough in magnitude with maximal phase lag. In this way II,may obtain larger phase lag, and therefore decrease faster, without encircling or intersecting the f,. regions of the corresponding frequencies. The choice for i1 should therefore be such that it is not surrounding or cutting the -Al, regions, and that L,+hll values are as small as possible with minimal phase lag. +
I
In the second step'the f, regions are found and is shaped to "sneak" through these without en::rcling or intersecting them. In the general nth order case assume without loss for i=l,n-1 The of generality that leil
I
.
A second order case, d@jlj has already been solved
, is
by the previous method
taken.
The A elemen* are:A, ks/s+a, h =ks/s+b and ni=ck2s2/(s+a) (s+b) &e values k and c vary independently from 0.5 to 1 and those of a and b from 1 to 2 The low frequency requirements on g, and R, are as follows: the mgritudeof!?., must be 100 at 0 rps, 10 at 2 rps, while that of 11, must be 5 at 0 rps, and 1.5 at 1 rps.
.
OF
.
in this case 12 I a I , the regions of -A,, are first considere& This is done in Fig. 2 for 5 different frequencies. R, is then shaped so that will be stable over the entire range of kiues and that it will acquire as high phase lead as possible, especially in the high frequency region where k, is supposed to "sneak" through the resulting regions of f,= (-x,~R,+A~)/(R,+A,,)
As
.
R,
basically a sin e loop shaping problem. In each step, say the i , it is possible to increase the bandwidth of ti , to some extent, in order to economize on R+ which has higher bindwidth. This transparen=$ is useful to permit trade-off between the ti in proportion to the individual sensor noise strengths. The advantage of the present technique over the foner method is demonstrated in an example of a two input two output system.
! k .
Example
has been chosen to be
REFERENCES (1) Brockett, R. W., "Poles Zeroes and Feedback: State Space Interpretation", IEEE Trans. Automatic Control Vol. AC-10, pp. 129-135, April 1965. (2) Wonham, W. M., "On Pole Assignment in MultiInput Controllable Linear Systems", IEEE Trans. Automatic Control Vol. AC-12, pp. 660665, December 1967. (3) Pearson, J. B. and C. Y. Ding, "Compensator Design for Multivariable Linear Systems", IEEE Trans. Automatic Control Vol. AC-14, pp. 130134, April 1969.
and is shown in Fig. 2.
(4)
Next, the resulting boundaries of f, are found and shown in Fig. 3. The choice for L?, prevents the regions of f, for frequencies of 5 rps and more, from sliding toward the smaller phase lag region and so enables I R, I . to decrease by 10 db per octave, still guaranteeing large stability margins.
(5) Paul, C. A., "Pole Specifications in Decoupled System", Int. J. Control Vol. 15, No. 4, pp. 651-664, 1972.
Rl
, which
is also shown in Fig. 3 is
It is of interest to compare the present design which has been proved, by a computer solution, to be stable, with the former design. Fig. 4 shows the Bode plot of both designs for R, and demonstrates the superiority of the present method. S W Y AND a3NCLUSIONS The ability of state space formalism to cope with practical system design is considered. It is shown that although this formulation introduces useful analytic concepts, it fails to deal with the whole problem of system design. Concepts of design, such as pole assignment, decoupling and eigensensitivits are treated separately without taking other factors such as noise effects and significant parameters ignorance into account. It is shown that the state space formulation artificially increases the problem complexity by introducing many redundant variables that increase the dimension of the matrices without actually improving the design. It is also demonstrated that the inadequate results found by this formalism can be easily obtained using the transfer function approach. Confined to the latter, a new method for stability design, in the pres ence of significant plant parameter ignorance, is introduced. The method handles the nominal loop functions Ili one by one such that each step is
Howze, J. W. and J. B. Pearson, "Decoupling and Arbitrary Pole Placement in Linear Systems Using Output Feedback", IEEE Trans. Automatic Control Vol. AC-15, pp. 660-663,December 1970.
(6) Morgan, Jr., B. S., "The Synthesis of Linear Multlvariable Systems by State Variable Feedback," IEEE Trans. Automatic Control Vol. AC-9 pp. 405-411, October 1964. (7) Morgan, Jr., B. S., "Sensitivity Analysis and Synthesis of Multivariable Systems", IEEE Trans. Automatic Control Vol. AC-11, pp. 506512, July 1966. (8) Reddy, D. C., "Parameter Sensitivity of Linear Time Invariant, Dynamic Systems", Report R-315, Coordinated Science Laboratory, University of Illinois, Urbana, Illinois, September 1966.
(9) Mantey, P. E., "Eigenvalue Sensitivity and State-Variable Selection", IEEE Trans. Automatic Control, Vol. AC-13, pp. 263-209, June 1968. (10) Reddy, D. C., "Eigenfunction Sensitivity and Parameter Variation Problem", Int. J. Control, Vol. 9, No. 5, pp. 561-568, 1969. (11) Crossley, T. R. and B. Porter, "Eigenvalue and Eigenvector Sensitivities in Linear Systems Theory", Int. J. Control, Vol. 10, No. 2, pp. 163-170, 1969.
Fdb, Peter L. and W. A. hblovich, "Decoupling in the Design and Synthesis of Multivariable Control Systems", IEEE Trans. Automatic Control, Vol. AC-12, pp. 651-659, December 1967. Gilbert, E. 0. "The Decoupling of Multivariable Systems by State Feedback", SIAM J. Control, Vol. 7, pp. 50-64, 1969. McLane, P. J. and E. J. Davison, "Disturbance Localization and Decoupling in Stationary Linear Multivariable Systems", IEEE Trans. Automatic Control, Vol. AC-15, pp. 133-134, February 1970. brse, A. S. and W. M. Wonham, "Status of Noninteracting Control", IEEE Trans. Automatic Control, Vol. AC-16, pp. 568-581, December 1971. Gilbert, E. G. and J. R. Pionidmy, "A Computer Program for the Synthesis of Decoupled Multivariable Feedback Systems", IEEE Trans. Automatic Control, Vol. AC-14, pp. 652-659, December 1969. Horowitz, I., Synthesis of Feedback Systems, Academic Press, 1963, (31. 10.
FIG. 2:
Shaping
of
R,
for the stability
(112+A22)
Ding, C. Y., F. N. Brasch, Jr. and J. B. Pearson, "On Multivariable Systems", IEEE Trans. Automatic Control, Vol. AC-15, pp. 9697, February 1970. Sidi, M. and I. Horowitz, "Synthesis of Feedback Systems with Large Plant Ignorance for Prescribed Time-Domain Tolerances", to appear in Int. J. of Control.
Time in seconds
FIG. 1:
Two responses which are essentially similar
F I G . 3:
for the stability
1
30
-
former design
a 1
(d
z
-10-20
-
-30
-
FIG. 4 :
Bode p l o t o f t h e two d e s i g n s f o r R, -