- Email: [email protected]
Robust performance for systems with component-bounded signals
Aukmotica,
Pergamon
ooos1098@4)00117-0
Vol. 31. No. 3. pp. 471-475, 1995 Ekvier Science Ltd in Great Britain. All rights reserved om-1098/95 $9.50 + 0.00
Copyr<t Q 1995
Printed
Brief Paper
Robust Performance for Systems with Componentbounded Signals* JOR-YAN Key Words-Control analysis.
WONGt
and DOUGLAS
system analysis; robust control; performance
--This paper presents a non-conservative of performance for systems whose multivariable inputs and performance outputs consist of arbitrary of independently norm-bounded components. This is expressed as the structured singular value appropriate interconnection system.
measure external numbers measure of an
P. LOOZEt bounds; stability criteria; system
the underlying signal norm and the underlying vector norm of the signal-space norm. Application of the usual L2 and H, norms to vector-valued signals assume that the underlying vector norm is the Euclidean norm. More general norms of multivariable signals using the p-norm for the underlying vector norm and the q-norm for the underlying signal norm, and the implications of such models on the induced operator norms that determine performance have been considered by Wilson (1991). In general, any signal-space norm that provides a useful representation of the properties of the desired performance signals or external input signals can be employed. The underlying vector norms of the signal models that have been discussed require that the signal sets relate the elements of the vector-valued signals.* In many problems, however, a system may possess groups of external input signals that are best represented independently of each other. Likewise, it may be more appropriate to require that groups of performance output vectors be individually bounded. Examples of this situation are numerous. For instance, in a large-scale interconnected system that is composed of many subsystems, each local subsystem may have its own set of external inputs and performance outputs that are specified independently of thd other subsystems (Looze and- Wong, 1990). In a standard feedback svstem (Dovle and Steip 1981j, the commands, plant distuibances‘ani measureme;; noises represent different sources of external inputs that may be independently specified. Similarly, the error signals and control signals represent two performance outputs that may be required to be bounded independently of each other. The objective of this paper is to consider the robust performance problem for systems in which groups of input and performance signals are to be bounded independently. In terms of the standard robust performance framework (Doyle, 1985), the overall external input vector and the overall performance output vector will be composed of component vectors that are individually norm-bounded. We shall refer to signals of this type as component-bounded signals. The principal result of this paper develops a nonconservative measure of performance for systems with component-bounded signals. Specifically, it will be shown that both nominal performance and robust performance for such systems can be characterized by the structured singular value of an appropriate interconnection system. For the special case when the numbers of component vectors in the input and output are both one, our results specialize to the usual robust performance condition of Doyle et al. (1982). Thus, the results in this paper generalize those given by Doyle et al. (1982). The basis of the performance result of this paper is a characterization of the norm induced by the component-
1. Inrroduction The principal objective of robust control system design is to achieve closed-loop performance in the presence of modeling errors and uncertainties. Mathematical modeling of the performance objectives, external signals, and dynamic model uncertainties can have a significant impact on the analysis of robust performance. For linear systems, dynamic uncertainties are typically represented as transfer functions with bounded H, norms. Performance objectives (the desired performance output signals) and external input signals are usually represented as sets determined by signal norms (see Doyle and Stein, 1981; Doyle et al., 1982; Doyle, 1985; Doyle and Packard, 1987; Khammash and Pearson, 1991; Wilson, 1989). Common representations for these sets are specified as signals that have bounded Lz or H, signal norms. For these norms, the induced norm on the operator that maps external inputs into performance outputs is the H, operator norm. As shown by Doyle et al., (1982, 1991), this representation of performance is equivalent to robust stability with respect to an uncertain dynamic feedback connection from performance outputs to external inputs. Thus, tools for analyzing robust stability can also be used to determine robust performance. In particular, the structured singular value (Doyle, 1982) has been commonly used for this purpose. Other signal models may provide more accurate representations of certain robust performance problems. Signals that have persistent excitation have been represented as elements of 1, (for discrete-time systems) (Khammash and Pearson, 1991). When the desired performance and allowable external input signal sets are represented in this manner, the induced operator norm on the transfer function takes the form of the II norm of its kernel. Again, it has been shown that this robust performance problem is equivalent to a robust stability problem when the uncertainties are represented appropriately. The norms used to represent multivariable signals as norm-bounded sets must account for both the behavior of the signal as a function of the independent variable, and the vector space to which the signal belongs at each value of the independent variable. We shall refer to these, respectively, as *Received 19 October 1992; revised 3 November 1993; received in final form 14 June 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. F. Curtain under the direction of Editor Huibert Kwakemaak. Corresponding author Professor Douglas P. Looze. Tel. +l 413 545 0973; E-mail [email protected]. TDepartment of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A.
$ An exception is when the underlying vector norm is the m-norm, in which case each element is bounded independently. The situation in which the underlying vector norm is the m-norm is a special case of the component-bounded signal model. 471
472
Brief
bounded signals. This characterization will be developed using the Z&norm as the underlying signal norm. The &-norm is a frequency-domain characterization of the signal that bounds the power spectrum of the signal. A popular alternative to this characterization is the &-signal norm, which has the interpretation of bounding the total energy of the signal. It is straightforward to verify that the results of this paper remain unchanged when the underlying signal norm is the Lz-norm, although the proof that the performance condition is necessary is more complicated. This paper is organized as follows: Section 2 formulates the component-bounded robust performance problem. Section 3 characterizes the solution of this problem in terms of the structured singular value of a modified interconnection system. In Section 4, the result is demonstrated by evaluating performance bounds of a control system for a high purity distillation column (Skogestad et al., 1988). 2. Problem formulation The robust performance analysis problem for the system of Fig. 1 is to determine whether the performance signal e is an acceptable response for every allowable external input v and every allowable model uncertainty A,. Specifically, let V be the set of possible external input signals, let D, be the set of possible model errors and let E be the set of acceptable system responses. The system of Fig. -1 is said to have robust performance if es E
VWE V,
VA,e D,.
(1)
E and V are typically selected to be unit balls in Banach spaces of signals. Selections of the particular norm used by the input signal space and the output signal space should be based on modeling considerations: which norms best represent the important physical properties of the system. Typically, the Banach spaces that have been used are spaces of square-integrable (Lz) or bounded (L,) time functions, spaces of functions with Laplace transforms that are bounded in the closed right half-plane (H,), or spaces of stochastic processes with finite covariances. Function spaces with less usual norms have also been considered by Wilson (1991). In each of these cases, the robust performance condition (1) can be represented in terms of an induced norm on the operator F,(M, A,) from external inputs to external outputs:
IlF,(M, A,)II < 1 VA, E D,.
(2)
As noted in the introduction, it can be important to consider bounding groups of signals independently of each other. The norms considered in this paper are based on this premise. Assume that the input signals v and the performance signals e can be partitioned into n and m groups each of which is to be (components), respectively, independently bounded. That is, let the input space If and the performance space ‘i5be defined as
Papers
Define the block norms of v E V and e E g as
where ll.ll_ denotes the norm on RH, and 11.11z denotes the usual Euclidean norm. With these definitions of vector norms, the block-induced norm of an operator F: Y+ g is
IIFllB = max IIWlo. IMIS1 Note that the block-induced norm depends on the component structures of both v and e. In particular, when n=m=l, l\Flla is the usual norm on e”” (i.e. the supremum over frequency of the largest singular value); when n = n and m = III, IIFlln is the supremum over frequency of the infinity matrix norm of F&J). Throughout this paper, we shall assume that the set of allowable model uncertainty is D, = {A, E HF+“ul:
~~AU~~_~ l},
(5)
where H, is the space of functions whose Laplace transforms are analytic and bounded in the closed right half-plane. Define V and E to the unit balls in Sr and $, respectively. Then the robust performance condition (1) can be expressed as a norm bound, as in (2):
IIFuOW~,h<1 The robust performance
VA, ED”.
(6)
analysis problem is to determine
whether (6) holds. 3. A structured singular-valued condition for robust performance The principal contribution of this paper is to express the robust performance condition (6) in terms of the structured singular value. As is shown by Doyle et al. (1982), when there are single input and output blocks (i.e. n = 1 and m = l), the robust performance condition (6) is equivalent to requiring that the structured singular value (with respect to the uncertainty structure (5) augmented with a performance block whose dimensions are compatible with the input and performance signals) be bounded below one at all frequencies. The main result of this section will show that condition (6) can be determined by a similar test with simple modifications to the interconnection system and the uncertainty structure. Before stating the main result of this section, we shall introduce some required notation. The structured singular value of a matrix M defined with respect to a set X is given by
FXUW =
1 -1
~~x{C(A):det(I-MA)XO} ifdet(I-MA)#O
WAEX,
where it is assumed that the dimensions of the matrix M and the perturbations A are such that the indicated operations are appropriate. Given the system of Fig. 1, with the sets of external inputs and performance outputs (3) and (4) partition the system operator M conformably as
M= e Fig. 1. The upper linear fractional transformation
F,(M, A,).
Brief Papers and M 22.1E RHzjx”. Also define the suboperator Mj obtained by considering only the uncertainty and jth component outputs: M,=
M,, M2l.j
M12 M22.j
I
Let T, be the mapping that replicates blocks of Mj: I ““I I;=
? . [ ;
0
n times the lower
1
I!? . . I;,j
473
techniques (Doyle, 1985; Balas er al., 1991) to design compensators to achieve robust performance objectives for systems whose signals are component bounded. 4. Example Consider the simplified linear model of a high-purity distillation column that has been studied by Skogestad et nl. (1988). The transfer function of the system is 1 P(s) = 75s+l
0.878 -0.864 [ 1.082 -1.096 1
The system is to be controlled to reject output disturbances (see Fig. 2). The performance weighting on the output is W(s) = f
(“!)I.
Finally, define the structure set S, as S, = {A:A = diag (A,, A,,, . . . , A,,,), A,, E C”U~~“U~, A,, E C”@‘J, i = 1, . . , n}. Theorem 1. Consider the system shown in Fig. 1 with the signal component structure defined by (3) and (4) and the uncertainty structure defined by (5). The condition (6) will be satisfied if and only if
Proof
See the Appendix.
The condition for robust performance for systems with component-bounded signals provided by Theorem 1 actually requires that robust performance be satisfied for each of the output components individually. This is a direct consequence of the structure of the component norm on the performance signal. The condition that must be satisfied for each performance component is a robust stability condition for a related system that contains the portion of the transfer function that supplies the uncertainty output plus n replicas of the portion of the transfer function that generates the performance signal component. The robust performance condition (7) can be rewritten by interchanging the maximum and supremum functions:
This form shows that the operator norm induced by the component signal norms can be represented as finding the largest value (over frequency) of a matrix norm also induced by the component structure. In particular, if the uncertainty is not present (A, = 0) and if each of the block components has dimension one (i.e. each signal is to be bounded individually) then the matrix norm is just the matrix infinity norm:
The weighting on the disturbance signal is the identity matrix. The compensator that will be used for this system is given by 75s+l K(s) = 2.64-- s
100 s+lOO
[ 1 1 0
0 -1.
Although the source of the compensator is unimportant for the demonstration of the robust performance analysis, it was obtained by augmenting the diagonal PI compensator given in Skogestad et al. (1988) with a low-pass filter having cutoff frequency at 108 rad s-’ and a gain factor of 1.1. We shall consider the problem of measuring nominal performance to highlight the effects of the block norm representations of the signals. That is, we shall assume that there is no plant uncertainty. In the standard model, the input signals are bounded together to form one input block, and the output signals are bounded together to form one output block. Nominal performance is then determined by the largest singular value of the weighted sensitivity function: tiW(jo)[I
+ P(jo)K(jo)]-‘}
< 1.
(8)
We wish to compare the standard formulation in which the disturbance and performance signals are represented by a Euclidean norm bound at each frequency with a formulation in which each of the components is bounded individually. If the weighting function for the component-bounded input is reduced by a factor of fi, every signal in the component-bounded set is then also included in the Euclidean norm-bounded set. Thus, if the standard formulation with single input and performance components satisfies (8) then the system with individually bounded signals will also satisfy the performance objective. However, as the examples will show, it is possible for the standard formulation to violate (8) and the component-bounded formulation to satisfy the performance objective. We shall consider three cases. The first assumes that both the input signal and the output signal are component bounded. In this case, the performance weighting function is taken to be W,,(s) = flW(s).
where Mzzji is the (j, i) element of M,,. Thus, an immediate consequence of this theorem is the expression of the matrix infinity norm in terms of the structured singular value. Note, however, that the presence of the uncertainty block or of components with dimensions greater than one prevent the decomposition of the induced norm into the maximum (over column components) of a sum (over row components) of norms. Although (5) assumes a single block of dynamic uncertainty, the result of the theorem extends readily to block-diagonal uncertainty structures through obvious modification of the structure sets Sj. Real parametric uncertainty can be induced by using a real structured singular value in (6). Finally, it should be noted that the expression of the robust performance objective in terms of the structured singular value should allow the application of p-synthesis
The Euclidean performance bound and the component bound given by (7) are shown in Fig. 3. Because all the components in both the disturbance and performance signals are scalars, the resulting bound should be the norm of the weighted sensitivity induced by the vector infinity norm. This bound, also shown in Fig. 3, coincides exactly with the bound given by (5). Note that the condition obtained by overbounding the input signal set with a Euclidean bound indicates that the performance objective might not be satisfied. However, the component-bounded norm shows that the performance condition is, in fact, satisfied. The next two cases consider component bounds on the input and output signals individually. The performance weighting functions when the input signal is componentbounded (W,(S)) and when the performance signal is component-bounded (W,(s)) are taken as W,(S) = fl W(s),
W,(S) = W(s).
Brief Papers
474
K(s)
__)
P(s)
Fig. 2. Block diagram of unity feedback control system.
Fig. 3. Performance
bounds for the Euclidean si nal model (-a--.--), the input and output signal model (-) and t I!e vector-induced norm (+ + +).
Again, W,(s) guarantees that every allowable componentbounded signal is contained in the Euclidean-bounded set, and W,(s) guarantees that every desired performance signal in the Euclidean-bounded performance set is contained in the component-bounded performance set. The Euclidean performance bound and the component performance bounds given by (7) are shown in Fig. 4. Note that the performance analysis that is obtained by overbounding the problem with a Euclidean norm can range from tight to very conservative. 5. Summary This paper provides a measure of robust performance that allows groups of signals that constitute the external inputs or the performance outputs to be bounded independently of each other. The measure is given in terms of a standard structured singular-value condition. The interconnection system and uncertainty structure that determine the structured singular-value condition are derived from the
10-t
original interconnection and uncertainty structure of the system, and from the component structure of the input and performance signals. References Bales, G. J., J. C. Doyle, K. Glover, A. Packard and R. Smith (1991). p-Analysis and Synthesis Toolbox: Matlab Functions for the Analysis and Design of Robust Control Systems. The MathWorks, Inc. and MUSYN Inc, Natick, MA, U.S.A. Doyle, J. C. (1982). Analysis of feedback systems with structured uncertainties. IEEE Proc., l29,242-250. Doyle, J. C. (1985). Structured uncertainty in control system design. In Proc. 24th IEEE Conf. on Decision and Control, Fort Lauderdale, FL, pp. 260-265. Doyle, J. and A. Packard (1987). Uncertain multivariable signals from a state space perspective. In Proc. 1987 American Control Conf, Minneapolis, MN, pp. 2147-2152.
100
Frequency(radkc) Fig. 4. Performance
bounds for the Euclidean signal model (-. - *-) and the individual input (-) output (---) component signal models.
and
Brief Papers
475
Doyle, J. C., A. Packard and K. Zhou (1991). Review of LFTs, LMIs, and p. In Proc. 30th IEEE Co& on Decision AP 4 and Control, Brighton, U.K., pp. 1227-1232. Doyle, J. C. and G. Stein (1981). Multivariable feedback design: concepts for a classical/modem synthesis. IEEE Trans. Autom. Control, AC--4-16. Doyle, J. C., J. E. Wall, Jr. and G. Stein (1982). Performance and robustness analysis for structured uncertainty. In Proc. 21~1IEEE Conf. on Decision and Control, Orlando, FL, pp. 629-636. Khammash, M. and J. B. Pearson, Jr. (1991). Performance robustness of discrete-time systems with structured uncertainty. IEEE Trans. Autom. Control, AC-36, 398-412. Looze, D. P. and J. Y. Wong (1990). Robust performance of large scale integrated systems. In Proc. 1990 American Cokrol ConjI, S-k Diego, CA. Skoeestad. S.. M. Morari and J. C. Dovle (19881. Robust Fig. A.l. Illustration of proof of Theorem 1. control of ‘ill-conditioned plants: high-puiity distillation columns. IEEE Trans. Autom. Control, AC-33,1092-1105. Wilson, D. A. (1989). Convolution and Hankel norms for Thus, the system of Fig. 1 will have robust performance if linear systems. IEEE Trans. Autom. Control, AC-34, and only if the system of Fig. A.1 (with A, = 0) has robust 94-97. performance with respect to the uncertainties A, E D, and A, E Dpi, and inputs in the set {VE HZ’: llV11_ I 1). Define Y
,
,
Appendix-Proof of Theorem 1 First, note that the structure of the output norm implies each of the output components can be considered independently. That is, we can consider whether each of the systems of Fig. 1 with M replaced by M, and m = mj satisfies the robust performance objective. Thus, we assume, without loss of generality, that m = 1. Note that the set of allowable input vectors V={[vT
v: ...
v~~:v~EH~,I~v~[[,~~
Vi=l,...,n)
D, = {A, E HE*“: /A&
5 l}.
The equivalence between robust performance and robust stability (see Doyle et al., 1982) implies that the system in the gray box in Fig. A.1 will have robust performance if and only if the system of Fig. A.1 is robustly stable with respect to the uncertainties A,, ED,, Ai E Dpi and Api and 4 E D,. Form the matrix product A, = A, A, and note that Api E Dpr. The system shown in Fig. A.1 IS stable for all Ap E Dp, n} and Au E 0, if and only if it is also {AiEDpi;i=l,..., stable for all {A, E D*:i = 1, . . . , n} and A, E D,. Define ra,
0 1
ra,
0
...
0 1
is equivalently expressed as
IIAill,s 1, i = 1,. . . , n; i E I-E, Ilill- 551). Define D, as o,={Ai~H~Xm:IIAill,~l},
i=l,...,
n.
Noting that A = AdT, M in Fig. A.1 is robustly stable with respect to A if and only if TM is robustly stable with respect to A,,. But, from Doyle et al. (1982), robust stability of TM with respect to A., is equivalent to (7) being satisfied (for m = 1). The result for multiple blocks follows from earlier comments.
Pergamon
ooos1098@4)00117-0
Vol. 31. No. 3. pp. 471-475, 1995 Ekvier Science Ltd in Great Britain. All rights reserved om-1098/95 $9.50 + 0.00
Copyr<t Q 1995
Printed
Brief Paper
Robust Performance for Systems with Componentbounded Signals* JOR-YAN Key Words-Control analysis.
WONGt
and DOUGLAS
system analysis; robust control; performance
--This paper presents a non-conservative of performance for systems whose multivariable inputs and performance outputs consist of arbitrary of independently norm-bounded components. This is expressed as the structured singular value appropriate interconnection system.
measure external numbers measure of an
P. LOOZEt bounds; stability criteria; system
the underlying signal norm and the underlying vector norm of the signal-space norm. Application of the usual L2 and H, norms to vector-valued signals assume that the underlying vector norm is the Euclidean norm. More general norms of multivariable signals using the p-norm for the underlying vector norm and the q-norm for the underlying signal norm, and the implications of such models on the induced operator norms that determine performance have been considered by Wilson (1991). In general, any signal-space norm that provides a useful representation of the properties of the desired performance signals or external input signals can be employed. The underlying vector norms of the signal models that have been discussed require that the signal sets relate the elements of the vector-valued signals.* In many problems, however, a system may possess groups of external input signals that are best represented independently of each other. Likewise, it may be more appropriate to require that groups of performance output vectors be individually bounded. Examples of this situation are numerous. For instance, in a large-scale interconnected system that is composed of many subsystems, each local subsystem may have its own set of external inputs and performance outputs that are specified independently of thd other subsystems (Looze and- Wong, 1990). In a standard feedback svstem (Dovle and Steip 1981j, the commands, plant distuibances‘ani measureme;; noises represent different sources of external inputs that may be independently specified. Similarly, the error signals and control signals represent two performance outputs that may be required to be bounded independently of each other. The objective of this paper is to consider the robust performance problem for systems in which groups of input and performance signals are to be bounded independently. In terms of the standard robust performance framework (Doyle, 1985), the overall external input vector and the overall performance output vector will be composed of component vectors that are individually norm-bounded. We shall refer to signals of this type as component-bounded signals. The principal result of this paper develops a nonconservative measure of performance for systems with component-bounded signals. Specifically, it will be shown that both nominal performance and robust performance for such systems can be characterized by the structured singular value of an appropriate interconnection system. For the special case when the numbers of component vectors in the input and output are both one, our results specialize to the usual robust performance condition of Doyle et al. (1982). Thus, the results in this paper generalize those given by Doyle et al. (1982). The basis of the performance result of this paper is a characterization of the norm induced by the component-
1. Inrroduction The principal objective of robust control system design is to achieve closed-loop performance in the presence of modeling errors and uncertainties. Mathematical modeling of the performance objectives, external signals, and dynamic model uncertainties can have a significant impact on the analysis of robust performance. For linear systems, dynamic uncertainties are typically represented as transfer functions with bounded H, norms. Performance objectives (the desired performance output signals) and external input signals are usually represented as sets determined by signal norms (see Doyle and Stein, 1981; Doyle et al., 1982; Doyle, 1985; Doyle and Packard, 1987; Khammash and Pearson, 1991; Wilson, 1989). Common representations for these sets are specified as signals that have bounded Lz or H, signal norms. For these norms, the induced norm on the operator that maps external inputs into performance outputs is the H, operator norm. As shown by Doyle et al., (1982, 1991), this representation of performance is equivalent to robust stability with respect to an uncertain dynamic feedback connection from performance outputs to external inputs. Thus, tools for analyzing robust stability can also be used to determine robust performance. In particular, the structured singular value (Doyle, 1982) has been commonly used for this purpose. Other signal models may provide more accurate representations of certain robust performance problems. Signals that have persistent excitation have been represented as elements of 1, (for discrete-time systems) (Khammash and Pearson, 1991). When the desired performance and allowable external input signal sets are represented in this manner, the induced operator norm on the transfer function takes the form of the II norm of its kernel. Again, it has been shown that this robust performance problem is equivalent to a robust stability problem when the uncertainties are represented appropriately. The norms used to represent multivariable signals as norm-bounded sets must account for both the behavior of the signal as a function of the independent variable, and the vector space to which the signal belongs at each value of the independent variable. We shall refer to these, respectively, as *Received 19 October 1992; revised 3 November 1993; received in final form 14 June 1994. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor R. F. Curtain under the direction of Editor Huibert Kwakemaak. Corresponding author Professor Douglas P. Looze. Tel. +l 413 545 0973; E-mail [email protected]. TDepartment of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003, U.S.A.
$ An exception is when the underlying vector norm is the m-norm, in which case each element is bounded independently. The situation in which the underlying vector norm is the m-norm is a special case of the component-bounded signal model. 471
472
Brief
bounded signals. This characterization will be developed using the Z&norm as the underlying signal norm. The &-norm is a frequency-domain characterization of the signal that bounds the power spectrum of the signal. A popular alternative to this characterization is the &-signal norm, which has the interpretation of bounding the total energy of the signal. It is straightforward to verify that the results of this paper remain unchanged when the underlying signal norm is the Lz-norm, although the proof that the performance condition is necessary is more complicated. This paper is organized as follows: Section 2 formulates the component-bounded robust performance problem. Section 3 characterizes the solution of this problem in terms of the structured singular value of a modified interconnection system. In Section 4, the result is demonstrated by evaluating performance bounds of a control system for a high purity distillation column (Skogestad et al., 1988). 2. Problem formulation The robust performance analysis problem for the system of Fig. 1 is to determine whether the performance signal e is an acceptable response for every allowable external input v and every allowable model uncertainty A,. Specifically, let V be the set of possible external input signals, let D, be the set of possible model errors and let E be the set of acceptable system responses. The system of Fig. -1 is said to have robust performance if es E
VWE V,
VA,e D,.
(1)
E and V are typically selected to be unit balls in Banach spaces of signals. Selections of the particular norm used by the input signal space and the output signal space should be based on modeling considerations: which norms best represent the important physical properties of the system. Typically, the Banach spaces that have been used are spaces of square-integrable (Lz) or bounded (L,) time functions, spaces of functions with Laplace transforms that are bounded in the closed right half-plane (H,), or spaces of stochastic processes with finite covariances. Function spaces with less usual norms have also been considered by Wilson (1991). In each of these cases, the robust performance condition (1) can be represented in terms of an induced norm on the operator F,(M, A,) from external inputs to external outputs:
IlF,(M, A,)II < 1 VA, E D,.
(2)
As noted in the introduction, it can be important to consider bounding groups of signals independently of each other. The norms considered in this paper are based on this premise. Assume that the input signals v and the performance signals e can be partitioned into n and m groups each of which is to be (components), respectively, independently bounded. That is, let the input space If and the performance space ‘i5be defined as
Papers
Define the block norms of v E V and e E g as
where ll.ll_ denotes the norm on RH, and 11.11z denotes the usual Euclidean norm. With these definitions of vector norms, the block-induced norm of an operator F: Y+ g is
IIFllB = max IIWlo. IMIS1 Note that the block-induced norm depends on the component structures of both v and e. In particular, when n=m=l, l\Flla is the usual norm on e”” (i.e. the supremum over frequency of the largest singular value); when n = n and m = III, IIFlln is the supremum over frequency of the infinity matrix norm of F&J). Throughout this paper, we shall assume that the set of allowable model uncertainty is D, = {A, E HF+“ul:
~~AU~~_~ l},
(5)
where H, is the space of functions whose Laplace transforms are analytic and bounded in the closed right half-plane. Define V and E to the unit balls in Sr and $, respectively. Then the robust performance condition (1) can be expressed as a norm bound, as in (2):
IIFuOW~,h<1 The robust performance
VA, ED”.
(6)
analysis problem is to determine
whether (6) holds. 3. A structured singular-valued condition for robust performance The principal contribution of this paper is to express the robust performance condition (6) in terms of the structured singular value. As is shown by Doyle et al. (1982), when there are single input and output blocks (i.e. n = 1 and m = l), the robust performance condition (6) is equivalent to requiring that the structured singular value (with respect to the uncertainty structure (5) augmented with a performance block whose dimensions are compatible with the input and performance signals) be bounded below one at all frequencies. The main result of this section will show that condition (6) can be determined by a similar test with simple modifications to the interconnection system and the uncertainty structure. Before stating the main result of this section, we shall introduce some required notation. The structured singular value of a matrix M defined with respect to a set X is given by
FXUW =
1 -1
~~x{C(A):det(I-MA)XO} ifdet(I-MA)#O
WAEX,
where it is assumed that the dimensions of the matrix M and the perturbations A are such that the indicated operations are appropriate. Given the system of Fig. 1, with the sets of external inputs and performance outputs (3) and (4) partition the system operator M conformably as
M= e Fig. 1. The upper linear fractional transformation
F,(M, A,).
Brief Papers and M 22.1E RHzjx”. Also define the suboperator Mj obtained by considering only the uncertainty and jth component outputs: M,=
M,, M2l.j
M12 M22.j
I
Let T, be the mapping that replicates blocks of Mj: I ““I I;=
? . [ ;
0
n times the lower
1
I!? . . I;,j
473
techniques (Doyle, 1985; Balas er al., 1991) to design compensators to achieve robust performance objectives for systems whose signals are component bounded. 4. Example Consider the simplified linear model of a high-purity distillation column that has been studied by Skogestad et nl. (1988). The transfer function of the system is 1 P(s) = 75s+l
0.878 -0.864 [ 1.082 -1.096 1
The system is to be controlled to reject output disturbances (see Fig. 2). The performance weighting on the output is W(s) = f
(“!)I.
Finally, define the structure set S, as S, = {A:A = diag (A,, A,,, . . . , A,,,), A,, E C”U~~“U~, A,, E C”@‘J, i = 1, . . , n}. Theorem 1. Consider the system shown in Fig. 1 with the signal component structure defined by (3) and (4) and the uncertainty structure defined by (5). The condition (6) will be satisfied if and only if
Proof
See the Appendix.
The condition for robust performance for systems with component-bounded signals provided by Theorem 1 actually requires that robust performance be satisfied for each of the output components individually. This is a direct consequence of the structure of the component norm on the performance signal. The condition that must be satisfied for each performance component is a robust stability condition for a related system that contains the portion of the transfer function that supplies the uncertainty output plus n replicas of the portion of the transfer function that generates the performance signal component. The robust performance condition (7) can be rewritten by interchanging the maximum and supremum functions:
This form shows that the operator norm induced by the component signal norms can be represented as finding the largest value (over frequency) of a matrix norm also induced by the component structure. In particular, if the uncertainty is not present (A, = 0) and if each of the block components has dimension one (i.e. each signal is to be bounded individually) then the matrix norm is just the matrix infinity norm:
The weighting on the disturbance signal is the identity matrix. The compensator that will be used for this system is given by 75s+l K(s) = 2.64-- s
100 s+lOO
[ 1 1 0
0 -1.
Although the source of the compensator is unimportant for the demonstration of the robust performance analysis, it was obtained by augmenting the diagonal PI compensator given in Skogestad et al. (1988) with a low-pass filter having cutoff frequency at 108 rad s-’ and a gain factor of 1.1. We shall consider the problem of measuring nominal performance to highlight the effects of the block norm representations of the signals. That is, we shall assume that there is no plant uncertainty. In the standard model, the input signals are bounded together to form one input block, and the output signals are bounded together to form one output block. Nominal performance is then determined by the largest singular value of the weighted sensitivity function: tiW(jo)[I
+ P(jo)K(jo)]-‘}
< 1.
(8)
We wish to compare the standard formulation in which the disturbance and performance signals are represented by a Euclidean norm bound at each frequency with a formulation in which each of the components is bounded individually. If the weighting function for the component-bounded input is reduced by a factor of fi, every signal in the component-bounded set is then also included in the Euclidean norm-bounded set. Thus, if the standard formulation with single input and performance components satisfies (8) then the system with individually bounded signals will also satisfy the performance objective. However, as the examples will show, it is possible for the standard formulation to violate (8) and the component-bounded formulation to satisfy the performance objective. We shall consider three cases. The first assumes that both the input signal and the output signal are component bounded. In this case, the performance weighting function is taken to be W,,(s) = flW(s).
where Mzzji is the (j, i) element of M,,. Thus, an immediate consequence of this theorem is the expression of the matrix infinity norm in terms of the structured singular value. Note, however, that the presence of the uncertainty block or of components with dimensions greater than one prevent the decomposition of the induced norm into the maximum (over column components) of a sum (over row components) of norms. Although (5) assumes a single block of dynamic uncertainty, the result of the theorem extends readily to block-diagonal uncertainty structures through obvious modification of the structure sets Sj. Real parametric uncertainty can be induced by using a real structured singular value in (6). Finally, it should be noted that the expression of the robust performance objective in terms of the structured singular value should allow the application of p-synthesis
The Euclidean performance bound and the component bound given by (7) are shown in Fig. 3. Because all the components in both the disturbance and performance signals are scalars, the resulting bound should be the norm of the weighted sensitivity induced by the vector infinity norm. This bound, also shown in Fig. 3, coincides exactly with the bound given by (5). Note that the condition obtained by overbounding the input signal set with a Euclidean bound indicates that the performance objective might not be satisfied. However, the component-bounded norm shows that the performance condition is, in fact, satisfied. The next two cases consider component bounds on the input and output signals individually. The performance weighting functions when the input signal is componentbounded (W,(S)) and when the performance signal is component-bounded (W,(s)) are taken as W,(S) = fl W(s),
W,(S) = W(s).
Brief Papers
474
K(s)
__)
P(s)
Fig. 2. Block diagram of unity feedback control system.
Fig. 3. Performance
bounds for the Euclidean si nal model (-a--.--), the input and output signal model (-) and t I!e vector-induced norm (+ + +).
Again, W,(s) guarantees that every allowable componentbounded signal is contained in the Euclidean-bounded set, and W,(s) guarantees that every desired performance signal in the Euclidean-bounded performance set is contained in the component-bounded performance set. The Euclidean performance bound and the component performance bounds given by (7) are shown in Fig. 4. Note that the performance analysis that is obtained by overbounding the problem with a Euclidean norm can range from tight to very conservative. 5. Summary This paper provides a measure of robust performance that allows groups of signals that constitute the external inputs or the performance outputs to be bounded independently of each other. The measure is given in terms of a standard structured singular-value condition. The interconnection system and uncertainty structure that determine the structured singular-value condition are derived from the
10-t
original interconnection and uncertainty structure of the system, and from the component structure of the input and performance signals. References Bales, G. J., J. C. Doyle, K. Glover, A. Packard and R. Smith (1991). p-Analysis and Synthesis Toolbox: Matlab Functions for the Analysis and Design of Robust Control Systems. The MathWorks, Inc. and MUSYN Inc, Natick, MA, U.S.A. Doyle, J. C. (1982). Analysis of feedback systems with structured uncertainties. IEEE Proc., l29,242-250. Doyle, J. C. (1985). Structured uncertainty in control system design. In Proc. 24th IEEE Conf. on Decision and Control, Fort Lauderdale, FL, pp. 260-265. Doyle, J. and A. Packard (1987). Uncertain multivariable signals from a state space perspective. In Proc. 1987 American Control Conf, Minneapolis, MN, pp. 2147-2152.
100
Frequency(radkc) Fig. 4. Performance
bounds for the Euclidean signal model (-. - *-) and the individual input (-) output (---) component signal models.
and
Brief Papers
475
Doyle, J. C., A. Packard and K. Zhou (1991). Review of LFTs, LMIs, and p. In Proc. 30th IEEE Co& on Decision AP 4 and Control, Brighton, U.K., pp. 1227-1232. Doyle, J. C. and G. Stein (1981). Multivariable feedback design: concepts for a classical/modem synthesis. IEEE Trans. Autom. Control, AC--4-16. Doyle, J. C., J. E. Wall, Jr. and G. Stein (1982). Performance and robustness analysis for structured uncertainty. In Proc. 21~1IEEE Conf. on Decision and Control, Orlando, FL, pp. 629-636. Khammash, M. and J. B. Pearson, Jr. (1991). Performance robustness of discrete-time systems with structured uncertainty. IEEE Trans. Autom. Control, AC-36, 398-412. Looze, D. P. and J. Y. Wong (1990). Robust performance of large scale integrated systems. In Proc. 1990 American Cokrol ConjI, S-k Diego, CA. Skoeestad. S.. M. Morari and J. C. Dovle (19881. Robust Fig. A.l. Illustration of proof of Theorem 1. control of ‘ill-conditioned plants: high-puiity distillation columns. IEEE Trans. Autom. Control, AC-33,1092-1105. Wilson, D. A. (1989). Convolution and Hankel norms for Thus, the system of Fig. 1 will have robust performance if linear systems. IEEE Trans. Autom. Control, AC-34, and only if the system of Fig. A.1 (with A, = 0) has robust 94-97. performance with respect to the uncertainties A, E D, and A, E Dpi, and inputs in the set {VE HZ’: llV11_ I 1). Define Y
,
,
Appendix-Proof of Theorem 1 First, note that the structure of the output norm implies each of the output components can be considered independently. That is, we can consider whether each of the systems of Fig. 1 with M replaced by M, and m = mj satisfies the robust performance objective. Thus, we assume, without loss of generality, that m = 1. Note that the set of allowable input vectors V={[vT
v: ...
v~~:v~EH~,I~v~[[,~~
Vi=l,...,n)
D, = {A, E HE*“: /A&
5 l}.
The equivalence between robust performance and robust stability (see Doyle et al., 1982) implies that the system in the gray box in Fig. A.1 will have robust performance if and only if the system of Fig. A.1 is robustly stable with respect to the uncertainties A,, ED,, Ai E Dpi and Api and 4 E D,. Form the matrix product A, = A, A, and note that Api E Dpr. The system shown in Fig. A.1 IS stable for all Ap E Dp, n} and Au E 0, if and only if it is also {AiEDpi;i=l,..., stable for all {A, E D*:i = 1, . . . , n} and A, E D,. Define ra,
0 1
ra,
0
...
0 1
is equivalently expressed as
IIAill,s 1, i = 1,. . . , n; i E I-E, Ilill- 551). Define D, as o,={Ai~H~Xm:IIAill,~l},
i=l,...,
n.
Noting that A = AdT, M in Fig. A.1 is robustly stable with respect to A if and only if TM is robustly stable with respect to A,,. But, from Doyle et al. (1982), robust stability of TM with respect to A., is equivalent to (7) being satisfied (for m = 1). The result for multiple blocks follows from earlier comments.