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Minimization of the Bode sensitivity integral
Systems & Control Letters 40 (2000) 191–195
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Minimization of the Bode sensitivity integral M.E. Halpern ∗ , R.J. Evans Department of Electrical and Electronic Engineering, University of Melbourne, Victoria 3010, Australia Received 1 July 1997; accepted 16 March 2000
Abstract In this paper we present two new results on Bode-type integrals. Firstly, we obtain, for a given scalar or multivariable continuous-time plant, the inÿmum of the Bode sensitivity integral which can be obtained with any stabilizing controller. The result involves the unstable plant poles and, perhaps surprisingly, a subset of the plant nonminimum phase zeros. Secondly, we obtain an apparently new expression for the Bode integral for the complementary sensitivity for a stable discrete-time c 2000 Elsevier Science B.V. All rights reserved. scalar system. Keywords: Bode sensitivity integral; Fundamental limitations; Strong stabilization; Feedback systems
1. Introduction When considering feedback controller design for a given plant, it is useful to know the extent to which achievable performance is limited by the plant. Such limitations are sometimes called fundamental limitations [10] and they apply, irrespective of the particular controller used. In some cases, they involve unstable plant poles, nonminimum phase plant zeros and time delays. Many results exposing the nature of these limitations are in [6], some more recent extensions are in [3,7,8,10,11]. An important result, not fundamental in the sense above, since it depends on the controller as well as the plant, is the Bode sensitivity integral, obtained by Bode [1] for stable open-loop systems and then extended by Freudenberg and Looze [5] to cover unstable open-loop systems as well. The extended version
Supported by the Australian Research Council. Corresponding author. Tel.: +61-3-8344-6791; fax: +61-3-8344-6678. E-mail addresses: [email protected] (M.E. Halpern), [email protected] (R.J. Evans). ∗
is as follows. Consider the standard unity feedback arrangement with an open-loop part PC where P(s) is the plant and C(s) is the controller. If the closed-loop system is stable and PC has relative degree greater than or equal to two, then the sensitivity function S = 1=(1 + PC) satisÿes the following relation: Z ∞ X log|S(j!)| d! = Rpi ; (1) 0
where the pi are all the unstable (closed right-half plane (CRHP)) open-loop poles (i.e. the poles of PC such that Rpi ¿0) and R denotes their real parts. We assume throughout this work that the relative degree of PC is greater than or equal to two and that P is strictly proper. This result (1) allows easy computation of logarithmic sensitivity integral for a given plant and stabilizing controller, but it does not indicate the limitation on sensitivity caused by the plant alone, nor does the e ect of plant nonminimum phase zeros appear directly. This last point is of interest since in any stable closed-loop system, the sensitivity function satisÿes interpolation constraints at all the plant
c 2000 Elsevier Science B.V. All rights reserved. 0167-6911/00/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 0 ) 0 0 0 2 3 - 2
192
M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
nonminimum phase zeros, namely S(zi ) = 1 for all plant zeros zi : Rzi ¿0; as well as satisfying S(pi ) = 0 for all unstable plant poles pi : Rpi ¿0. In this note we determine, for any given plant, how small (1) can be made through selection of a stabilizing controller. We denote the inÿmal value as JS , that is, we set Z JS =
inf
stabilising C(s)
0
∞
log|S(j!)| d!:
(2)
The result makes use of work on strong stabilization by Youla et al. [13]. We show that the value of JS involves only the unstable plant poles and a distinguished subset of the plant’s real nonminimum phase zeros. In other words, the inÿmal value of the integral is a fundamental quantity. We also ÿnd that in cases where the inÿmum is not attained, a design approaching the inÿmum has many undesirable features. We note that a Bode integral for the complementary sensitivity function T =PC=(1+PC) has been obtained in [8,10]. A second contribution of this paper is the use of Jensen’s Formula (see, e.g. [9,14]) to obtain an apparently new expression for the complementary sensitivity integral for scalar discrete-time systems.
3. Results 3.1. Sensitivity integral minimization for continuous scalar plant We require the following results, which are direct consequences of Theorem 1. Corollary 1. In any stable closed-loop system; PC satisÿes the PIP; i.e.; between each pair of real unstable zeros of PC lie an even number of real unstable poles of PC. Corollary 2. The smallest number of unstable controller poles sucient to stabilize a plant P is the smallest number required to make PC satisfy the PIP. We next distinguish a useful subset of the nonminimum phase zeros of a plant. Deÿnition. Denote each real plant zero at s¿0 which has an odd number of plant poles to its right before the next real unstable zero as i ; i = 1; 2; : : : ; l. Set = {i }li=1 . For example, for the plant P(s) =
2. Preliminaries The computation of JS , the inÿmum of the log sensitivity integral, makes use of the notion of strong stabilizability. Recall [13] that a strongly stabilizable plant is one that can be stabilized by a stable controller, and that all strictly proper strongly stabilizable plants are characterized according to the following theorem. See also [12]. Theorem 1. A strictly proper rational plant is strongly stabilizable if and only if it satisÿes the parity interlacing property; (PIP) such that between each pair of real unstable plant zeros; including those at s = ∞ lie an even number of real unstable plant poles. In [13], the notion is also extended to multivariable plants, and is as above, but with real poles and real zeros at s¿0 counted according to their McMillan degree, and with the additional condition |P(s)C(s)| 6≡ 0.
(s − 1:0)(s − 2:0) (s + 1:0)2 (s − 4:0)
there are no unstable poles between the zeros at s = 1:0; 2:0, but there is one between the zeros at s=2:0; ∞ so that l = 1 and 1 = 2:0; = {2:0}. Finally, before stating the main result, we rewrite the right-hand side of (2) to separate out the contribution from the plant and the controller. We set JS = JP + JC ;
P where the contribution of the plant is JP = R i , where the i ’s with R i ¿0 are the plant unstable P poles; and the controller contribution is JC = Rÿi , where the ÿi ’s with Rÿi ¿0 are the unstable poles of the controller. Clearly, for a given plant, the minimization of (1) involves choosing a stabilizing controller to make JC as small as possible. We present the main result next. Theorem 2. For a rational strictly proper stabilizable plant P(s); either (a) or (b) holds. (a) If P(s) is strongly stabilizable; JS = JP
M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
and the inÿmum in (2) is attained using any stable stabilizing controller. (b) If P(s) is not strongly stabilizable; J S = JP +
l X
i
i=1
and the inÿmum is not attained by any stabilizing compensator. Proof. Firstly, P(s) is either strongly stabilizable, or it is not. (a) By the deÿnition of strong stabilizability, there exists a stable stabilizing controller for the plant so that JC = 0 is possible. (b) In order to make PC satisfy the parity interlacing property, the number of open-loop poles of PC between each pair of real unstable zeros of PC must be made into an even number. This can be done only by the incorporation of unstable controller poles. The way to do this while minimising the sum of the real parts of the unstable controller poles is, at each occurrence of an odd number of plant poles between a pair of real unstable plant zeros, to add one real controller pole arbitrarily close to and greater than the smaller valued zero, which will be an element of . This gives the desired result ! l X X i + i : JS = i=1
Achieving the inÿmum would require l cancellations between CRHP plant zeros and controller poles to take place. Such cancellations preclude internal stability. Remarks. (1) For a rational strictly proper plant P(s) with no unstable hidden modes, JS = 0 if and only if P(s) has no poles in the ORHP (s: Rs ¿ 0). (2) It seems surprising that some plant CRHP zeros contribute to the value of JS , and other do not, even though at every CRHP zero, zi of P(s); log |S(zi )|=0. (3) As noted above, for plants which are not strongly stabilizable, the inÿmum is not attained. Even just approaching the inÿmum will require unstable near cancellations which are associated with poor stability robustness against plant uncertainty. Examples. The plant,
193
(s − 1)(s − 2) (s + 1)2 (s − 4) is not strongly stabilizable sinceP it has one pole P between zeros at s=2 and ∞. Then, i =4 and i =2 so that JP = 4; JC = 2 giving JS = 6. Making the plant strongly stabilizable by adding another pole at s = 4 gives (s − 1)(s − 2) ; P(s) = (s + 1)2 (s − 4)2 for which JS = JP = 8. P(s) =
3.2. Comparison with H∞ norm bounds A consequence of the fact that not all the nonminimum phase plant zeros contribute to JS is that, for any given plant, the value of JS does not indicate how large the minimum H∞ norm of the sensitivity function can be, since unstable near pole-zero cancellations can occur without a ecting JS . However, there are other criteria which are better described by JS than by the minimum H∞ norm of the sensitivity function. We illustrate this by examining the e ect of the relative positions of unstable poles and zeros for a simple plant. Assume we have a strictly proper plant with one unstable pole 1 ¿ 0 and one ÿnite nonminimum phase zero, z1 ¿ 0. In [8], it is suggested, based on considerations of bandwidth associated with pole and zero locations, that a plant with 1 ¿ z1 is very dicult to control. For this plant, we ÿnd using Theorem 2 that JS = ( 1 + z1 ). On the other hand, a plant which should be easier to control has z1 ¿ 1 giving JS = 1 (note here 1 is the smaller of z1 ; 1 ). In contrast to the behaviour of JS , which takes rather di erent values for these two plants, the minimum achievable H∞ norm of the sensitivity or the complementary sensitivity is the same for each plant. In fact, for each plant 1 + z1 1 + z1 ; minkT k∞ = ; minkSk∞ = | 1 − z1 | | 1 − z1 | so swapping the value of 1 and z1 has no e ect on the minimum achievable H∞ norms of these transfer functions. The fact that minkSk∞ = minkT k∞ is not of concern here. Several properties relating to maximal stability robustness depend on the minimal values of kSk∞ and kT k∞ . For example, maximum allowable H∞ bounded multiplicative plant uncertainty as well as the maximum achievable gain margin [4] both depend on the minimum achievable value of kT k∞ .
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M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
3.3. Sensitivity integral minimization for multivariable plant We begin by reproducing a result for the multivariable case from [6]. Here the plant and compensator transfer functions are matrices. We assume that P(s) ∈ Cn×m ; C(s) ∈ Cm×n , where Cn×m is the set of complex matrices with n rows and m columns. The sensitivity function is S =(I +PC)−1 . Then, a multivariable version of (1) [6, p. 89] is: assume the open-loop transfer function PC has entries which are rational functions with at least two more poles than zeros. Then if the closed-loop system is stable, the determinant of the sensitivity function must satisfy Z ∞ X log |det S(j!)| d! = Rpi ; (3) 0
where the pi ’s are all the poles, including multiplicities, of PC in the open right half plane (ORHP). Taken together with the multivariable result on strong stabilization from [13], stated previously, the minimum value of the integral (3) is found in the same way as for the scalar case shown in Theorem 2.
f(0) 6= 0. If 0 ¡ r ¡ R and if z1 ; z2 ; : : : ; zm are the zeros of f(z) in |z| ¡ r, listed according to their multiplicities, then Z 2 1 log |f(re j )| d = log |f(0)| 2 0 m X r ; (4) + log |zk | k=1
2
where j = −1. Since the functions we are interested in are analytic for some R ¿ 1, we are able to set r = 1 to obtain Z 2 1 log |f(e j )| d = log |f(0)| 2 0 m X 1 : (5) log + |zk | k=1
3.4. Scalar discrete-time systems
3.4.1. Sensitivity integral Although the discrete-time sensitivity integral is obtained in [11,10], we illustrate here how the result falls out directly from (5): Z 2 X 1 1 ; (6) log |S(e j )| d = log 2 0 |zk |
Here we make use of the following notation involving polynomials A; B; G; F. The plant is
where the zk are the unstable poles of PC, that is to say those roots of polynomial AF with magnitude less than one.
P(z) =
z q B(z) ; A(z)
where q¿1 is the plant transport delay, B(z) = b0 + b1 z+· · ·+bnb z nb ; b0 6= 0; A(z)=1+a1 z+· · ·+ana z na ; the controller is G(z) ; C(z) = F(z) where G(z) = g0 + g1 z + · · · + gng z ng ; F(z) = 1 + f1 z +· · ·+fnf z nf , where na ; nb ; ng and nf are suitable integers. With this notation convention, stable poles are at values of z: |z| ¿ 1. Sensitivity integrals for scalar discrete-time systems have been obtained in [11,10], using Poisson’s Integral. Here we use Jensen’s Formula, see [9,14] which gives a result for the sensitivity integral rather directly and also allows an apparently new result to be obtained for the complementary sensitivity integral. We note that Jensen’s Formula has been referred to [2] in the context of the continuous sensitivity integral, but we are unaware of its use to produce results for discrete-time systems. Jensen’s Formula is as follows [14]. Suppose f(z) is analytic for |z| ¡ R and that
k
3.4.2. Complementary sensitivity integral The complementary sensitivity function is z q BG : (7) T (z) = AF + z q BG Due to the plant delay of q samples, we have T (0)=0, precluding direct use of Jensen’s Formula. Following [9] we set T 0 (z) = T (z)=z q and apply (5) to T 0 , obtaining, since the value of the integral is una ected by dividing T by z q : Z 2 1 log |T (e j )| d = log |T 0 (0)| 2 0 X 1 ; (8) log + |zk | k
where the zk are the nonminimum phase zeros (i.e. |zk | ¡ 1) of BG. This result appears to be new and di ers from the frequency weighted integral in [10, p. 80]. Note the ÿrst term, log|T 0 (0)| in (8). There is no corresponding term in the sensitivity integral (6) because the plant delay q¿1 ensures that S(0) = 1.
M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
In contrast, the value of T 0 (0) is not ÿxed, but depends on the controller. For any q¿1; T 0 (0) = b0 g0 . In the case where q = 1; b0 g0 can be expressed simply in terms of open-loop poles (all zeros of AF) and closed-loop poles (all zeros of V deÿned next). Denoting the closed-loop characteristic polynomial by V (z) = 1 + v1 z + · · · + vnv z nv , we have AF + z q BG = V:
(9)
Equating coecients of z on both sides gives (10) b0 g0 = v1 − a1 − f1 : P P Note that a1 = − 1=(plant P poles); f1 = − 1= (controller poles) and v1 =− 1=(closed-loop poles), so that X X 1=(plant poles) + 1=(controller poles) b 0 g0 = X − 1=(closed-loop poles): (11) Thus if q = 1, then T 0 (0) can be considered as a measure of the total pole shift between open-loop and closed-loop. 4. Conclusions Although the Bode sensitivity integral involves all the unstable poles of both plant and controller, we have shown that the inÿmal value of the integral, over all stabilising controllers, depends only on the unstable plant poles and on a distinguished subset of the plant’s real nonminimum phase zeros, namely those lying immediately to the left of an odd number of real unstable plant poles, unseparated by any real plant zeros. In addition, we have obtained an apparently new expression for the Bode complementary sensitivity integral for scalar discrete-time systems.
195
References [1] H.W. Bode, Network Analysis and Feedback Ampliÿer Design, Van Nostrand, New York, 1945. [2] S. Boyd, C.A. Desoer, Subharmonic functions and performance bounds on linear time-invariant feedback systems, IMA J. Math. Control. Inform. 2 (1985) 153–170. [3] J. Chen, Sensitivity integral relations and design trade-o s in linear multivariable feedback systems, IEEE Trans. Automat. Control 40 (1995) 1700–1716. [4] J.C. Doyle, B.A. Francis, A.R. Tannenbaum, Feedback Control Theory, Macmillan, New York, 1992. [5] J.S. Freudenberg, D.P. Looze, Right half plane poles and zeros and design trade-o s in feedback systems, IEEE Trans. Automat. Control AC-30 (1985) 555–565. [6] J.S. Freudenberg, D.P. Looze, Frequency Domain Properties of Scalar and Multivariable Feedback Systems, Lecture Notes in Control and Information Science, Vol. 104, Springer, Berlin, 1988. [7] G.I. Gomez, G.C. Goodwin, Generalization of integral constraints on sensitivity to time-delay systems, IEEE Trans. Automat. Control 43 (1998) 1008–1012. [8] R.H. Middleton, Trade-o s in linear control system design, Automatica 27 (1991) 281–292. [9] W. Rudin, Real and Complex Analysis, McGraw-Hill, Singapore, 1987. [10] M.M. Seron, J.H. Braslavsky, G.C. Goodwin, Fundamental Limitations in Filtering and Control, Springer, London, 1997. [11] H.K. Sung, S. Hara, Properties of sensitivity and complementary sensitivity funcitons in single-input single-output digital control systems, Int. J. Control 48 (1988) 2419–2439. [12] M. Vidyasagar, Control Systems Synthesis: A Factorization Approach, MIT Press, Cambridge, MA, 1985. [13] D.C. Youla, J.J. Bongiorno Jr., C.N. Lu, Single-loop feedback-stabilization of linear multivariable dynamical plants, Automatica 10 (1974) 159–173. R 2 [14] R.M. Young, On Jensen’s formula and log |1 − ei | dÂ, 0 Amer. Math. Mon. 1 (1986) 44–45.
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Minimization of the Bode sensitivity integral M.E. Halpern ∗ , R.J. Evans Department of Electrical and Electronic Engineering, University of Melbourne, Victoria 3010, Australia Received 1 July 1997; accepted 16 March 2000
Abstract In this paper we present two new results on Bode-type integrals. Firstly, we obtain, for a given scalar or multivariable continuous-time plant, the inÿmum of the Bode sensitivity integral which can be obtained with any stabilizing controller. The result involves the unstable plant poles and, perhaps surprisingly, a subset of the plant nonminimum phase zeros. Secondly, we obtain an apparently new expression for the Bode integral for the complementary sensitivity for a stable discrete-time c 2000 Elsevier Science B.V. All rights reserved. scalar system. Keywords: Bode sensitivity integral; Fundamental limitations; Strong stabilization; Feedback systems
1. Introduction When considering feedback controller design for a given plant, it is useful to know the extent to which achievable performance is limited by the plant. Such limitations are sometimes called fundamental limitations [10] and they apply, irrespective of the particular controller used. In some cases, they involve unstable plant poles, nonminimum phase plant zeros and time delays. Many results exposing the nature of these limitations are in [6], some more recent extensions are in [3,7,8,10,11]. An important result, not fundamental in the sense above, since it depends on the controller as well as the plant, is the Bode sensitivity integral, obtained by Bode [1] for stable open-loop systems and then extended by Freudenberg and Looze [5] to cover unstable open-loop systems as well. The extended version
Supported by the Australian Research Council. Corresponding author. Tel.: +61-3-8344-6791; fax: +61-3-8344-6678. E-mail addresses: [email protected] (M.E. Halpern), [email protected] (R.J. Evans). ∗
is as follows. Consider the standard unity feedback arrangement with an open-loop part PC where P(s) is the plant and C(s) is the controller. If the closed-loop system is stable and PC has relative degree greater than or equal to two, then the sensitivity function S = 1=(1 + PC) satisÿes the following relation: Z ∞ X log|S(j!)| d! = Rpi ; (1) 0
where the pi are all the unstable (closed right-half plane (CRHP)) open-loop poles (i.e. the poles of PC such that Rpi ¿0) and R denotes their real parts. We assume throughout this work that the relative degree of PC is greater than or equal to two and that P is strictly proper. This result (1) allows easy computation of logarithmic sensitivity integral for a given plant and stabilizing controller, but it does not indicate the limitation on sensitivity caused by the plant alone, nor does the e ect of plant nonminimum phase zeros appear directly. This last point is of interest since in any stable closed-loop system, the sensitivity function satisÿes interpolation constraints at all the plant
c 2000 Elsevier Science B.V. All rights reserved. 0167-6911/00/$ - see front matter PII: S 0 1 6 7 - 6 9 1 1 ( 0 0 ) 0 0 0 2 3 - 2
192
M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
nonminimum phase zeros, namely S(zi ) = 1 for all plant zeros zi : Rzi ¿0; as well as satisfying S(pi ) = 0 for all unstable plant poles pi : Rpi ¿0. In this note we determine, for any given plant, how small (1) can be made through selection of a stabilizing controller. We denote the inÿmal value as JS , that is, we set Z JS =
inf
stabilising C(s)
0
∞
log|S(j!)| d!:
(2)
The result makes use of work on strong stabilization by Youla et al. [13]. We show that the value of JS involves only the unstable plant poles and a distinguished subset of the plant’s real nonminimum phase zeros. In other words, the inÿmal value of the integral is a fundamental quantity. We also ÿnd that in cases where the inÿmum is not attained, a design approaching the inÿmum has many undesirable features. We note that a Bode integral for the complementary sensitivity function T =PC=(1+PC) has been obtained in [8,10]. A second contribution of this paper is the use of Jensen’s Formula (see, e.g. [9,14]) to obtain an apparently new expression for the complementary sensitivity integral for scalar discrete-time systems.
3. Results 3.1. Sensitivity integral minimization for continuous scalar plant We require the following results, which are direct consequences of Theorem 1. Corollary 1. In any stable closed-loop system; PC satisÿes the PIP; i.e.; between each pair of real unstable zeros of PC lie an even number of real unstable poles of PC. Corollary 2. The smallest number of unstable controller poles sucient to stabilize a plant P is the smallest number required to make PC satisfy the PIP. We next distinguish a useful subset of the nonminimum phase zeros of a plant. Deÿnition. Denote each real plant zero at s¿0 which has an odd number of plant poles to its right before the next real unstable zero as i ; i = 1; 2; : : : ; l. Set = {i }li=1 . For example, for the plant P(s) =
2. Preliminaries The computation of JS , the inÿmum of the log sensitivity integral, makes use of the notion of strong stabilizability. Recall [13] that a strongly stabilizable plant is one that can be stabilized by a stable controller, and that all strictly proper strongly stabilizable plants are characterized according to the following theorem. See also [12]. Theorem 1. A strictly proper rational plant is strongly stabilizable if and only if it satisÿes the parity interlacing property; (PIP) such that between each pair of real unstable plant zeros; including those at s = ∞ lie an even number of real unstable plant poles. In [13], the notion is also extended to multivariable plants, and is as above, but with real poles and real zeros at s¿0 counted according to their McMillan degree, and with the additional condition |P(s)C(s)| 6≡ 0.
(s − 1:0)(s − 2:0) (s + 1:0)2 (s − 4:0)
there are no unstable poles between the zeros at s = 1:0; 2:0, but there is one between the zeros at s=2:0; ∞ so that l = 1 and 1 = 2:0; = {2:0}. Finally, before stating the main result, we rewrite the right-hand side of (2) to separate out the contribution from the plant and the controller. We set JS = JP + JC ;
P where the contribution of the plant is JP = R i , where the i ’s with R i ¿0 are the plant unstable P poles; and the controller contribution is JC = Rÿi , where the ÿi ’s with Rÿi ¿0 are the unstable poles of the controller. Clearly, for a given plant, the minimization of (1) involves choosing a stabilizing controller to make JC as small as possible. We present the main result next. Theorem 2. For a rational strictly proper stabilizable plant P(s); either (a) or (b) holds. (a) If P(s) is strongly stabilizable; JS = JP
M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
and the inÿmum in (2) is attained using any stable stabilizing controller. (b) If P(s) is not strongly stabilizable; J S = JP +
l X
i
i=1
and the inÿmum is not attained by any stabilizing compensator. Proof. Firstly, P(s) is either strongly stabilizable, or it is not. (a) By the deÿnition of strong stabilizability, there exists a stable stabilizing controller for the plant so that JC = 0 is possible. (b) In order to make PC satisfy the parity interlacing property, the number of open-loop poles of PC between each pair of real unstable zeros of PC must be made into an even number. This can be done only by the incorporation of unstable controller poles. The way to do this while minimising the sum of the real parts of the unstable controller poles is, at each occurrence of an odd number of plant poles between a pair of real unstable plant zeros, to add one real controller pole arbitrarily close to and greater than the smaller valued zero, which will be an element of . This gives the desired result ! l X X i + i : JS = i=1
Achieving the inÿmum would require l cancellations between CRHP plant zeros and controller poles to take place. Such cancellations preclude internal stability. Remarks. (1) For a rational strictly proper plant P(s) with no unstable hidden modes, JS = 0 if and only if P(s) has no poles in the ORHP (s: Rs ¿ 0). (2) It seems surprising that some plant CRHP zeros contribute to the value of JS , and other do not, even though at every CRHP zero, zi of P(s); log |S(zi )|=0. (3) As noted above, for plants which are not strongly stabilizable, the inÿmum is not attained. Even just approaching the inÿmum will require unstable near cancellations which are associated with poor stability robustness against plant uncertainty. Examples. The plant,
193
(s − 1)(s − 2) (s + 1)2 (s − 4) is not strongly stabilizable sinceP it has one pole P between zeros at s=2 and ∞. Then, i =4 and i =2 so that JP = 4; JC = 2 giving JS = 6. Making the plant strongly stabilizable by adding another pole at s = 4 gives (s − 1)(s − 2) ; P(s) = (s + 1)2 (s − 4)2 for which JS = JP = 8. P(s) =
3.2. Comparison with H∞ norm bounds A consequence of the fact that not all the nonminimum phase plant zeros contribute to JS is that, for any given plant, the value of JS does not indicate how large the minimum H∞ norm of the sensitivity function can be, since unstable near pole-zero cancellations can occur without a ecting JS . However, there are other criteria which are better described by JS than by the minimum H∞ norm of the sensitivity function. We illustrate this by examining the e ect of the relative positions of unstable poles and zeros for a simple plant. Assume we have a strictly proper plant with one unstable pole 1 ¿ 0 and one ÿnite nonminimum phase zero, z1 ¿ 0. In [8], it is suggested, based on considerations of bandwidth associated with pole and zero locations, that a plant with 1 ¿ z1 is very dicult to control. For this plant, we ÿnd using Theorem 2 that JS = ( 1 + z1 ). On the other hand, a plant which should be easier to control has z1 ¿ 1 giving JS = 1 (note here 1 is the smaller of z1 ; 1 ). In contrast to the behaviour of JS , which takes rather di erent values for these two plants, the minimum achievable H∞ norm of the sensitivity or the complementary sensitivity is the same for each plant. In fact, for each plant 1 + z1 1 + z1 ; minkT k∞ = ; minkSk∞ = | 1 − z1 | | 1 − z1 | so swapping the value of 1 and z1 has no e ect on the minimum achievable H∞ norms of these transfer functions. The fact that minkSk∞ = minkT k∞ is not of concern here. Several properties relating to maximal stability robustness depend on the minimal values of kSk∞ and kT k∞ . For example, maximum allowable H∞ bounded multiplicative plant uncertainty as well as the maximum achievable gain margin [4] both depend on the minimum achievable value of kT k∞ .
194
M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
3.3. Sensitivity integral minimization for multivariable plant We begin by reproducing a result for the multivariable case from [6]. Here the plant and compensator transfer functions are matrices. We assume that P(s) ∈ Cn×m ; C(s) ∈ Cm×n , where Cn×m is the set of complex matrices with n rows and m columns. The sensitivity function is S =(I +PC)−1 . Then, a multivariable version of (1) [6, p. 89] is: assume the open-loop transfer function PC has entries which are rational functions with at least two more poles than zeros. Then if the closed-loop system is stable, the determinant of the sensitivity function must satisfy Z ∞ X log |det S(j!)| d! = Rpi ; (3) 0
where the pi ’s are all the poles, including multiplicities, of PC in the open right half plane (ORHP). Taken together with the multivariable result on strong stabilization from [13], stated previously, the minimum value of the integral (3) is found in the same way as for the scalar case shown in Theorem 2.
f(0) 6= 0. If 0 ¡ r ¡ R and if z1 ; z2 ; : : : ; zm are the zeros of f(z) in |z| ¡ r, listed according to their multiplicities, then Z 2 1 log |f(re j )| d = log |f(0)| 2 0 m X r ; (4) + log |zk | k=1
2
where j = −1. Since the functions we are interested in are analytic for some R ¿ 1, we are able to set r = 1 to obtain Z 2 1 log |f(e j )| d = log |f(0)| 2 0 m X 1 : (5) log + |zk | k=1
3.4. Scalar discrete-time systems
3.4.1. Sensitivity integral Although the discrete-time sensitivity integral is obtained in [11,10], we illustrate here how the result falls out directly from (5): Z 2 X 1 1 ; (6) log |S(e j )| d = log 2 0 |zk |
Here we make use of the following notation involving polynomials A; B; G; F. The plant is
where the zk are the unstable poles of PC, that is to say those roots of polynomial AF with magnitude less than one.
P(z) =
z q B(z) ; A(z)
where q¿1 is the plant transport delay, B(z) = b0 + b1 z+· · ·+bnb z nb ; b0 6= 0; A(z)=1+a1 z+· · ·+ana z na ; the controller is G(z) ; C(z) = F(z) where G(z) = g0 + g1 z + · · · + gng z ng ; F(z) = 1 + f1 z +· · ·+fnf z nf , where na ; nb ; ng and nf are suitable integers. With this notation convention, stable poles are at values of z: |z| ¿ 1. Sensitivity integrals for scalar discrete-time systems have been obtained in [11,10], using Poisson’s Integral. Here we use Jensen’s Formula, see [9,14] which gives a result for the sensitivity integral rather directly and also allows an apparently new result to be obtained for the complementary sensitivity integral. We note that Jensen’s Formula has been referred to [2] in the context of the continuous sensitivity integral, but we are unaware of its use to produce results for discrete-time systems. Jensen’s Formula is as follows [14]. Suppose f(z) is analytic for |z| ¡ R and that
k
3.4.2. Complementary sensitivity integral The complementary sensitivity function is z q BG : (7) T (z) = AF + z q BG Due to the plant delay of q samples, we have T (0)=0, precluding direct use of Jensen’s Formula. Following [9] we set T 0 (z) = T (z)=z q and apply (5) to T 0 , obtaining, since the value of the integral is una ected by dividing T by z q : Z 2 1 log |T (e j )| d = log |T 0 (0)| 2 0 X 1 ; (8) log + |zk | k
where the zk are the nonminimum phase zeros (i.e. |zk | ¡ 1) of BG. This result appears to be new and di ers from the frequency weighted integral in [10, p. 80]. Note the ÿrst term, log|T 0 (0)| in (8). There is no corresponding term in the sensitivity integral (6) because the plant delay q¿1 ensures that S(0) = 1.
M.E. Halpern, R.J. Evans / Systems & Control Letters 40 (2000) 191–195
In contrast, the value of T 0 (0) is not ÿxed, but depends on the controller. For any q¿1; T 0 (0) = b0 g0 . In the case where q = 1; b0 g0 can be expressed simply in terms of open-loop poles (all zeros of AF) and closed-loop poles (all zeros of V deÿned next). Denoting the closed-loop characteristic polynomial by V (z) = 1 + v1 z + · · · + vnv z nv , we have AF + z q BG = V:
(9)
Equating coecients of z on both sides gives (10) b0 g0 = v1 − a1 − f1 : P P Note that a1 = − 1=(plant P poles); f1 = − 1= (controller poles) and v1 =− 1=(closed-loop poles), so that X X 1=(plant poles) + 1=(controller poles) b 0 g0 = X − 1=(closed-loop poles): (11) Thus if q = 1, then T 0 (0) can be considered as a measure of the total pole shift between open-loop and closed-loop. 4. Conclusions Although the Bode sensitivity integral involves all the unstable poles of both plant and controller, we have shown that the inÿmal value of the integral, over all stabilising controllers, depends only on the unstable plant poles and on a distinguished subset of the plant’s real nonminimum phase zeros, namely those lying immediately to the left of an odd number of real unstable plant poles, unseparated by any real plant zeros. In addition, we have obtained an apparently new expression for the Bode complementary sensitivity integral for scalar discrete-time systems.
195
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