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On the frequency response of scalar discrete-time systems
Automatica 35 (1999) 1843}1853
Technical Communique
On the frequency response of scalar discrete-time systems夽 Bernardo A. LeoH n de la Barra *, RauH l Prieto Department of Electrical Engineering, Universidad de Chile, P.O. Box 412-3, Santiago, Chile Synapsis S.A., Divisio& n de Servicios Enersis, Catedral 1284, Chile Received 3 June 1998; received in "nal form 8 March 1999
Abstract A detailed understanding of the gain and phase characteristics of discrete-time zeros and poles is obtained. This new knowledge extends and complements previous results available in the control systems and digital signal processing literature. The results introduced in the paper also substantiate the view that discrete-time zeros and poles have much more intricate frequency domain features than their continuous-time counterparts. A simple loop-shaping example is used to illustrate the feasibility of discrete-time control system design developed entirely in the z-plane. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Amplitude response; Bode diagrams; Discrete-time systems; Frequency-response characteristics; Frequency-response methods; Phase response; Z transformation
1. Introduction It is surprising that a thorough study of the frequency response of discrete-time zeros and poles is not available in the control systems and digital signal processing literature. Note that the control literature provides a number of reasons for this study not having been completed before. As stroK m and Wittenmark (1997) stated that frequency curves for discrete-time systems were more di$cult to draw since the pulse transfer functions were not rational functions in ih, but in e F. Franklin et al. (1998) pointed out that the amplitude plots did not approach the simple asymptotes used in the hand-plotting procedures developed by Bode (for the continuous time). Ogata (1995) went further by stating that the direct application of discrete-time frequency response methods was not worthy of consideration. It is fair to say that the above comments were quite relevant at a time when handplotting procedures were the main way of constructing such plots. On the other hand, since Nyquist loci can nowadays be displayed in polar, Bode or Nichols form within seconds
夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato. *Corresponding author. Tel.: #56-2-6784215; fax #56-2-6953881. E-mail address: [email protected] (B.A. LeoH n de la Barra)
by means of state-of-the-art graphical software and given that synthesis is also becoming increasingly automated, we argue that it is appropriate to shift attention from the means of constructing such loci to their interpretation. In fact, the results to be presented in this paper will be central to gaining insight into this interpretation. The authors also believe that there is a need for analysis and design methods which can be carried out entirely in a discrete-time domain. To the best of the authors' knowledge only Haack and Tomizuka (1991), Menq and Chen (1993), Menq and Xia (1993), Tomizuka (1987, 1989, 1993a,b), and Xia and Menq (1995) have explicitly used some speci"c features of the frequency response gain and phase characteristics of non-minimum-phase discrete-time zeros (in a precision tracking control setting). Not much else seems to be available in the systems and digital signal processing literature where Jackson (1991), Oppenheim and Schafer (1999), Oppenheim and Willsky (1997), Proakis and Manolakis (1996), Strum and Kirk (1988), and Zelniker and Taylor (1994), have mainly concentrated on graphically studying the frequency response of "rst- and second-order discrete-time systems for given locations of the zeros and poles, providing only partial insight into some of the basic features of discrete-time frequency responses. In this paper, we go further by studying in great detail the gain and phase characteristics of various types of discrete-time zeros, i.e., real and complex, minimum and
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 6 2 - X
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non-minimum-phase zeros are included. For this purpose we consider the polynomials described by z!a r(z, a)& , 1!a
a31!+1,
(1.1)
and z!2o cos(d) ) z#o c(z, d, o)& , 1#o!2o cos(d)
(1.2)
respectively, where 1 denotes the set of real numbers, 0(d(p, and o'0. Note that assuming dO0, dOp, and oO0 guarantees that the roots of c(z,d, o) are indeed di!erent complex conjugate numbers located at z"oe B and z"oe\ B, respectively, where i denotes the imaginary unit. By gaining a deep understanding of the frequency-response characteristics of these "rst- and second-order polynomials we will be in a position to develop signi"cant insight into the frequency response of general discrete-time linear time-invariant systems described by linear constant-coe$cient di!erence equations. In particular, the results in this paper will provide insight into the design of simple digital "lters by zeropole placement; see Proakis and Manolakis (1996) for some speci"c applications. Note that we will mainly concentrate on studying the frequency response of polynomials r(z, a) and c(z, d, o), paying special attention to the maximum and minimum values of their gain and angle characteristics. Corresponding statements for real and complex conjugate discrete-time poles can be easily obtained just by considering r\(z, a) and c\(z, d, o), respectively. As it will be seen, the results to be presented in the paper will substantiate the view that discrete-time zeros and poles exhibit much more intricate frequency domain features than their continuous-time counterparts. 2. Continuous-time gain and phase characteristics The frequency-response characteristics of the continuous-time counterparts of polynomials (1.1) and (1.2) are well known and constitute the backbone of some important frequency domain analysis and design methods, see e.g., Kuo (1995), and Franklin et al. (1994). Indeed, some of the advantages of working with frequency response in terms of continuous-time Bode diagrams have been explicitly stated in the control literature, see e.g., Franklin et al. (1998). 2.1. Gain and phase characteristics of real continuous-time zeros
Fig. 1. Gain and phase characteristics of a real continuous-time zero.
has a frequency response r (iu, b) whose magnitude increases monotonically with u, u'0. On the other hand, if b(0 (b'0), the phase of r (iu, b) increases (decreases) monotonically with u from 0 to p/2 (0 to !p/2), respectively. Fig. 1 illustrates both characteristics. 2.2. Gain and phase characteristics of complex conjugate continuous-time zeros Likewise, it is also known that s#2fu ) s#u L L , "f"(1, u '0, c (s, f, u )& (2.2) L L u L has a frequency response c (iu, f, u ) whose magnitude L increases monotonically from its DC gain value for all u, u'0, if and only if "f"51/(2. On the other hand, if "f" (1/(2, "c (iu, f, u )" "rst decreases monotonically L from its DC gain value to c &min +"c (iu, f, u )"," T SY L "c (iu\ , f, u )""2"f"(1!f(1, where u\ &u (1!2f, L L and then increases monotonically for all u'u\ . If f50 (f(0), the phase of c (iu, f, u ) increases (deL creases) monotonically with u, u'0, from 0 to p (0 to !p), respectively. Fig. 2 illustrates both characteristics. In the next section we will return to the discrete-time domain and develop an in-depth study of the frequencyresponse characteristics of polynomials r(z, a) and c(z, d, o). We will show that these properties are not as easily characterized as their continuous-time counterparts.
For comparison purposes with the discrete-time results that will be presented in Section 3 recall that
3. Discrete-time gain and phase characteristics
b!s r (s, b)& , bO0, b
The frequency response of polynomials r(z, a) and c(z, d, o) is obtained by making z"e F, h31. Note
(2.1)
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Fig. 3. ""r"" and r for !104a45, aO1. T
Fig. 2. Gain and phase characteristics of complex conjugate continuous-time zeros.
also that we can restrict ourselves to considering 04h4p. 3.1. Gain characteristic of real discrete-time zeros The following proposition describes the behaviour of the maximum and minimum gain values of r(e F, a) as a function of a. Proposition 1. ""r"" &max +"r(e F, a)", is given by XFXp
1, a40, ""r"" " ">?", a31>!+1,, \? and r &min +"r(e F, a)", is given by T XFXp ">?", a40, r " \? T 1, a31>!+1,,
(3.1)
In Fig. 3 we illustrate Eqs. (3.1) and (3.2) for !104a45, aO1. We can see from Fig. 3 that positive real discrete-time zeros may exhibit substantial amplifying e!ects, and that the `closera the zero is to z"1, the larger the value of ""r"" . On the other hand, non positive real discrete-time zeros do not display amplifying e!ects at any frequency. Note also that negative real discrete-time zeros may exhibit signi"cant attenuating e!ects, and that the `closera the zero is to z"!1, the larger the attenuation level. It can also be veri"ed that for a31>!+1,, "r(e F, a)" increases monotonically from "r(1, a)""r "1 to "r(!1, a)""""r"" '1, and that for T a(0, "r(e F, a)" decreases monotonically from "r(1, a)""""r"" "1 to "r(!1, a)""r (1, respectively. It T is then clear that positive (negative) real discrete-time zeros exhibit high (low) pass characteristics, respectively. Note that the latter behaviour has no counterpart in continuous time, where all real zeros display amplifying features at all frequencies, cf. top Fig. 1. In the top Figs. 4 and 5 we illustrate "r(e F, a)" for a number of values of a, "a"(1, and "a"'1, respectively. Note that the statements in the previous paragraph can also be veri"ed from these "gures, where a linear (as opposed to logarithmic) scale is being used on the horizontal axis.
(3.2)
respectively, where 1> denotes the set of positive real numbers. Proof. The proposition follows easily from a direct analysis of "r(e F, a)". 䊐
Since "r(e F, a)","r(e F, a\)", aO0, the indices ""r"" and r asso T ciated with a are the same as those of its mirror image a\. Note also that normalization at DC in Eq. (1.1) is not possible for a"1, and that "e F!1" monotonically increases from zero to two when h goes from 0 to p.
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2. If a"!1, arg+r(e F,!1), grows linearly from arg+r(1,!1),"0 to arg+r(!1,!1),"p/2, having a discontinuity of height !p at h"p. 3. If a"1, arg+[cos(h)!1]#i sin(h), grows linearly from p/2 to p on [0, p], having discontinuities of heights p and !p at h"0, and h"p, respectively. 4. For real discrete-time zeros satisfying a'1 (a(!1), arg+r(e F, a), decreases (increases) monotonically from arg+r(1, a),"0 to its minimum (maximum) value, which is located at h"arccos(a\), and is given by
arctan
!sign(a) (a!1
,
(3.4)
respectively. Finally, arg+r(e F, a), increases (decreases) monotonically from the latter value to arg+r(!1,a),"0, respectively. Fig. 4. Gain and phase of r(e F, a) for di!erent values of a, "a"(1.
Proof. The proposition follows easily from a direct analysis of the four quadrant inverse tangent function in Eq. (3.3). 䊐 The statements in Proposition 2 can be veri"ed from the bottom Figs. 4 and 5, respectively. Note also that for those real discrete-time zeros satisfying a'1 (a(!1), the `closera the zero is to z"1 (z"!1) the larger the peak value of its phase lag (lead), respectively, cf. Eq. (3.4). 3.3. Gain characteristic of complex conjugate discrete-time zeros The following theorem describes the behaviour of the maximum and minimum gain values of c(e F, d, o) as a function of d and o, for 0(d(p, and o'0, respectively. Fig. 5. Gain and phase of r(e F, a) for di!erent values of a, "a"'1.
3.2. Phase characteristic of real discrete-time zeros We have that
arg+r(e F, a),"arg
cos(h)!a sin(h) #i , aO1, 1!a 1!a
(3.3)
where the phase operator arg+ ) , corresponds to the four quadrant inverse tangent measured in radians. The following proposition describes the behaviour of arg+r(e F, a), as a function of a. Proposition 2. 1. For real discrete-time zeros satisfying "a"(1, arg+r(e F, a), increases monotonically from arg+r(1, a),"0 to arg+r(!1, a),"p.
Theorem 1. ""c"" &max +"c(e F, d, o)", is given by XFXp >M>M B, 0(d(p/2, ""c"" " >M\M B (3.5) 1, p/24d(p
and c &min +"c(e F, d, o)", is given by T XFXp 1, (d, o)3(G 6G ), M \ B , (d, o)3(G 6G ), (3.6) c" T >M\M B >M >M B, (d, o)3(G 6G ), >M\M B respectively, where the regions G , l"1,2, 6, are illusJ trated in Fig. 6, (d, o)3G means that oe B3G , and I I R[ ) ] (I[ ) ]) denotes real (imaginary) part, respectively. The regions G and G are bounded by the curve r""sec(a)"!"tan(a)", and the regions G and G are bounded by its mirror image r""sec(a)"#"tan(a)", 04a42p, respectively, where r and a denote the standard polar coordinates. Finally, the regions G and G are bounded by the unit circle and the above curves, respectively.
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Proof. It can be easily seen that (d/dh)+"c(e F, d, o)", vanishes if and only if 4o sin(h)[(1#o)cos(d)!2o cos(h)]"0.
(3.7)
Note that Eq. (3.7) has roots at h"0, h"p, and
1 1 h &arccos o# cos(d) , T 2 o
(3.8)
respectively. For h to be a valid solution, d and o need to T satisfy
1 1 o# cos(d) 41. 2 o
It can be shown that "(o#1/o)cos(d)""1 if and only if o""sec (d)"$"tan(d)". These loci have been illustrated in Fig. 6, where they were used to de"ne regions G , l"1,2, 6. It is also straightforward to verify that J " (o#)cos(d)"(1 if and only if oe B lies strictly outside M these curves, i.e., in regions G or G . By evaluating "c(e F, d, o)" at h"h , with T (d, o)3(G 6G ), it follows that "c(e FT, d, o)"""(o!1) ) sin(d)"/(1#o!2o cos(d))(1. The theorem follows by combining this latter result with the fact that "c(1, d, o)",1, and "c(!1, d, o)""(1#o#2o cos(d))/ (1#o!2o cos(d)), respectively. 䊐 It is clear from Eq. (3.5) that for 0(d(p/2, ""c"" "(1#o#2o cos(d))/(1#o!2o c os(d)) is strictly larger than one, determining that all complex conjugate discrete-time zeros having positive
Fig. 7. ""c"" for !34R[z], I[z]43.
real part exhibit amplifying e!ects at some frequencies. On the other hand, complex conjugate discrete-time zeros having non-positive real part display attenuating e!ects at all frequencies. To gain further insight into Eq. (3.5) in Fig. 7 we illustrate ""c"" for !34R[z], I[z]43. From Fig. 7 we verify that complex conjugate discrete-time zeros `closea to z"1 may display signi"cant amplifying effects. Since "c(e F, d, o)","c(e F, d, o\)", oO0, the indices ""c"" and c associated with (d, o) are the same as those of T its mirror image (d, o\). To get additional insight into Eq. (3.6) in Figs. 8}10 we take a closer look at the behaviour of c . From Figs. 8}10 T we see that complex conjugate discrete-time zeros `closea to z"!1 may display signi"cant attenuating e!ects. In particular, Eqs. (3.5) and (3.6) also reveal that it is possible to choose the pair (d, o) in such a way that "c(e F, d, o)" meets given values in ""c"" and c . T The following corollary characterizes the overall behaviour of "c(e F, d, o)", 04h4p, as a function of d and o. Corollary 1. 1. If (d, o)3(G 6G ), "c(e F, d, o)" increases monotonically from "c(1, d, o)""c "1 to "c(!1, d, o)""""c"" . T 2. If (d, o)3(G 6G ), "c(e F, d, o)" decreases monotonically from "c(1, d, o)""""c"" "1 to "c(!1, d, o)""c . T 3. If (d, o)3(G 6G ), "c(e F, d, o)" decreases monotonically from "c(1, d, o)""1 to "c(e FT, d, o)""c , and then T increases monotonically from this value to "c(!1, d, o)""(1#o#2o cos(d))/(1#o!2o cos( d)), where the angle h was given in Eq. (3.8). T In particular, if (d, o)3(G>6G>), where the super script # denotes the right half of the corresponding
Fig. 6. Key regions in the characterization of c . T
Note that the contour line associated to ""c"" , ""c"" '1, is a circle centered at ((""c"" #1)/(""c"" !1), 0) having radius 2(""c"" /(""c"" !1).
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Fig. 10. A view of c from region G . T
Fig. 8. A few contour lines for c , c (1. T T
Fig. 11. Gain and phase of c(e F, d, o) for distinctive values of (d, o), o(1. Fig. 9. An `aeriala view of the overall behaviour of c . T
region, "c(!1, d, o)""""c"" '1. On the other hand, if (d, o)3(G\6G\), where the superscript ! denotes the left-half of the corresponding region, "c(1, d, o)"" ""c"" "1. Proof. It follows as a direct consequence of Theorem 1 and its proof. 䊐
3.3.1. Distinctive frequency response magnitudes To get a better feel for the statements in Corollary 1, in the top Figs. 11 and 12 we illustrate representatives of the four distinctive frequency response magnitudes of polynomial (1.2). They qualitatively describe the behaviour of ""c"" and c according to Eqs. (3.5) and (3.6), respectively. T It is known from Corollary 1 that only the frequency response magnitude of those complex conjugate
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3.4.1. Sign behaviour of the real part of c(e F, d, o) It can easily be seen that R[c(1, d, o)],1, and that R[c(!1, d, o)]"(1#o#2o cos(d))/(1#o!2o cos( d)) represents the ratio between the squared distances from the complex numbers oe! B to z"!1 and z"1, respectively. In particular, if o cos(d)50 it follows that R[c(!1, d, o)]51. On the other hand, if o cos(d)(0 it is clear that 0(R[c(!1, d, o)](1. To determine the sign of R[c(e F, d, o)], for 0(h(p, we will study the behaviour of the numerator polynomial in Eq. (3.9) which can be seen to have roots at o cos(d) (ocos(d)!2(o!1) cos(h )& $ , 2 2
(3.11)
and a discriminant *(d, o)&ocos(d)!2(o!1) which can be shown to be negative outside the ellipse de"ned by x #y"1, 2
Fig. 12. Gain and phase of c(e F, d, o) for distinctive values of (d, o), o'1.
discrete-time zeros lying in region (G 6G ) displays what could be called a `notcha e!ect, where the notch frequency is given by Eq. (3.8). On the other hand, only those complex conjugate discrete-time zeros lying in region (G 6G ) are associated with a monotonically de creasing frequency response magnitude. Note that the latter behaviour has no counterpart in continuous time, where all complex conjugate zeros display amplifying features at some frequencies, cf. top Fig. 2. 3.4. Phase characteristic of complex conjugate discretetime zeros We will characterize the behaviour of arg+c(e F, d, o), by studying the sign of R[c(e F, d, o)] and I[c(e F, d, o)], for 04h4p, and as a function of d and o. Note that the real and imaginary parts of c(e F, d, o) are explicitly given by R[c(e F, d, o)] 2 cos(h)!2o cos(d) cos(h)#(o!1) " 1#o!2o cos(d)
(3.9)
and 2+cos(h)!o cos(d), sin(h) I[c(e F, d, o)]" , 1#o!2o cos(d)
(3.12)
where x and y are the standard rectangular coordinates. On the other hand, it is clear that *(d, o),0, and *(d, o)'0, on and inside the above ellipse, respectively. Note that the roots in Eq. (3.11) coincide on the ellipse and are given by o cos(d) cos(h )"cos(h )" , 2
(3.13)
which is a valid solution since "x"""o cos(d)"4(2 on the ellipse. It is also straightforward to verify that Eq. (3.11) de"nes valid solutions when (d, o) is strictly inside the ellipse. Thus, it has been established that R[c(e F, d, o)], 04h4p, is non-negative on and outside the ellipse (3.12). On the other hand, R[c(e F, d, o)] is negative on (h , h ) if (d, o) is strictly inside the ellipse. 3.4.2. Sign behaviour of the imaginary part of c(e F,d,o) Determining the sign of I[c(e F, d, o)] will turn out to be very simple. It is obvious from Eq. (3.10) that I[c(1, d, o)]"I[c(!1, d, o)],0. It is also clear that I[c(e F, d, o)]"0, where h &arccos+o cos(d), if and only if "o cos(d)"41. In particular, if "o cos(d)"(1, I[c(e F, d, o)] will be positive on (0, h ) and negative on (h , p), respectively. Note also that if o cos(d)51, I[c(e F, d, o)] will be non-positive for all h3[0, p]. On the other hand, if ocos(d)4!1, I[c(e F, d, o)] will be non-negative for all h3[0, p].
(3.10)
respectively. The conditions on the sign behaviour of the real and imaginary parts of c(e F, d, o) which will be obtained in this section will de"ne seven regions of the complex plane where arg+c(e F, d, o), displays di!erent qualitative and quantitative behaviour.
3.4.3. Overall behaviour of arg+c(e F,d,o), and key regions involved The conditions obtained in Sections 3.4.1 and 3.4.2 on the sign behaviour of the real and imaginary parts of c(e F, d, o), respectively, de"ne seven regions of the complex plane where arg+c(e F, d, o), displays di!erent
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2. If (d, o)3(P 6P ), arg+c(e F, d, o), is positive for 0(h(p, monotonically increasing on (0, h ), and monotonically decreasing on (h , p), respectively, where h &arccos [o#3]cos(d)#((o!1)([o!9]cos(d)#8) . 4o
Fig. 13. Key regions in the characterization of arg+c(e F, d, o),.
qualitative and quantitative behaviour. These regions are denoted by P , i"1,2, 7, and are illustrated in Fig. 13, G where we can identify the ellipse described by Eq. (3.12), the unit circle, together with the lines R[z]"1, and R[z]"!1, respectively. Note also that regions P , P , and P extend towards "z"PR. The following theorem characterizes the overall behaviour of arg+c(e F, d, o), as a function of d and o, and for 04h4p. In the theorem below we use u &min [arg+c(e F, d, o),],
XFXp and u &max [arg+c(e F, d, o),],
XFXp respectively. Recall that the phase behaviour in continuous-time is quite simple to describe, cf. Section 2.2. This is in sharp contrast with the intricate behaviour that the phase of a complex pair exhibits in discrete time. Theorem 2 substantiates this latter statement. Theorem 2. arg+c(1, d, o),,0 for all (d, o). If (d, o) is strictly outside P (the unit disk), arg+c(!1, d, o),,0. 1. If (d, o)3(P 6P ), arg+c(e F, d, o), is negative for 0(h(p, monotonically decreasing on (0, h ), and monotonically increasing on (h , p), respectively, where h &arccos+([o#3]cos(d) !((o!1)([o!9]cos(d)#8))/4o,. If (d, o)3P , !p/24u 40, and if (d, o)3P ,
!p4u 4!p/2, respectively.
If (d, o)3P , 04u 4p/2, and if (d, o)3P ,
p/24u 4p, respectively.
3. If (d, o)3P , arg+c(e F, d, o), is monotonically increasing on (0, h ), monotonically decreasing on (h , h ), and monotonically increasing on (h , p), respectively. Also 04u 4p/2, and !p/24u 40, respectively. In
particular, arg+c(e F, d, o), is non-negative on [0, h ] and non-positive on [h , p], respectively, where h was intro duced in Section 3.4.2. 4. If (d, o)3P , arg+c(e F, d, o), is monotonically increasing on (0, h ), monotonically decreasing on (h , h ), and monotonically increasing on (h , p), respectively. If (d, o)3P>, 04u 4p/2, and !p4u 4!p/2,
respectively. On the other hand, if (d, o)3P\, p/24u 4p, and !p/24u 40, respectively. In
particular, arg+c(e F, d, o), is non-negative on [0, h ] and non-positive on [h , p], respectively. 5. If (d, o)3P , arg+c(e F, d, o), is positive on (0, p) and monotonically increasing from 0 to 2p. In particular, complex conjugate discrete-time zeros located on the unit circle display the same behaviour but arg+c(e F, d, o), also exhibits a discontinuity of height p at h"d. Proof. Since arg+c(e F, d, o), is the four-quadrant inverse tangent function, which increases with h, 04h4p, it su$ces to study the ratio I[c(e F, d, o)] 2+cos(h)!o cos(d),sin(h) " . R[c(e F, d, o)] 2 cos(h)!2o cos(d)cos(h)#(o!1) It can be easily seen that (d/dh)+I[c(e F, d, o)]/ R[c(e F, d, o)], vanishes if and only if 2ocos(h)!o(o#3)cos(d) cos(h) #+1#o[2cos(d)!1],"0.
(3.14)
Note that Eq. (3.14) has roots at h &
arccos
[o#3]cos(d)G((o!1)([o!9]cos(d)#8) . 4o (3.15)
It is straightforward to verify that h +h , is not a valid solution for Eq. (3.14) if (d,o)3(P 6P ) +(P 6P ),, re spectively. Note that these regions are associated to o cos(d)(!1 +o cos(d)'1,, respectively, which can be
B.A. Leo& n de la Barra, R. Prieto / Automatica 35 (1999) 1843}1853
shown to determine that the argument in Eq. (3.15) takes values outside the interval [!1, 1]. On the other hand, it can be seen that h and h are both well de"ned by Eq. (3.15) if (d, o)3(P 6P ), where "o cos(d)"41, and o51. Note also that h and h are not valid solutions if (d, o)3P , where o(1. The growth conditions on arg+c(e F, d, o), can be established after a straightforward analysis of the sign of the polynomial in Eq. (3.14) for 04h4p, taking into account the particular behaviour at h"0>, h"h , h"h , and h"p\, respectively. Note that the zeros in region P are those which are both outside the ellipse, i.e., they satisfy R[c(e F, d, o)]50, and meet o cos(d)51, i.e., they also satisfy I[c(e F, d, o)]40. These conditions determine that the complex number c(e F, d, o), (d, o)3P , lies in the fourth quadrant, for 04h4p, which translates into !p/24arg+c(e F, d, o),40. Similar arguments can be used to determine the bounds on arg+c(e F, d, o), in the remaining regions. 䊐 The statements in Theorem 2 can be veri"ed from the bottom Figs. 11 and 12 where representatives of the distinctive phase characteristics of polynomial (1.2) have been illustrated. It is also evident from Theorem 2 that complex conjugate discrete-time zeros located strictly outside the unit circle may display both negative and positive phase shift. This obviously has no counterpart in continuous time, where all open right half plane complex conjugate zeros exhibit negative phase shift at all frequencies, cf. bottom Fig. 2.
4. An illustration of loop-shaping in the Z-domain Consider the unity feedback sampled-data control system illustrated in Fig. 14, which corresponds to Example 4-12 in Ogata (1995). Note that the open-loop plant is given by G(z)&0.01873 )
(z#0.9356) , (z!1)(z!0.8187)
when the sampling period satis"es ¹"0.2. The cascade controller is a "rst-order phase-lead compensator described by
H(z)&K
1!b z!a , 1!a z!b
(4.1)
where !1(b(a(1, and KO0, respectively. It is required to design H(z) such that the phase margin is 503, the gain margin is at least 10 dB, and the static velocity error constant K is 2 Hz, respectively. T It is well known that K "lim T X
(z!1)G(z)H(z) 1 " lim [K(z!1)G(z)], ¹z ¹ X
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Fig. 14. Sampled-data control system in Example 4-12 (Ogata, 1995).
Fig. 15. Bode diagram of K ) G(z).
which leads to K"2. From Fig. 15 the phase margin can be seen to be 31.23, determining that the additional phase-lead angle required from the compensator is 18.83. Considering the shift to the right in the gain crossover frequency due to the application of the lead compensator, we may assume that , the maximum phase-lead angle K needed, is approximately 283+0.489 rad. Note also that arg+H(e F), reaches its maximum positive value, arctan+(a!b)/((1!a)(1!b),, at hK &arccos+(a#b)/(1#ab),. Thus, making " K arctan+(a!b)/((1!a)(1!b), it follows that a!sin( ) K . b" 1!asin( ) K The new gain crossover frequency hK will be determined graphically from Fig. 15 according to
"K" (1#b)(1!a) "K ) G(e FK )"" K " "H(e F)" (1#a)(1!b)
1!sin( ) 1!sin(0.489) K" 1#sin( ) 1#sin(0.489) K +0.6+!4.43 (dB), "
This means that 9.23 has been added to compensate for the shift in the gain crossover frequency.
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tinuous-time counterparts. A simple loop-shaping example was used to illustrate the feasibility of discretetime control system design developed completely in the z-plane.
Acknowledgements
Fig. 16. Final Bode diagram.
leading to hK +0.1087p+0.341 rad. Using the fact that the phase-lead compensator's zero and pole are given by 1#sin( !hK ) K a" +0.812 cos(hK )#sin( ) K and cos( )!sin(hK ) K +0.554, b" cos( !hK ) K respectively, the controller (4.1) is explicitly given by
H(z)"2
1!0.554 z!0.812 z!0.812 "4.74 . 1!0.812 z!0.554 z!0.554
Fig. 16 illustrates the resulting Bode diagram, where the gain and phase margins can be seen to be 14.46 dB and 503, respectively. Note that the previous design procedure was completely developed in the Z-domain. Thus, there was no need for bilinear transformations which are known to bring along frequency axis distortions (Ogata, 1995). In fact, the above procedure closely reproduced the solution in Ogata (1995) with the added bene"t of being focused on a single design domain.
5. Conclusions A thorough understanding of the gain and phase characteristics of discrete-time zeros and poles has been obtained. This new knowledge has extended and complemented previous results available in the control systems and digital signal processing literature. The results presented in this paper have also substantiated the view that discrete-time zeros and poles exhibit much more intricate frequency domain features than their con-
The authors gratefully acknowledge the support of the ComisioH n Nacional de InvestigacioH n CientmH "ca y TecnoloH gica (CONICYT, Chile) through FONDECYT grant 1970783. The "rst author is also grateful for the continuing support of FundacioH n Andes, Chile, through its `Programa de EstadmH as de InvestigacioH n en el Extranjeroa. The authors also acknowledge insightful comments provided by Abbas Emami-Naeini, Graham Goodwin, Anna So!mH a HauksdoH ttir and Rick Middleton.
References As stroK m, K.J., & Wittenmark, B. (1997). Computer-controlled systems: Theory and design (3rd ed., pp. 305}306). Englewood Cli!s, NJ: Prentice-Hall. Franklin, G.F., Powell, J.D., & Emami-Naeini, A. (1994). Feedback control of dynamic systems (3rd ed., pp. 337}467, 613}615). Reading, MA: Addison-Wesley. Franklin, G.F., Powell, J.D., & Workman, M. (1998). Digital control of dynamic systems (3rd ed., pp. 31}41, 234}264). Reading, MA: Addison-Wesley. Haack, B., & Tomizuka, M. (1991). The e!ect of adding zeroes to feedforward controllers. ASME Journal of Dynamic Systems, Measurement, and Control, 113(1), 6}10. Jackson, L.B. (1991). Signals, systems and transforms (pp. 334}337). Reading, MA: Addison-Wesley. Kuo, B.C. (1995). Automatic control systems (7th ed., pp. 387}449, 539}663, 864}886). Englewood Cli!s, NJ: Prentice-Hall. Menq, C.-H., & Chen, J.-J. (1993). Precision tracking control of discrete time nonminimum-phase systems. ASME Journal of Dynamic Systems, Measurement, and Control, 115(2), 238}245. Menq, C.-H., & Xia, J.Z. (1993). Characterization and compensation of discrete-time non-minimum-phase systems for precision tracking control. International Journal of Systems Science, 24(6), 1189}1205. Ogata, K. (1995). Discrete-time control systems (2nd ed., pp. 174}182, 225}242). Englewood Cli!s, NJ: Prentice-Hall. Oppenheim, A.V., & Schafer, R. (with Buck, J.R.) (1999). Discrete-time signal processing (2nd ed., pp. 253}270). Englewood Cli!s, NJ: Prentice-Hall. Oppenheim, A.V., & Willsky, A.S. (with Nawab, S.H.) (1997). Signals & systems (2nd ed., pp. 461}472, 763}767). Englewood Cli!s, NJ: Prentice-Hall. Proakis, J.G., & Manolakis, D.G. (1996). Digital signal processing principles, algorithms and applications (3rd ed., pp. 305}354). Englewood Cli!s, NJ: Prentice-Hall. Strum, R., & Kirk, D. (1988). First principles of discrete systems and digital signal processing (pp. 262}263). Reading, MA: Addison-Wesley. Tomizuka, M. (1987). Zero-phase error tracking controller for digital control. ASME Journal of Dynamic Systems, Measurement, and Control, 109(1), 65}68. Tomizuka, M. (1989). Design of digital tracking controllers for manufacturing applications. Manufacturing Review, 2(2), 134}141.
B.A. Leo& n de la Barra, R. Prieto / Automatica 35 (1999) 1843}1853 Tomizuka, M. (1993a). On compensation of phase shift in discrete-time linear controller design. In M. Jamshidi, M. Mansour, & B.D.O. Anderson, Fundamentals of Discrete-Time Systems (pp. 449}456). TSI Press. Tomizuka, M. (1993b). On the design of digital tracking controllers. ASME Journal of Dynamic Systems, Measurement, and Control, 115(2), 412}418.
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Xia, J.Z., & Menq, C.-H. (1995). Precision tracking control of non-minimum phase systems with zero phase error. International Journal of Control, 61(4), 791}807. Zelniker, G., & Taylor, F.J. (1994). Advanced digital signal processing theory and applications (pp. 52}64). New York: Marcel Dekker.
Technical Communique
On the frequency response of scalar discrete-time systems夽 Bernardo A. LeoH n de la Barra *, RauH l Prieto Department of Electrical Engineering, Universidad de Chile, P.O. Box 412-3, Santiago, Chile Synapsis S.A., Divisio& n de Servicios Enersis, Catedral 1284, Chile Received 3 June 1998; received in "nal form 8 March 1999
Abstract A detailed understanding of the gain and phase characteristics of discrete-time zeros and poles is obtained. This new knowledge extends and complements previous results available in the control systems and digital signal processing literature. The results introduced in the paper also substantiate the view that discrete-time zeros and poles have much more intricate frequency domain features than their continuous-time counterparts. A simple loop-shaping example is used to illustrate the feasibility of discrete-time control system design developed entirely in the z-plane. 1999 Elsevier Science Ltd. All rights reserved. Keywords: Amplitude response; Bode diagrams; Discrete-time systems; Frequency-response characteristics; Frequency-response methods; Phase response; Z transformation
1. Introduction It is surprising that a thorough study of the frequency response of discrete-time zeros and poles is not available in the control systems and digital signal processing literature. Note that the control literature provides a number of reasons for this study not having been completed before. As stroK m and Wittenmark (1997) stated that frequency curves for discrete-time systems were more di$cult to draw since the pulse transfer functions were not rational functions in ih, but in e F. Franklin et al. (1998) pointed out that the amplitude plots did not approach the simple asymptotes used in the hand-plotting procedures developed by Bode (for the continuous time). Ogata (1995) went further by stating that the direct application of discrete-time frequency response methods was not worthy of consideration. It is fair to say that the above comments were quite relevant at a time when handplotting procedures were the main way of constructing such plots. On the other hand, since Nyquist loci can nowadays be displayed in polar, Bode or Nichols form within seconds
夽 This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Editor Peter Dorato. *Corresponding author. Tel.: #56-2-6784215; fax #56-2-6953881. E-mail address: [email protected] (B.A. LeoH n de la Barra)
by means of state-of-the-art graphical software and given that synthesis is also becoming increasingly automated, we argue that it is appropriate to shift attention from the means of constructing such loci to their interpretation. In fact, the results to be presented in this paper will be central to gaining insight into this interpretation. The authors also believe that there is a need for analysis and design methods which can be carried out entirely in a discrete-time domain. To the best of the authors' knowledge only Haack and Tomizuka (1991), Menq and Chen (1993), Menq and Xia (1993), Tomizuka (1987, 1989, 1993a,b), and Xia and Menq (1995) have explicitly used some speci"c features of the frequency response gain and phase characteristics of non-minimum-phase discrete-time zeros (in a precision tracking control setting). Not much else seems to be available in the systems and digital signal processing literature where Jackson (1991), Oppenheim and Schafer (1999), Oppenheim and Willsky (1997), Proakis and Manolakis (1996), Strum and Kirk (1988), and Zelniker and Taylor (1994), have mainly concentrated on graphically studying the frequency response of "rst- and second-order discrete-time systems for given locations of the zeros and poles, providing only partial insight into some of the basic features of discrete-time frequency responses. In this paper, we go further by studying in great detail the gain and phase characteristics of various types of discrete-time zeros, i.e., real and complex, minimum and
0005-1098/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved. PII: S 0 0 0 5 - 1 0 9 8 ( 9 9 ) 0 0 0 6 2 - X
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non-minimum-phase zeros are included. For this purpose we consider the polynomials described by z!a r(z, a)& , 1!a
a31!+1,
(1.1)
and z!2o cos(d) ) z#o c(z, d, o)& , 1#o!2o cos(d)
(1.2)
respectively, where 1 denotes the set of real numbers, 0(d(p, and o'0. Note that assuming dO0, dOp, and oO0 guarantees that the roots of c(z,d, o) are indeed di!erent complex conjugate numbers located at z"oe B and z"oe\ B, respectively, where i denotes the imaginary unit. By gaining a deep understanding of the frequency-response characteristics of these "rst- and second-order polynomials we will be in a position to develop signi"cant insight into the frequency response of general discrete-time linear time-invariant systems described by linear constant-coe$cient di!erence equations. In particular, the results in this paper will provide insight into the design of simple digital "lters by zeropole placement; see Proakis and Manolakis (1996) for some speci"c applications. Note that we will mainly concentrate on studying the frequency response of polynomials r(z, a) and c(z, d, o), paying special attention to the maximum and minimum values of their gain and angle characteristics. Corresponding statements for real and complex conjugate discrete-time poles can be easily obtained just by considering r\(z, a) and c\(z, d, o), respectively. As it will be seen, the results to be presented in the paper will substantiate the view that discrete-time zeros and poles exhibit much more intricate frequency domain features than their continuous-time counterparts. 2. Continuous-time gain and phase characteristics The frequency-response characteristics of the continuous-time counterparts of polynomials (1.1) and (1.2) are well known and constitute the backbone of some important frequency domain analysis and design methods, see e.g., Kuo (1995), and Franklin et al. (1994). Indeed, some of the advantages of working with frequency response in terms of continuous-time Bode diagrams have been explicitly stated in the control literature, see e.g., Franklin et al. (1998). 2.1. Gain and phase characteristics of real continuous-time zeros
Fig. 1. Gain and phase characteristics of a real continuous-time zero.
has a frequency response r (iu, b) whose magnitude increases monotonically with u, u'0. On the other hand, if b(0 (b'0), the phase of r (iu, b) increases (decreases) monotonically with u from 0 to p/2 (0 to !p/2), respectively. Fig. 1 illustrates both characteristics. 2.2. Gain and phase characteristics of complex conjugate continuous-time zeros Likewise, it is also known that s#2fu ) s#u L L , "f"(1, u '0, c (s, f, u )& (2.2) L L u L has a frequency response c (iu, f, u ) whose magnitude L increases monotonically from its DC gain value for all u, u'0, if and only if "f"51/(2. On the other hand, if "f" (1/(2, "c (iu, f, u )" "rst decreases monotonically L from its DC gain value to c &min +"c (iu, f, u )"," T SY L "c (iu\ , f, u )""2"f"(1!f(1, where u\ &u (1!2f, L L and then increases monotonically for all u'u\ . If f50 (f(0), the phase of c (iu, f, u ) increases (deL creases) monotonically with u, u'0, from 0 to p (0 to !p), respectively. Fig. 2 illustrates both characteristics. In the next section we will return to the discrete-time domain and develop an in-depth study of the frequencyresponse characteristics of polynomials r(z, a) and c(z, d, o). We will show that these properties are not as easily characterized as their continuous-time counterparts.
For comparison purposes with the discrete-time results that will be presented in Section 3 recall that
3. Discrete-time gain and phase characteristics
b!s r (s, b)& , bO0, b
The frequency response of polynomials r(z, a) and c(z, d, o) is obtained by making z"e F, h31. Note
(2.1)
B.A. Leo& n de la Barra, R. Prieto / Automatica 35 (1999) 1843}1853
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Fig. 3. ""r"" and r for !104a45, aO1. T
Fig. 2. Gain and phase characteristics of complex conjugate continuous-time zeros.
also that we can restrict ourselves to considering 04h4p. 3.1. Gain characteristic of real discrete-time zeros The following proposition describes the behaviour of the maximum and minimum gain values of r(e F, a) as a function of a. Proposition 1. ""r"" &max +"r(e F, a)", is given by XFXp
1, a40, ""r"" " ">?", a31>!+1,, \? and r &min +"r(e F, a)", is given by T XFXp ">?", a40, r " \? T 1, a31>!+1,,
(3.1)
In Fig. 3 we illustrate Eqs. (3.1) and (3.2) for !104a45, aO1. We can see from Fig. 3 that positive real discrete-time zeros may exhibit substantial amplifying e!ects, and that the `closera the zero is to z"1, the larger the value of ""r"" . On the other hand, non positive real discrete-time zeros do not display amplifying e!ects at any frequency. Note also that negative real discrete-time zeros may exhibit signi"cant attenuating e!ects, and that the `closera the zero is to z"!1, the larger the attenuation level. It can also be veri"ed that for a31>!+1,, "r(e F, a)" increases monotonically from "r(1, a)""r "1 to "r(!1, a)""""r"" '1, and that for T a(0, "r(e F, a)" decreases monotonically from "r(1, a)""""r"" "1 to "r(!1, a)""r (1, respectively. It T is then clear that positive (negative) real discrete-time zeros exhibit high (low) pass characteristics, respectively. Note that the latter behaviour has no counterpart in continuous time, where all real zeros display amplifying features at all frequencies, cf. top Fig. 1. In the top Figs. 4 and 5 we illustrate "r(e F, a)" for a number of values of a, "a"(1, and "a"'1, respectively. Note that the statements in the previous paragraph can also be veri"ed from these "gures, where a linear (as opposed to logarithmic) scale is being used on the horizontal axis.
(3.2)
respectively, where 1> denotes the set of positive real numbers. Proof. The proposition follows easily from a direct analysis of "r(e F, a)". 䊐
Since "r(e F, a)","r(e F, a\)", aO0, the indices ""r"" and r asso T ciated with a are the same as those of its mirror image a\. Note also that normalization at DC in Eq. (1.1) is not possible for a"1, and that "e F!1" monotonically increases from zero to two when h goes from 0 to p.
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2. If a"!1, arg+r(e F,!1), grows linearly from arg+r(1,!1),"0 to arg+r(!1,!1),"p/2, having a discontinuity of height !p at h"p. 3. If a"1, arg+[cos(h)!1]#i sin(h), grows linearly from p/2 to p on [0, p], having discontinuities of heights p and !p at h"0, and h"p, respectively. 4. For real discrete-time zeros satisfying a'1 (a(!1), arg+r(e F, a), decreases (increases) monotonically from arg+r(1, a),"0 to its minimum (maximum) value, which is located at h"arccos(a\), and is given by
arctan
!sign(a) (a!1
,
(3.4)
respectively. Finally, arg+r(e F, a), increases (decreases) monotonically from the latter value to arg+r(!1,a),"0, respectively. Fig. 4. Gain and phase of r(e F, a) for di!erent values of a, "a"(1.
Proof. The proposition follows easily from a direct analysis of the four quadrant inverse tangent function in Eq. (3.3). 䊐 The statements in Proposition 2 can be veri"ed from the bottom Figs. 4 and 5, respectively. Note also that for those real discrete-time zeros satisfying a'1 (a(!1), the `closera the zero is to z"1 (z"!1) the larger the peak value of its phase lag (lead), respectively, cf. Eq. (3.4). 3.3. Gain characteristic of complex conjugate discrete-time zeros The following theorem describes the behaviour of the maximum and minimum gain values of c(e F, d, o) as a function of d and o, for 0(d(p, and o'0, respectively. Fig. 5. Gain and phase of r(e F, a) for di!erent values of a, "a"'1.
3.2. Phase characteristic of real discrete-time zeros We have that
arg+r(e F, a),"arg
cos(h)!a sin(h) #i , aO1, 1!a 1!a
(3.3)
where the phase operator arg+ ) , corresponds to the four quadrant inverse tangent measured in radians. The following proposition describes the behaviour of arg+r(e F, a), as a function of a. Proposition 2. 1. For real discrete-time zeros satisfying "a"(1, arg+r(e F, a), increases monotonically from arg+r(1, a),"0 to arg+r(!1, a),"p.
Theorem 1. ""c"" &max +"c(e F, d, o)", is given by XFXp >M>M B, 0(d(p/2, ""c"" " >M\M B (3.5) 1, p/24d(p
and c &min +"c(e F, d, o)", is given by T XFXp 1, (d, o)3(G 6G ), M \ B , (d, o)3(G 6G ), (3.6) c" T >M\M B >M >M B, (d, o)3(G 6G ), >M\M B respectively, where the regions G , l"1,2, 6, are illusJ trated in Fig. 6, (d, o)3G means that oe B3G , and I I R[ ) ] (I[ ) ]) denotes real (imaginary) part, respectively. The regions G and G are bounded by the curve r""sec(a)"!"tan(a)", and the regions G and G are bounded by its mirror image r""sec(a)"#"tan(a)", 04a42p, respectively, where r and a denote the standard polar coordinates. Finally, the regions G and G are bounded by the unit circle and the above curves, respectively.
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Proof. It can be easily seen that (d/dh)+"c(e F, d, o)", vanishes if and only if 4o sin(h)[(1#o)cos(d)!2o cos(h)]"0.
(3.7)
Note that Eq. (3.7) has roots at h"0, h"p, and
1 1 h &arccos o# cos(d) , T 2 o
(3.8)
respectively. For h to be a valid solution, d and o need to T satisfy
1 1 o# cos(d) 41. 2 o
It can be shown that "(o#1/o)cos(d)""1 if and only if o""sec (d)"$"tan(d)". These loci have been illustrated in Fig. 6, where they were used to de"ne regions G , l"1,2, 6. It is also straightforward to verify that J " (o#)cos(d)"(1 if and only if oe B lies strictly outside M these curves, i.e., in regions G or G . By evaluating "c(e F, d, o)" at h"h , with T (d, o)3(G 6G ), it follows that "c(e FT, d, o)"""(o!1) ) sin(d)"/(1#o!2o cos(d))(1. The theorem follows by combining this latter result with the fact that "c(1, d, o)",1, and "c(!1, d, o)""(1#o#2o cos(d))/ (1#o!2o cos(d)), respectively. 䊐 It is clear from Eq. (3.5) that for 0(d(p/2, ""c"" "(1#o#2o cos(d))/(1#o!2o c os(d)) is strictly larger than one, determining that all complex conjugate discrete-time zeros having positive
Fig. 7. ""c"" for !34R[z], I[z]43.
real part exhibit amplifying e!ects at some frequencies. On the other hand, complex conjugate discrete-time zeros having non-positive real part display attenuating e!ects at all frequencies. To gain further insight into Eq. (3.5) in Fig. 7 we illustrate ""c"" for !34R[z], I[z]43. From Fig. 7 we verify that complex conjugate discrete-time zeros `closea to z"1 may display signi"cant amplifying effects. Since "c(e F, d, o)","c(e F, d, o\)", oO0, the indices ""c"" and c associated with (d, o) are the same as those of T its mirror image (d, o\). To get additional insight into Eq. (3.6) in Figs. 8}10 we take a closer look at the behaviour of c . From Figs. 8}10 T we see that complex conjugate discrete-time zeros `closea to z"!1 may display signi"cant attenuating e!ects. In particular, Eqs. (3.5) and (3.6) also reveal that it is possible to choose the pair (d, o) in such a way that "c(e F, d, o)" meets given values in ""c"" and c . T The following corollary characterizes the overall behaviour of "c(e F, d, o)", 04h4p, as a function of d and o. Corollary 1. 1. If (d, o)3(G 6G ), "c(e F, d, o)" increases monotonically from "c(1, d, o)""c "1 to "c(!1, d, o)""""c"" . T 2. If (d, o)3(G 6G ), "c(e F, d, o)" decreases monotonically from "c(1, d, o)""""c"" "1 to "c(!1, d, o)""c . T 3. If (d, o)3(G 6G ), "c(e F, d, o)" decreases monotonically from "c(1, d, o)""1 to "c(e FT, d, o)""c , and then T increases monotonically from this value to "c(!1, d, o)""(1#o#2o cos(d))/(1#o!2o cos( d)), where the angle h was given in Eq. (3.8). T In particular, if (d, o)3(G>6G>), where the super script # denotes the right half of the corresponding
Fig. 6. Key regions in the characterization of c . T
Note that the contour line associated to ""c"" , ""c"" '1, is a circle centered at ((""c"" #1)/(""c"" !1), 0) having radius 2(""c"" /(""c"" !1).
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Fig. 10. A view of c from region G . T
Fig. 8. A few contour lines for c , c (1. T T
Fig. 11. Gain and phase of c(e F, d, o) for distinctive values of (d, o), o(1. Fig. 9. An `aeriala view of the overall behaviour of c . T
region, "c(!1, d, o)""""c"" '1. On the other hand, if (d, o)3(G\6G\), where the superscript ! denotes the left-half of the corresponding region, "c(1, d, o)"" ""c"" "1. Proof. It follows as a direct consequence of Theorem 1 and its proof. 䊐
3.3.1. Distinctive frequency response magnitudes To get a better feel for the statements in Corollary 1, in the top Figs. 11 and 12 we illustrate representatives of the four distinctive frequency response magnitudes of polynomial (1.2). They qualitatively describe the behaviour of ""c"" and c according to Eqs. (3.5) and (3.6), respectively. T It is known from Corollary 1 that only the frequency response magnitude of those complex conjugate
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3.4.1. Sign behaviour of the real part of c(e F, d, o) It can easily be seen that R[c(1, d, o)],1, and that R[c(!1, d, o)]"(1#o#2o cos(d))/(1#o!2o cos( d)) represents the ratio between the squared distances from the complex numbers oe! B to z"!1 and z"1, respectively. In particular, if o cos(d)50 it follows that R[c(!1, d, o)]51. On the other hand, if o cos(d)(0 it is clear that 0(R[c(!1, d, o)](1. To determine the sign of R[c(e F, d, o)], for 0(h(p, we will study the behaviour of the numerator polynomial in Eq. (3.9) which can be seen to have roots at o cos(d) (ocos(d)!2(o!1) cos(h )& $ , 2 2
(3.11)
and a discriminant *(d, o)&ocos(d)!2(o!1) which can be shown to be negative outside the ellipse de"ned by x #y"1, 2
Fig. 12. Gain and phase of c(e F, d, o) for distinctive values of (d, o), o'1.
discrete-time zeros lying in region (G 6G ) displays what could be called a `notcha e!ect, where the notch frequency is given by Eq. (3.8). On the other hand, only those complex conjugate discrete-time zeros lying in region (G 6G ) are associated with a monotonically de creasing frequency response magnitude. Note that the latter behaviour has no counterpart in continuous time, where all complex conjugate zeros display amplifying features at some frequencies, cf. top Fig. 2. 3.4. Phase characteristic of complex conjugate discretetime zeros We will characterize the behaviour of arg+c(e F, d, o), by studying the sign of R[c(e F, d, o)] and I[c(e F, d, o)], for 04h4p, and as a function of d and o. Note that the real and imaginary parts of c(e F, d, o) are explicitly given by R[c(e F, d, o)] 2 cos(h)!2o cos(d) cos(h)#(o!1) " 1#o!2o cos(d)
(3.9)
and 2+cos(h)!o cos(d), sin(h) I[c(e F, d, o)]" , 1#o!2o cos(d)
(3.12)
where x and y are the standard rectangular coordinates. On the other hand, it is clear that *(d, o),0, and *(d, o)'0, on and inside the above ellipse, respectively. Note that the roots in Eq. (3.11) coincide on the ellipse and are given by o cos(d) cos(h )"cos(h )" , 2
(3.13)
which is a valid solution since "x"""o cos(d)"4(2 on the ellipse. It is also straightforward to verify that Eq. (3.11) de"nes valid solutions when (d, o) is strictly inside the ellipse. Thus, it has been established that R[c(e F, d, o)], 04h4p, is non-negative on and outside the ellipse (3.12). On the other hand, R[c(e F, d, o)] is negative on (h , h ) if (d, o) is strictly inside the ellipse. 3.4.2. Sign behaviour of the imaginary part of c(e F,d,o) Determining the sign of I[c(e F, d, o)] will turn out to be very simple. It is obvious from Eq. (3.10) that I[c(1, d, o)]"I[c(!1, d, o)],0. It is also clear that I[c(e F, d, o)]"0, where h &arccos+o cos(d), if and only if "o cos(d)"41. In particular, if "o cos(d)"(1, I[c(e F, d, o)] will be positive on (0, h ) and negative on (h , p), respectively. Note also that if o cos(d)51, I[c(e F, d, o)] will be non-positive for all h3[0, p]. On the other hand, if ocos(d)4!1, I[c(e F, d, o)] will be non-negative for all h3[0, p].
(3.10)
respectively. The conditions on the sign behaviour of the real and imaginary parts of c(e F, d, o) which will be obtained in this section will de"ne seven regions of the complex plane where arg+c(e F, d, o), displays di!erent qualitative and quantitative behaviour.
3.4.3. Overall behaviour of arg+c(e F,d,o), and key regions involved The conditions obtained in Sections 3.4.1 and 3.4.2 on the sign behaviour of the real and imaginary parts of c(e F, d, o), respectively, de"ne seven regions of the complex plane where arg+c(e F, d, o), displays di!erent
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B.A. Leo& n de la Barra, R. Prieto / Automatica 35 (1999) 1843}1853
2. If (d, o)3(P 6P ), arg+c(e F, d, o), is positive for 0(h(p, monotonically increasing on (0, h ), and monotonically decreasing on (h , p), respectively, where h &arccos [o#3]cos(d)#((o!1)([o!9]cos(d)#8) . 4o
Fig. 13. Key regions in the characterization of arg+c(e F, d, o),.
qualitative and quantitative behaviour. These regions are denoted by P , i"1,2, 7, and are illustrated in Fig. 13, G where we can identify the ellipse described by Eq. (3.12), the unit circle, together with the lines R[z]"1, and R[z]"!1, respectively. Note also that regions P , P , and P extend towards "z"PR. The following theorem characterizes the overall behaviour of arg+c(e F, d, o), as a function of d and o, and for 04h4p. In the theorem below we use u &min [arg+c(e F, d, o),],
XFXp and u &max [arg+c(e F, d, o),],
XFXp respectively. Recall that the phase behaviour in continuous-time is quite simple to describe, cf. Section 2.2. This is in sharp contrast with the intricate behaviour that the phase of a complex pair exhibits in discrete time. Theorem 2 substantiates this latter statement. Theorem 2. arg+c(1, d, o),,0 for all (d, o). If (d, o) is strictly outside P (the unit disk), arg+c(!1, d, o),,0. 1. If (d, o)3(P 6P ), arg+c(e F, d, o), is negative for 0(h(p, monotonically decreasing on (0, h ), and monotonically increasing on (h , p), respectively, where h &arccos+([o#3]cos(d) !((o!1)([o!9]cos(d)#8))/4o,. If (d, o)3P , !p/24u 40, and if (d, o)3P ,
!p4u 4!p/2, respectively.
If (d, o)3P , 04u 4p/2, and if (d, o)3P ,
p/24u 4p, respectively.
3. If (d, o)3P , arg+c(e F, d, o), is monotonically increasing on (0, h ), monotonically decreasing on (h , h ), and monotonically increasing on (h , p), respectively. Also 04u 4p/2, and !p/24u 40, respectively. In
particular, arg+c(e F, d, o), is non-negative on [0, h ] and non-positive on [h , p], respectively, where h was intro duced in Section 3.4.2. 4. If (d, o)3P , arg+c(e F, d, o), is monotonically increasing on (0, h ), monotonically decreasing on (h , h ), and monotonically increasing on (h , p), respectively. If (d, o)3P>, 04u 4p/2, and !p4u 4!p/2,
respectively. On the other hand, if (d, o)3P\, p/24u 4p, and !p/24u 40, respectively. In
particular, arg+c(e F, d, o), is non-negative on [0, h ] and non-positive on [h , p], respectively. 5. If (d, o)3P , arg+c(e F, d, o), is positive on (0, p) and monotonically increasing from 0 to 2p. In particular, complex conjugate discrete-time zeros located on the unit circle display the same behaviour but arg+c(e F, d, o), also exhibits a discontinuity of height p at h"d. Proof. Since arg+c(e F, d, o), is the four-quadrant inverse tangent function, which increases with h, 04h4p, it su$ces to study the ratio I[c(e F, d, o)] 2+cos(h)!o cos(d),sin(h) " . R[c(e F, d, o)] 2 cos(h)!2o cos(d)cos(h)#(o!1) It can be easily seen that (d/dh)+I[c(e F, d, o)]/ R[c(e F, d, o)], vanishes if and only if 2ocos(h)!o(o#3)cos(d) cos(h) #+1#o[2cos(d)!1],"0.
(3.14)
Note that Eq. (3.14) has roots at h &
arccos
[o#3]cos(d)G((o!1)([o!9]cos(d)#8) . 4o (3.15)
It is straightforward to verify that h +h , is not a valid solution for Eq. (3.14) if (d,o)3(P 6P ) +(P 6P ),, re spectively. Note that these regions are associated to o cos(d)(!1 +o cos(d)'1,, respectively, which can be
B.A. Leo& n de la Barra, R. Prieto / Automatica 35 (1999) 1843}1853
shown to determine that the argument in Eq. (3.15) takes values outside the interval [!1, 1]. On the other hand, it can be seen that h and h are both well de"ned by Eq. (3.15) if (d, o)3(P 6P ), where "o cos(d)"41, and o51. Note also that h and h are not valid solutions if (d, o)3P , where o(1. The growth conditions on arg+c(e F, d, o), can be established after a straightforward analysis of the sign of the polynomial in Eq. (3.14) for 04h4p, taking into account the particular behaviour at h"0>, h"h , h"h , and h"p\, respectively. Note that the zeros in region P are those which are both outside the ellipse, i.e., they satisfy R[c(e F, d, o)]50, and meet o cos(d)51, i.e., they also satisfy I[c(e F, d, o)]40. These conditions determine that the complex number c(e F, d, o), (d, o)3P , lies in the fourth quadrant, for 04h4p, which translates into !p/24arg+c(e F, d, o),40. Similar arguments can be used to determine the bounds on arg+c(e F, d, o), in the remaining regions. 䊐 The statements in Theorem 2 can be veri"ed from the bottom Figs. 11 and 12 where representatives of the distinctive phase characteristics of polynomial (1.2) have been illustrated. It is also evident from Theorem 2 that complex conjugate discrete-time zeros located strictly outside the unit circle may display both negative and positive phase shift. This obviously has no counterpart in continuous time, where all open right half plane complex conjugate zeros exhibit negative phase shift at all frequencies, cf. bottom Fig. 2.
4. An illustration of loop-shaping in the Z-domain Consider the unity feedback sampled-data control system illustrated in Fig. 14, which corresponds to Example 4-12 in Ogata (1995). Note that the open-loop plant is given by G(z)&0.01873 )
(z#0.9356) , (z!1)(z!0.8187)
when the sampling period satis"es ¹"0.2. The cascade controller is a "rst-order phase-lead compensator described by
H(z)&K
1!b z!a , 1!a z!b
(4.1)
where !1(b(a(1, and KO0, respectively. It is required to design H(z) such that the phase margin is 503, the gain margin is at least 10 dB, and the static velocity error constant K is 2 Hz, respectively. T It is well known that K "lim T X
(z!1)G(z)H(z) 1 " lim [K(z!1)G(z)], ¹z ¹ X
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Fig. 14. Sampled-data control system in Example 4-12 (Ogata, 1995).
Fig. 15. Bode diagram of K ) G(z).
which leads to K"2. From Fig. 15 the phase margin can be seen to be 31.23, determining that the additional phase-lead angle required from the compensator is 18.83. Considering the shift to the right in the gain crossover frequency due to the application of the lead compensator, we may assume that , the maximum phase-lead angle K needed, is approximately 283+0.489 rad. Note also that arg+H(e F), reaches its maximum positive value, arctan+(a!b)/((1!a)(1!b),, at hK &arccos+(a#b)/(1#ab),. Thus, making " K arctan+(a!b)/((1!a)(1!b), it follows that a!sin( ) K . b" 1!asin( ) K The new gain crossover frequency hK will be determined graphically from Fig. 15 according to
"K" (1#b)(1!a) "K ) G(e FK )"" K " "H(e F)" (1#a)(1!b)
1!sin( ) 1!sin(0.489) K" 1#sin( ) 1#sin(0.489) K +0.6+!4.43 (dB), "
This means that 9.23 has been added to compensate for the shift in the gain crossover frequency.
B.A. Leo& n de la Barra, R. Prieto / Automatica 35 (1999) 1843}1853
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tinuous-time counterparts. A simple loop-shaping example was used to illustrate the feasibility of discretetime control system design developed completely in the z-plane.
Acknowledgements
Fig. 16. Final Bode diagram.
leading to hK +0.1087p+0.341 rad. Using the fact that the phase-lead compensator's zero and pole are given by 1#sin( !hK ) K a" +0.812 cos(hK )#sin( ) K and cos( )!sin(hK ) K +0.554, b" cos( !hK ) K respectively, the controller (4.1) is explicitly given by
H(z)"2
1!0.554 z!0.812 z!0.812 "4.74 . 1!0.812 z!0.554 z!0.554
Fig. 16 illustrates the resulting Bode diagram, where the gain and phase margins can be seen to be 14.46 dB and 503, respectively. Note that the previous design procedure was completely developed in the Z-domain. Thus, there was no need for bilinear transformations which are known to bring along frequency axis distortions (Ogata, 1995). In fact, the above procedure closely reproduced the solution in Ogata (1995) with the added bene"t of being focused on a single design domain.
5. Conclusions A thorough understanding of the gain and phase characteristics of discrete-time zeros and poles has been obtained. This new knowledge has extended and complemented previous results available in the control systems and digital signal processing literature. The results presented in this paper have also substantiated the view that discrete-time zeros and poles exhibit much more intricate frequency domain features than their con-
The authors gratefully acknowledge the support of the ComisioH n Nacional de InvestigacioH n CientmH "ca y TecnoloH gica (CONICYT, Chile) through FONDECYT grant 1970783. The "rst author is also grateful for the continuing support of FundacioH n Andes, Chile, through its `Programa de EstadmH as de InvestigacioH n en el Extranjeroa. The authors also acknowledge insightful comments provided by Abbas Emami-Naeini, Graham Goodwin, Anna So!mH a HauksdoH ttir and Rick Middleton.
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