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Limit cycles analysis of reset control systems with reset band
Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Limit cycles analysis of reset control systems with reset band Alfonso Baños a,∗ , Sebastián Dormido b , Antonio Barreiro c a
Dpt. Informática y Sistemas, Univ. Murcia, 30071 Murcia, Spain
b
Dpt. Informática y Automática, U.N.E.D., 28040 Madrid, Spain
c
Dpt. Ingeniería de Sistemas y Automática, Univ. Vigo, 36200 Vigo, Spain
article
info
Article history: Received 27 November 2009 Accepted 25 July 2010 Keywords: Hybrid systems Stability of hybrid systems Impulsive systems Reset control systems
abstract The reset band is a simple idea, and a must in practice, to improve reset compensation by adding extra phase lead in a feedback loop. However, a formal treatment of how the reset band can affect stability and performance of a reset control system is still an open issue. This work approaches the problem of the existence and stability of limit cycles of reset control systems with reset band. A frequency domain approach is given by using standard methods based on the describing function. In addition, closed-form expressions have been obtained for the describing function of arbitrary order full reset compensators with and without reset band. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Reset control is a kind of impulsive/hybrid control in which some of the compensator states are set to zero at those instants in which its input is zero. Using [1] or the most recent work [2], reset systems (without external inputs) can be classified as autonomous systems with impulse effects, an area of systems theory that has been started to be developed in the last few years. However, in the control literature the development of reset control systems dates back to the work of [3], and was followed by the works [4,5], where very simple reset compensators, CI (Clegg integrator) and FORE (first order reset element) respectively, were analyzed and design procedures were developed. More general reset compensators, including partial and full reset compensation, have been studied for example in [6]. One of the main difficulties of reset compensation is that closed-loop stability may be not guaranteed if reset actions are not properly performed, and in fact it is well known that reset can unstabilize a base stable control system. Thus, stability of reset control systems is a main concern from a theoretical and practical point of view, and several recent works have approached the stability problem. See for example [6–8], and the works [9–11]. An important practical issue of reset control is that compensator implementation is usually done by using reset band. In addition, it has been noted that use of reset band may improve stability and performance in systems with time-delays [12], due to the phase lead characteristic that is common to reset compensators. Phase lead can be even improved by using reset band. However, a formal analysis of how the reset band can affect stability and performance of a reset control system is still an open issue. In this work, a frequency domain approach by means of the describing function will be used to approach an important question such as the existence and stability of limit cycles in reset control systems with reset band. The analysis with the describing function will follow standard methods (see for example [13]). Although it is well-known that the describing function may fail in some cases, it has been proved to be an efficient method and has been formally justified in many practical cases [14,15], including some types of switched systems [16].
∗
Corresponding author. E-mail addresses: [email protected] (A. Baños), [email protected] (S. Dormido), [email protected] (A. Barreiro).
1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.07.004
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A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
v
e
u P
C
y
Fig. 1. Reset compensator C and LTI plant P.
Fig. 2. Reset band.
The work is organized as follows. Section 2 introduces the problem setup, with a precise definition of a reset compensator with reset band. Note that single-input single-output systems are considered in this work. In Section 3, the describing function of reset control systems with reset band is calculated, in particular the describing function of CI and FORE with reset band are obtained. Section 4 approaches the existence and stability of limit cycles by using a describing functionbased analysis, and finally in Section 5 the describing function method is justified by exploring the harmonics structure of the sinuosidal response and the low pass condition. 2. Reset compensators with reset band The main motivation for the use of reset compensation is to improve the performance of a previously designed LTI control system, being the goal to reset some states of a base LTI compensator to improve control system performance, both in terms of velocity of response and relative stability. In general, these specifications will be impossible to achieve by means of LTI compensation. Reset control systems with reset band are feedback control systems (Fig. 1), where for the purposes of this work the plant P is described in general by a transfer function P (s), and the reset compensator C is given by the impulsive differential equation
x˙ r (t ) C : xr (t + ) v(t )
= = =
Ar xr (t ) + Br e(t ), Aρ xr (t ), Cr xr (t ) + Dr e(t )
(e(t ), e˙ (t )) ̸∈ Bδ (e(t ), e˙ (t )) ∈ Bδ
(1)
where the reset band surface Bδ is given by Bδ = {(x, y) ∈ R2 | (x = −δ ∧ y > 0) ∨ (x = δ ∧ y < 0)}, being δ some non-negative real number. In this way, the compensator states are reset at the instants in which its input is entering into − the reset band. In general, the reset band surface Bδ will consist of two reset lines B+ δ and Bδ in the plane, as shown in Fig. 2. In the particular case δ = 0, the standard reset compensator is obtained. On the other hand, if δ is big enough in relation to the error amplitude, then no reset action is produced and the reset compensator reduce to its base compensator, given by the transfer function Cbase (s) = Cr (sI − Ar )−1 Br + Dr .
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165
Fig. 3. FORE sinusoidal response.
3. Describing function The following notation will be used to distinguish between reset compensators with and without reset band: Cδ will denote a reset compensator with reset band Bδ , whereas C0 (or simply C ) is the standard reset compensator without reset band. Note that Cbase stands for the base reset compensator. By simplicity, the describing function (DF) of a FOREδ compensator is considered at a first instance. In addition, since DF computation will be based on FORE DF, this DF is considered first. 3.1. FORE K FORE is a simple reset compensator with a first order base compensator given by FOREbase (s) = s+ . As is well known a (see for example [13]), for a given system its DF is calculated as the balance between the fundamental component of its sinusoidal response and the sinousoidal input. The response v of the reset compensator FORE (with δ = 0 in (1)), to the sinusoidal input e(t ) = E sin(ωt ) is given in the s-domain simply by
Eω K (2) s + a s2 + ω 2 and the time response v(t ) will consist of two terms: one corresponding to the transient response given by the compensator modes, and other corresponding to the steady response given by the sinusoidal input modes. Since the reset instants will be given by the crossings of the sinusoidal input with zero, and thus they will be periodic with fundamental period 2π /ω (see Fig. 3), it turns out that v(t ) will be also periodic with that period. To compute the FORE describing function, a simple procedure is to add to the base frequency response the contribution of the transitory terms in v(t ) over a period, for example [0, 2ωπ ]. By symmetry (see Fig. 3), it will be computed by using the interval [0, πω ], where V (s) =
vt ( t ) =
KE ω a2 + ω 2
e−at .
(3)
The result is 1
π
π/ω 0
vt (t )e−jωt ωdt E 2j
=
j2K ω2
π
π (a + jω)(a2 + ω2 )
(1 + e−a ω )
(4)
and finally adding to the frequency response of the base system, the describing function of FORE results in FORE (ω) =
j2K ω2
π (a + jω)(a2 + ω2 )
π
(1 + e−a ω ) +
K a + jω
.
(5)
3.2. FOREδ To obtain the describing function with reset band, again the sinusoidal response needs to be calculated. In this case, note that since FOREδ is time-invariant, the FOREδ response to e(t ) = E sin(ωt ) over the period [−tδ , 2ωπ − tδ ] is the same (except a translation) that its response to e(t − tδ ) over the period [0, 2ωπ ] (see Fig. 4). Thus, FOREδ DF can be obtained as the contribution of the transitory terms in the response to e(t − tδ ) = E cos(ωtδ ) sin(ωt ) − E sin(ωtδ ) cos(ωt ), plus the base frequency response. But note that the transitory terms consists of two terms: one term v1t corresponding to the response to E cos(ωtδ ) sin(ωt ), and a second term vt2 with the response to −E sin(ωtδ ) cos(ωt ).
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Fig. 4. FORE with reset band sinusoidal response. KE cos(ωt )
By using (3), the term vt1 is directly given by vt1 = a2 +ω2 δ e−at , and thus using again symmetry arguments its harmonic balance (corresponding to the fundamental component) is given by 1
π 1 2π
π/ω 0
2π/ω 0
vt1 (t )e−jωt ωdt
e(t − tδ )e−jωt ωdt
=
π KE ω2 cos(ωtδ ) (1 + e−a ω ) π(a+jω)(a2 +ω2 ) . E −jωtδ e
On the other hand, the second transitory term is vt2 = − 1
π 1 2π
π/ω 0
2π/ω 0
vt2 (t )e−jωt ωdt
e(t − tδ )e−jωt ωdt
(6)
2j
=
KEa sin(ωtδ ) −at e , a2 +ω2
and its harmonic balance is now
π KEaω sin(ωtδ ) (1 + e−a ω ) π(a+jω)(a2 +ω2 ) . E −jωtδ e
(7)
2j
Finally, the FOREδ describing function is obtained by adding to (6) and (7) the base frequency response. After some simple manipulation the result is: FOREδ (E , ω) FOREbase (jω)
π
=1+
j2ω(1 + e−a ω )
π (a2 + ω2 )
ω 1 −
2 δ E
+a
δ
E
j sin−1 Eδ
e
(8)
K , and the identities sin(ωtδ ) = Eδ and cos(ωtδ ) = 1 − Eδ have been used. Note that in contrast where FOREbase (jω) = jω+ a to (5), FOREδ (E , ω) depends both on the magnitude and frequency of the input. In general, the expression (8) is valid for 0 ≤ Eδ ≤ 1. Obviously, for δ > E, no reset action is performed and thus the describing function will directly be the base frequency response. Note that the FOREδ introduces extra phase lead for frequencies over 1a , in comparison with FORE for some values of δ (Fig. 5). Since, in addition for increasing values of δ a significant phase lag is introduced, it seems that a proper election of δ should be in the range 0 < δ/E < 0.8. 2
3.3. Clegg integrator with reset band The describing function of the Clegg integrator with reset band, that will be referred to as CIδ , is obtained as a particular case of FOREδ for K = 1 and a = 0. The result is
CIδ (E , ω) =
1
1 + jω
j4
2 1 − Eδ
π
j sin−1 Eδ
e
.
(9)
Again, note that the reset band introduces extra phase lead for values 0 < δ/E < 0.8, and at every frequency (Fig. 6). 3.4. General reset compensator with reset band In the case in which the general reset compensator (1) has a state matrix Ar with real and distinct eigenvalues, the compensator may be implemented as a sum of different FOREb elements, and thus calculation of its DF is straightforward. In addition, partial reset may be implemented by considering the base frequency response of the non-reset element.
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167
0 δ = 0 (FORE) E
-10
δ = 0.6 E
δ = 0.2 E
δ = 0.8 E
Phase (degrees)
-20 -30 -40 δ = 0.9 E
-50 -60 -70
δ = 0.1 E
-80 -90
ω (rad/s)
1 a
Fig. 5. FOREδ (E , ω) Phase for different values of δ/E.
-20 -30 -40 -50 ∠CIb(E,ω ) -60 -70 -80 -90
0
0.1
0.2
0.3
0.4
0.5 δ E
0.6
0.7
0.8
0.9
1
Fig. 6. CId (E , ω) Phase for different values of δ/E.
On the other hand, although the derivation is somehow involved (see Appendix), a closed-form expression of the describing function can be obtained for the general case of full reset with reset band. The result is: Cδ (E , ω) = Cbase (jω) +
1
π
1
Cr e ω
Ar sin −1 Eδ
G
δ E
Ar π −1 δ , ω (I + e−Ar ω ) · e( ω −jI )(π−sin ( E )) Br
(10)
where G is given by −1 −1 G(x, ω) = {(jωI − Ar )−1 e−j sin (x) + (jωI + Ar )−1 ej sin (x) } · ω(jωI − Ar )−1 .
(11)
Using (10)–(11), for δ = 0 a general closed-form expression can also be obtained for the describing function C (jω) of a full reset compensator without reset band. It is given by C (jω) = C0 (E , ω) = Cbase (jω) +
1
π
1
π
Cr e ω Ar π F (0, ω)(I + e−Ar ω ) · e
Ar
ω −jI (π −π )
Br
(12)
and after simplification results in
ω π Cr {(jωI − Ar )−1 + (jωI + Ar )−1 } · (jωI − Ar )−1 (eAr ω + I )Br . (13) π Note that in general C (jω) will depend only on frequency ω, and Cδ (E , ω) both on the amplitude E and frequency ω. It C (jω) = Cbase (jω) +
should also be noted that these expressions appear to be new, and thus by itself they are an important contribution to the K reset control field. Obviously, expression for FOREδ as given in (8) is a particular case of (10) for Cbase (jω) = jω+ , Ar = a −a, Br = 1, Cr = K . In addition, CIδ FD expression in (9) is a particular case for K = 1 and a = 0.
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0.6 y
0.4
v
0.2 0 -0.2 -0.4 -0.6 -0.8
0
5
10
15
Time (s) Fig. 7. Time simulation of feedback system with FOREδ and first order plant (note that e = −y).
4. Limit cycles analysis Consider the feedback system of Fig. 1, the question now is to analyze the existence of limit cycles of reset control systems with reset band, by using the describing function of the reset compensator obtained above. A well-known method is to compute possible values of E and ω such that the describing function of the reset compensator Cδ (E , jω) verifies that 1 + Cδ (E , jω)P (jω) = 0, which is usually referred to as the harmonic balance principle. This is usually made by analyzing the crossings of the Nyquist plot of the plant P with plots of −1/C (E , jω) vs ω, for different values of E. It is well known that this method only gives approximate results, and that it works well only if the plant P (s) filters higher order harmonics of the periodic signal v in Fig. 1 (low pass condition). Alternatively, the above procedure is equivalent to finding the crossings of the critical locus −1/Cδ (E , ω) with the plant Nyquist plot P (jω), and then obtaining crossings for values of E and ω that will correspond to the amplitude and frequency of a limit cycle. In the following, two examples are developed for the reset compensators FOREδ and CIδ , respectively. Note that since, in general, their describing functions depend on the parameter δ/E, where δ/E ∈ (0, 1) defines the reset band in relation to the compensator input amplitude, if a limit cycle exists for some δ then there always exist limit cycles for any value of E such that δ/E ∈ (0, 1). Thus, a normalized reset band δ/E and describing function with amplitude E = 1 will be used. As a result, the critical locus −1/Cδ/E (1, ω) will be used to evaluate the crossings, and then values of δ/E and ω will be obtained instead. It turns out that for each given frequency ω, critical loci −1/Cδ/E (1, ω) are circle arcs in the Nyquist plane as δ/E goes from 0 to 1. 4.1. FOREδ with a first order plant A simple analysis shows that there are no limit cycles in the case corresponding to a feedback control system (Fig. 1) with a FOREδ reset compensator having a stable base compensator, and a first order stable plant P (s). The reason is that the after-reset state of the closed-loop system will always correspond to a zero compensator output, and an alternating value of the error δ, −δ, . . . is obtained. The only case in which a limit cycle may exist is in the case in which the error signal reaches the reset band from each one of the two values δ and −δ , and this is only possible if the base feedback system is unstable. In this example, a FOREδ compensator with an unstable base compensator FOREbase (s) = s−2 1 and δ = 0.1, and a first order plant P (s) = s+10.5 is considered. A simulation of the feedback system reveals (see Fig. 7) that a limit cycle exists with amplitude E = 0.20 and frequency ω = 1 rad/s. In Fig. 8 crossings of critical loci −1/FOREδ/E (1, ω), for ω = 0.5, 1, 1.5 rad/s, with P (jω) are shown. Note that after some computation it can be shown that critical loci are circle arcs with centers xc + iyc and radii rc , dependent on the frequency, given by xc (ω) = − yc (ω) = −
a + β(ω2 − a2 )
ω
K (1 − 2β a)
K
rc (ω) = −xc (ω) − a.
(14) (15) (16)
A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
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0.4
-2.4
-2
-1.6
-1.2
-0.8
-0.4
ω = 0.5 rad/s
0
0.4
0.8
1.2
1.6
2
2.4
-0.4
ω = 1 rad/s -0.8
ω = 1.5 rad/s δ/E
-1.2
δ/E = 0.68 ω = 1 rad/s
-1.6 Fig. 8. Crossings of critical loci −1/FOREδ/E (1, ω), for ω = 0.5, 1, 1.5 rad/s, with Nyquist plot of the first order plant.
1.5
1
y
v
0.5
0
-0.5
-1
-1.5
-2
0
2
4
6
8
10 Time (s)
12
14
16
18
20
Fig. 9. Crossings of critical locus −1/CIb (E , ω) with Nyquist plot of the plant.
Then, it is possible to evaluate numerically those crossings corresponding to a limit cycle, and in addition values of δ/E and ω. The result is that there exist limit cycles for any value of E such as δ/E = 0.68, and for ω = 1 rad/s. In this example, the amplitude will be E = δ/0.68 = 0.1/0.68 = 0.15. Note that the describing function method gives a very accurate prediction of the limit cycle. In addition, a perturbation method can be used to analyze the (local) stability of the limit cycle. It can be shown that for the limit cycle to be stable a sufficient condition is
∂ R(E , ω) ∂ I (E , ω) ∂ R(E , ω) ∂ I (E , ω) − >0 (17) ∂E ∂ω ∂ω ∂E at (E0 , ω0 ) = (0.15, 1), where R(E , ω) = Re(1 + FOREδ (E , ω)P (jω)) and I (E , ω) = Im(1 + FOREδ (E , ω)P (jω)). In this case, condition (17) can be checked numerically and thus the limit cycle is stable. 4.2. CIδ with a second order plant Consider again the feedback system of Fig. 1, in this case with a Clegg integrator CIδ as reset compensator with δ = 0.25, 1 and a second order plant P (s) = s+ . A simulation has been run first to analyze the existence of limit cycles, the result is s2 that there exist a limit cycle as shown in Fig. 9, with amplitude E = 0.5 and frequency ω = 0.92 rad/s. Let us consider now the use of the describing function method. Different critical loci −1/CIδ/E (1, ω), for ω = 0.1, 0.3, 0.5, 0.92 rad/s, are shown in Fig. 10. In this case, again critical loci are circle arcs where centers and radii depend on the frequency, and are given by (14)–(16) for a = 0, that is xc (ω) = − K2ωπ , yc = − ω , and rc (ω) = K2ωπ . After a numerical K evaluation of the different crossings, the result is that there exist a limit cycle for any value of E such as δ/E = 0.82, and
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0.4
-2
-1.6
-1.2
-0.8
-0.4
0
-0.4
0.4 ω = 0.1 rad/s
0.8
ω = 0.3 rad/s ω = 0.5 rad/s
-0.8 δ/E = 0.82 ω = 0.92 rad/s ω = 0.92 rad/s δ/E
-1.2
-1.6
Fig. 10. Crossings of critical locus −1/CIδ/E (1, ω) with Nyquist plot of the second order plant.
ω = 0.92 rad/s. For this example, the amplitude is E = δ/0.82 = 0.3. Again, the prediction of the limit cycle is very accurate. The deviation of the amplitude is consistent with the fact that signal v in Fig. 9 may have significant higher order harmonics. In addition, condition (17) is satisfied for (E0 , ω0 ) = (0.3, 0.92), and thus the limit cycle is also stable. 5. Justification of the describing function As is well known, in spite of the fact that the describing function technique gives only approximate results it has been found useful in many practical applications. Although a formal justification is clearly desirable, it turns out that the several analytical justifications that can be found in the literature are of limited use. The first work in this direction is [14], where it is shown that a limit cycle exist, using Fig. 1, if C is bounded and P is able to filter enough high order harmonics (low pass condition). Also, the works [15,16] develop formal results for nonlinear systems that are sector-bounded or exhibit a passivity condition, respectively. These results are not applicable to reset control systems, and in general an analytical treatment seems to be a hard open problem. In the following, the low pass condition will be explored for some particular types of reset control systems with reset band. In addition, the harmonics structure of the sinusoidal response of FOREδ will be explored (CIδ is a particular case). In general, the response of FOREδ to a sinusoidal input e(t ) = E sin(ω(t − tδ )) is periodic with fundamental period 2π /ω (Fig. 4). In addition, it is also symmetric and can be expressed as
KE (ω cos(ωtδ ) + a sin(ωtδ )) −at e + M (ω)E sin(ω(t − tδ ) + Φ (ω)) a2 + ω 2 v(t ) = π −y t − ω
π [ ω ] π 2π t ∈ , ω ω
t ∈ 0,
(18)
K | (see Fig. 11). where M (ω) and Φ (ω) are the magnitude and phase of | jω+ a
Now, computation of high order harmonics is relatively simple in this case since by symmetry only odd harmonics are different to zero. Therefore, by using the first n odd harmonics of FOREδ response as an approximation, a measure of the approximation can be defined as the L2 [0, 2ωπ ]-norm of the residual term. Then, v(t ) as given by (18) can be expressed by
v(t ) = vn (t ) + vˆ n (t ) where vn (t ) = shown that the ratio
‖ˆvn ‖ = ‖v‖
‖ˆvn ‖ ‖v‖
k=−n
a2k−1 ej(2k−1)ωt , and vˆ n (t ) = v(t ) − vn (t ). After some computation it may be
is given by
Im Ψ n +
Im Ψ
∑n
1 2
1 2
21 + i 2aω a
+ i 2ω
(19)
A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
171
y
0
0.25π
0.5π
0.75π
π
1.25π
1.5π
1.75π
2π
Fig. 11. (black) Sinusoidal response of FOREδ for ω = 1 rad/s, K = 1, and a = −1. (red) First plus third harmonics. (blue) Approximation with 20 harmonics. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
2
3
4
5 6 7 8 Number of odd harmonics (n)
9
10
‖ˆv ‖
n ) vs. n (number of odds harmonics). (blue) | ωa | = 1, (red) | ωa | = 0.5, (green) | ωa | = 0.25. (For interpretation of the references to Fig. 12. Ratio (1 − ‖v‖ colour in this figure legend, the reader is referred to the web version of this article.)
where Ψ is the digamma function. Fig. 12 shows a measure of how low-order harmonics give a good approximation of the ‖ˆvn ‖ sinusoidal FOREδ response. Note that in general the ratio ‖v‖ only depends on the parameters a and ω trough the quotient
| ωa |.
For the example of FOREδ plus a first order plant, the low-pass condition can be now evaluated by computing how the plant filter the harmonics of the FOREδ sinusoidal response. In Fig. 12, it is shown how for different values of |a/ω| the relative weight of the first harmonic is much more significative and it is a good approximation of the filtered FOREδ response. For the example of CIδ with a second order plant, the results are analogous (Figs. 13–14). Note that the harmonics structure of the FOREδ or the CIδ response is always given as shown in Figs. 12–14, then the lowpass condition can be evaluated for a given plant by computing how the harmonics are filtered, and then evaluating how the plant modifies the harmonics structure. For example, Fig. 12 shows how the relative error norm of the first harmonic
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A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
y
π
0.5π
0
1.5π
2π
2.5π
3π
3.5π
4π
4.5π
5π
5.5π
6π
x
Fig. 13. (black) Sinusoidal response of CIδ for ω = 0.92 rad/s. (green) First harmonic approximation. (red) First plus third harmonics. (blue) Approximation with 20 harmonics. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5 6 7 8 Number of odd harmonics (n)
9
10
‖ˆv ‖
n Fig. 14. Ratio (1 − ‖v‖ ) vs. n (number of odds harmonics). (blue) | ωa | = 1, (red) | ωa | = 0.5, (green) | ωa | = 0.25. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
approximating the FOREδ response is around 10%, and that the relative error considering both the first and third harmonic is around 5%, thus it may be expected that the describing function method works properly. The above analysis is general for feedback systems with FOREδ or CIδ compensators and arbitrary order plants with a given frequency response function (including systems with time delays). Although the procedure can be directly extended for higher order reset compensators that can be built as the parallel connection of several FOREδ and also LTI compensators (and thus partial reset compensators are allowed), in general computation of higher order harmonics it needs to be done for every specific case in order to evaluate the low-pass condition. 6. Conclusions The problem of existence and stability of limit cycles in reset control systems with reset band has been approached. It has been solved by using standard methods based on describing function analysis. After computing closed-form expressions
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for the describing function of general reset compensators with reset band, and in particular for FORE and CI compensators, the method has been applied in two particular cases. In general, it may be concluded that the method gives a very accurate prediction of existence and stability of limit cycles, that can be computed for values of the rate δ/E. Appendix. Describing function Cδ (E , ω) of a reset compensator with reset band A derivation of expression (10) will be sketched here. Assume that the base compensator Cbase (s) has a state-space realization given by (Ar , Br , Cr , 0), then to calculate the fundamental component of the Cδ (E , ω) sinousidal response it is sufficient to obtain its values for the interval [−tδ , πω − tδ ] (see Fig. 4), where ωtδ = sin−1 (δ/E ). In that interval, the reset compensator behaves exactly as its base system, then the response can be computed by integrating the base system equations. Equivalently, any interval of length π can be used. By simplicity, the interval [0, π] is used and also the phase Ψ = ωt is used as an independent variable, the result is that compensator output is given by:
v(Ψ ) =
Ψ E A 1 A Ψ Cr (−e r ω F (Ψ1 )e r ω + F (−Ψ ))Br , 2j E
2j
C r ( e− A r
Ψ1 ω
F (Ψ1 )eAr
Ψ ω
if Ψ ∈ [0, π − Ψ1 ) (A.1)
+ F (−Ψ ))Br ,
if Ψ ∈ [π − Ψ1 , π )
where F (Ψ ) = (jωI − Ar )−1 e−jΨ + (jωI + Ar )−1 ejΨ , and Ψ1 = sin−1 (δ/E ). Now, the fundamental component of v(Ψ ) is computed, and since the Fourier transform of the input is E /2j, the describing function is given by Cδ (E , ω) =
1
π
Cr
+ Cr e
Ψ1 eA r ω
F ( Ψ1 )
∫
π−Ψ1
eA r
Ψ ω
e− j Ψ d Ψ
0
−Ar
π −Ψ1 ω
F ( Ψ1 )
∫
π
π −Ψ1
Ψ eA r ω
e
−jΨ
dΨ + +Cr
π
∫
F (−Ψ )e
−j Ψ
dΨ
Br .
(A.2)
0
Finally, after calculating the integrals and simplificating the resulting expression (10) is obtained. Details are omitted by brevity. References [1] D.D. Bainov, P.S. Simeonov, Systems with impulse effect: stability, theory and applications, E. Horwood Ltd, 1989. [2] W.M. Haddad, V. Chellaboina, S.G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, 2006. [3] J.C. Clegg, A nonlinear integrator for servomechanisms, Trans. A. I. E. E. m, Part II 77 (1958) 41–42. [4] I.M. Horowitz, P. Rosenbaum, Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty, Internat. J. Control 24 (6) (1975) 977–1001. [5] K.R. Krishman, I.M. Horowitz, Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances, Internat. J. Control 19 (4) (1976) 689–706. [6] O. Beker, C.V. Hollot, Y. Chait, H. Han, Fundamental properties of reset control systems, Automatica 40 (2004) 905–915. [7] A. Baños, J. Carrasco, A. Barreiro, Reset times dependent stability of reset systems, in: European Control Conference, Greece, 2007, pp. 4792–4798. [8] A. Baños, A. Barreiro, Delay-independent stability of reset systems, IEEE Trans. Automat. Control 54 (2) (2009) 337–341. [9] D. Nešić, L. Zaccarian, A.R. Teel, Stability properties of reset systems, Automatica 44 (8) (2008) 2019–2026. [10] T. Loquen, S. Tarbouriech, C. Prieur, Stability of reset control systems with nonzero reference, in: IEEE Conf. on Decision and Control, Cancun, Mexico, 2008, pp. 3386–3391. [11] W.H.T.M. Aangenent, G. Witvoet, W.P.M.H. Heemels, M.J.G. van der Molengraft, M. Steinbuch, Performance analysis of reset control systems, Internat. J. Robust Nonlinear Control (2009) doi:10.1002/rnc.1502. [12] A. Vidal, A. Baños, QFT-based design of PI + CI compensators: application in process control, IEEE Mediterr. Conference Control Automation (2008) 806–811. [13] M. Vidyasagar, Nonlinear Systems Stability, Prentice-Hall, London, 1993. [14] A.R. Bergen, R.L. Franks, Justification of the describing function method, SIAM J. Control 9 (1971) 568–589. [15] A. Mees, A. Bergen, Describing functions revisited, IEEE Trans. Automat. Control 20 (4) (1975) 473–478. [16] S.R. Sanders, On limit cycles and the describing function method in periodically switched circuits, IEEE Trans. Circuits Syst.-I 20 (4) (1975) 473–478.
Contents lists available at ScienceDirect
Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
Limit cycles analysis of reset control systems with reset band Alfonso Baños a,∗ , Sebastián Dormido b , Antonio Barreiro c a
Dpt. Informática y Sistemas, Univ. Murcia, 30071 Murcia, Spain
b
Dpt. Informática y Automática, U.N.E.D., 28040 Madrid, Spain
c
Dpt. Ingeniería de Sistemas y Automática, Univ. Vigo, 36200 Vigo, Spain
article
info
Article history: Received 27 November 2009 Accepted 25 July 2010 Keywords: Hybrid systems Stability of hybrid systems Impulsive systems Reset control systems
abstract The reset band is a simple idea, and a must in practice, to improve reset compensation by adding extra phase lead in a feedback loop. However, a formal treatment of how the reset band can affect stability and performance of a reset control system is still an open issue. This work approaches the problem of the existence and stability of limit cycles of reset control systems with reset band. A frequency domain approach is given by using standard methods based on the describing function. In addition, closed-form expressions have been obtained for the describing function of arbitrary order full reset compensators with and without reset band. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction Reset control is a kind of impulsive/hybrid control in which some of the compensator states are set to zero at those instants in which its input is zero. Using [1] or the most recent work [2], reset systems (without external inputs) can be classified as autonomous systems with impulse effects, an area of systems theory that has been started to be developed in the last few years. However, in the control literature the development of reset control systems dates back to the work of [3], and was followed by the works [4,5], where very simple reset compensators, CI (Clegg integrator) and FORE (first order reset element) respectively, were analyzed and design procedures were developed. More general reset compensators, including partial and full reset compensation, have been studied for example in [6]. One of the main difficulties of reset compensation is that closed-loop stability may be not guaranteed if reset actions are not properly performed, and in fact it is well known that reset can unstabilize a base stable control system. Thus, stability of reset control systems is a main concern from a theoretical and practical point of view, and several recent works have approached the stability problem. See for example [6–8], and the works [9–11]. An important practical issue of reset control is that compensator implementation is usually done by using reset band. In addition, it has been noted that use of reset band may improve stability and performance in systems with time-delays [12], due to the phase lead characteristic that is common to reset compensators. Phase lead can be even improved by using reset band. However, a formal analysis of how the reset band can affect stability and performance of a reset control system is still an open issue. In this work, a frequency domain approach by means of the describing function will be used to approach an important question such as the existence and stability of limit cycles in reset control systems with reset band. The analysis with the describing function will follow standard methods (see for example [13]). Although it is well-known that the describing function may fail in some cases, it has been proved to be an efficient method and has been formally justified in many practical cases [14,15], including some types of switched systems [16].
∗
Corresponding author. E-mail addresses: [email protected] (A. Baños), [email protected] (S. Dormido), [email protected] (A. Barreiro).
1751-570X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.nahs.2010.07.004
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v
e
u P
C
y
Fig. 1. Reset compensator C and LTI plant P.
Fig. 2. Reset band.
The work is organized as follows. Section 2 introduces the problem setup, with a precise definition of a reset compensator with reset band. Note that single-input single-output systems are considered in this work. In Section 3, the describing function of reset control systems with reset band is calculated, in particular the describing function of CI and FORE with reset band are obtained. Section 4 approaches the existence and stability of limit cycles by using a describing functionbased analysis, and finally in Section 5 the describing function method is justified by exploring the harmonics structure of the sinuosidal response and the low pass condition. 2. Reset compensators with reset band The main motivation for the use of reset compensation is to improve the performance of a previously designed LTI control system, being the goal to reset some states of a base LTI compensator to improve control system performance, both in terms of velocity of response and relative stability. In general, these specifications will be impossible to achieve by means of LTI compensation. Reset control systems with reset band are feedback control systems (Fig. 1), where for the purposes of this work the plant P is described in general by a transfer function P (s), and the reset compensator C is given by the impulsive differential equation
x˙ r (t ) C : xr (t + ) v(t )
= = =
Ar xr (t ) + Br e(t ), Aρ xr (t ), Cr xr (t ) + Dr e(t )
(e(t ), e˙ (t )) ̸∈ Bδ (e(t ), e˙ (t )) ∈ Bδ
(1)
where the reset band surface Bδ is given by Bδ = {(x, y) ∈ R2 | (x = −δ ∧ y > 0) ∨ (x = δ ∧ y < 0)}, being δ some non-negative real number. In this way, the compensator states are reset at the instants in which its input is entering into − the reset band. In general, the reset band surface Bδ will consist of two reset lines B+ δ and Bδ in the plane, as shown in Fig. 2. In the particular case δ = 0, the standard reset compensator is obtained. On the other hand, if δ is big enough in relation to the error amplitude, then no reset action is produced and the reset compensator reduce to its base compensator, given by the transfer function Cbase (s) = Cr (sI − Ar )−1 Br + Dr .
A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
165
Fig. 3. FORE sinusoidal response.
3. Describing function The following notation will be used to distinguish between reset compensators with and without reset band: Cδ will denote a reset compensator with reset band Bδ , whereas C0 (or simply C ) is the standard reset compensator without reset band. Note that Cbase stands for the base reset compensator. By simplicity, the describing function (DF) of a FOREδ compensator is considered at a first instance. In addition, since DF computation will be based on FORE DF, this DF is considered first. 3.1. FORE K FORE is a simple reset compensator with a first order base compensator given by FOREbase (s) = s+ . As is well known a (see for example [13]), for a given system its DF is calculated as the balance between the fundamental component of its sinusoidal response and the sinousoidal input. The response v of the reset compensator FORE (with δ = 0 in (1)), to the sinusoidal input e(t ) = E sin(ωt ) is given in the s-domain simply by
Eω K (2) s + a s2 + ω 2 and the time response v(t ) will consist of two terms: one corresponding to the transient response given by the compensator modes, and other corresponding to the steady response given by the sinusoidal input modes. Since the reset instants will be given by the crossings of the sinusoidal input with zero, and thus they will be periodic with fundamental period 2π /ω (see Fig. 3), it turns out that v(t ) will be also periodic with that period. To compute the FORE describing function, a simple procedure is to add to the base frequency response the contribution of the transitory terms in v(t ) over a period, for example [0, 2ωπ ]. By symmetry (see Fig. 3), it will be computed by using the interval [0, πω ], where V (s) =
vt ( t ) =
KE ω a2 + ω 2
e−at .
(3)
The result is 1
π
π/ω 0
vt (t )e−jωt ωdt E 2j
=
j2K ω2
π
π (a + jω)(a2 + ω2 )
(1 + e−a ω )
(4)
and finally adding to the frequency response of the base system, the describing function of FORE results in FORE (ω) =
j2K ω2
π (a + jω)(a2 + ω2 )
π
(1 + e−a ω ) +
K a + jω
.
(5)
3.2. FOREδ To obtain the describing function with reset band, again the sinusoidal response needs to be calculated. In this case, note that since FOREδ is time-invariant, the FOREδ response to e(t ) = E sin(ωt ) over the period [−tδ , 2ωπ − tδ ] is the same (except a translation) that its response to e(t − tδ ) over the period [0, 2ωπ ] (see Fig. 4). Thus, FOREδ DF can be obtained as the contribution of the transitory terms in the response to e(t − tδ ) = E cos(ωtδ ) sin(ωt ) − E sin(ωtδ ) cos(ωt ), plus the base frequency response. But note that the transitory terms consists of two terms: one term v1t corresponding to the response to E cos(ωtδ ) sin(ωt ), and a second term vt2 with the response to −E sin(ωtδ ) cos(ωt ).
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Fig. 4. FORE with reset band sinusoidal response. KE cos(ωt )
By using (3), the term vt1 is directly given by vt1 = a2 +ω2 δ e−at , and thus using again symmetry arguments its harmonic balance (corresponding to the fundamental component) is given by 1
π 1 2π
π/ω 0
2π/ω 0
vt1 (t )e−jωt ωdt
e(t − tδ )e−jωt ωdt
=
π KE ω2 cos(ωtδ ) (1 + e−a ω ) π(a+jω)(a2 +ω2 ) . E −jωtδ e
On the other hand, the second transitory term is vt2 = − 1
π 1 2π
π/ω 0
2π/ω 0
vt2 (t )e−jωt ωdt
e(t − tδ )e−jωt ωdt
(6)
2j
=
KEa sin(ωtδ ) −at e , a2 +ω2
and its harmonic balance is now
π KEaω sin(ωtδ ) (1 + e−a ω ) π(a+jω)(a2 +ω2 ) . E −jωtδ e
(7)
2j
Finally, the FOREδ describing function is obtained by adding to (6) and (7) the base frequency response. After some simple manipulation the result is: FOREδ (E , ω) FOREbase (jω)
π
=1+
j2ω(1 + e−a ω )
π (a2 + ω2 )
ω 1 −
2 δ E
+a
δ
E
j sin−1 Eδ
e
(8)
K , and the identities sin(ωtδ ) = Eδ and cos(ωtδ ) = 1 − Eδ have been used. Note that in contrast where FOREbase (jω) = jω+ a to (5), FOREδ (E , ω) depends both on the magnitude and frequency of the input. In general, the expression (8) is valid for 0 ≤ Eδ ≤ 1. Obviously, for δ > E, no reset action is performed and thus the describing function will directly be the base frequency response. Note that the FOREδ introduces extra phase lead for frequencies over 1a , in comparison with FORE for some values of δ (Fig. 5). Since, in addition for increasing values of δ a significant phase lag is introduced, it seems that a proper election of δ should be in the range 0 < δ/E < 0.8. 2
3.3. Clegg integrator with reset band The describing function of the Clegg integrator with reset band, that will be referred to as CIδ , is obtained as a particular case of FOREδ for K = 1 and a = 0. The result is
CIδ (E , ω) =
1
1 + jω
j4
2 1 − Eδ
π
j sin−1 Eδ
e
.
(9)
Again, note that the reset band introduces extra phase lead for values 0 < δ/E < 0.8, and at every frequency (Fig. 6). 3.4. General reset compensator with reset band In the case in which the general reset compensator (1) has a state matrix Ar with real and distinct eigenvalues, the compensator may be implemented as a sum of different FOREb elements, and thus calculation of its DF is straightforward. In addition, partial reset may be implemented by considering the base frequency response of the non-reset element.
A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
167
0 δ = 0 (FORE) E
-10
δ = 0.6 E
δ = 0.2 E
δ = 0.8 E
Phase (degrees)
-20 -30 -40 δ = 0.9 E
-50 -60 -70
δ = 0.1 E
-80 -90
ω (rad/s)
1 a
Fig. 5. FOREδ (E , ω) Phase for different values of δ/E.
-20 -30 -40 -50 ∠CIb(E,ω ) -60 -70 -80 -90
0
0.1
0.2
0.3
0.4
0.5 δ E
0.6
0.7
0.8
0.9
1
Fig. 6. CId (E , ω) Phase for different values of δ/E.
On the other hand, although the derivation is somehow involved (see Appendix), a closed-form expression of the describing function can be obtained for the general case of full reset with reset band. The result is: Cδ (E , ω) = Cbase (jω) +
1
π
1
Cr e ω
Ar sin −1 Eδ
G
δ E
Ar π −1 δ , ω (I + e−Ar ω ) · e( ω −jI )(π−sin ( E )) Br
(10)
where G is given by −1 −1 G(x, ω) = {(jωI − Ar )−1 e−j sin (x) + (jωI + Ar )−1 ej sin (x) } · ω(jωI − Ar )−1 .
(11)
Using (10)–(11), for δ = 0 a general closed-form expression can also be obtained for the describing function C (jω) of a full reset compensator without reset band. It is given by C (jω) = C0 (E , ω) = Cbase (jω) +
1
π
1
π
Cr e ω Ar π F (0, ω)(I + e−Ar ω ) · e
Ar
ω −jI (π −π )
Br
(12)
and after simplification results in
ω π Cr {(jωI − Ar )−1 + (jωI + Ar )−1 } · (jωI − Ar )−1 (eAr ω + I )Br . (13) π Note that in general C (jω) will depend only on frequency ω, and Cδ (E , ω) both on the amplitude E and frequency ω. It C (jω) = Cbase (jω) +
should also be noted that these expressions appear to be new, and thus by itself they are an important contribution to the K reset control field. Obviously, expression for FOREδ as given in (8) is a particular case of (10) for Cbase (jω) = jω+ , Ar = a −a, Br = 1, Cr = K . In addition, CIδ FD expression in (9) is a particular case for K = 1 and a = 0.
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0.6 y
0.4
v
0.2 0 -0.2 -0.4 -0.6 -0.8
0
5
10
15
Time (s) Fig. 7. Time simulation of feedback system with FOREδ and first order plant (note that e = −y).
4. Limit cycles analysis Consider the feedback system of Fig. 1, the question now is to analyze the existence of limit cycles of reset control systems with reset band, by using the describing function of the reset compensator obtained above. A well-known method is to compute possible values of E and ω such that the describing function of the reset compensator Cδ (E , jω) verifies that 1 + Cδ (E , jω)P (jω) = 0, which is usually referred to as the harmonic balance principle. This is usually made by analyzing the crossings of the Nyquist plot of the plant P with plots of −1/C (E , jω) vs ω, for different values of E. It is well known that this method only gives approximate results, and that it works well only if the plant P (s) filters higher order harmonics of the periodic signal v in Fig. 1 (low pass condition). Alternatively, the above procedure is equivalent to finding the crossings of the critical locus −1/Cδ (E , ω) with the plant Nyquist plot P (jω), and then obtaining crossings for values of E and ω that will correspond to the amplitude and frequency of a limit cycle. In the following, two examples are developed for the reset compensators FOREδ and CIδ , respectively. Note that since, in general, their describing functions depend on the parameter δ/E, where δ/E ∈ (0, 1) defines the reset band in relation to the compensator input amplitude, if a limit cycle exists for some δ then there always exist limit cycles for any value of E such that δ/E ∈ (0, 1). Thus, a normalized reset band δ/E and describing function with amplitude E = 1 will be used. As a result, the critical locus −1/Cδ/E (1, ω) will be used to evaluate the crossings, and then values of δ/E and ω will be obtained instead. It turns out that for each given frequency ω, critical loci −1/Cδ/E (1, ω) are circle arcs in the Nyquist plane as δ/E goes from 0 to 1. 4.1. FOREδ with a first order plant A simple analysis shows that there are no limit cycles in the case corresponding to a feedback control system (Fig. 1) with a FOREδ reset compensator having a stable base compensator, and a first order stable plant P (s). The reason is that the after-reset state of the closed-loop system will always correspond to a zero compensator output, and an alternating value of the error δ, −δ, . . . is obtained. The only case in which a limit cycle may exist is in the case in which the error signal reaches the reset band from each one of the two values δ and −δ , and this is only possible if the base feedback system is unstable. In this example, a FOREδ compensator with an unstable base compensator FOREbase (s) = s−2 1 and δ = 0.1, and a first order plant P (s) = s+10.5 is considered. A simulation of the feedback system reveals (see Fig. 7) that a limit cycle exists with amplitude E = 0.20 and frequency ω = 1 rad/s. In Fig. 8 crossings of critical loci −1/FOREδ/E (1, ω), for ω = 0.5, 1, 1.5 rad/s, with P (jω) are shown. Note that after some computation it can be shown that critical loci are circle arcs with centers xc + iyc and radii rc , dependent on the frequency, given by xc (ω) = − yc (ω) = −
a + β(ω2 − a2 )
ω
K (1 − 2β a)
K
rc (ω) = −xc (ω) − a.
(14) (15) (16)
A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
169
0.4
-2.4
-2
-1.6
-1.2
-0.8
-0.4
ω = 0.5 rad/s
0
0.4
0.8
1.2
1.6
2
2.4
-0.4
ω = 1 rad/s -0.8
ω = 1.5 rad/s δ/E
-1.2
δ/E = 0.68 ω = 1 rad/s
-1.6 Fig. 8. Crossings of critical loci −1/FOREδ/E (1, ω), for ω = 0.5, 1, 1.5 rad/s, with Nyquist plot of the first order plant.
1.5
1
y
v
0.5
0
-0.5
-1
-1.5
-2
0
2
4
6
8
10 Time (s)
12
14
16
18
20
Fig. 9. Crossings of critical locus −1/CIb (E , ω) with Nyquist plot of the plant.
Then, it is possible to evaluate numerically those crossings corresponding to a limit cycle, and in addition values of δ/E and ω. The result is that there exist limit cycles for any value of E such as δ/E = 0.68, and for ω = 1 rad/s. In this example, the amplitude will be E = δ/0.68 = 0.1/0.68 = 0.15. Note that the describing function method gives a very accurate prediction of the limit cycle. In addition, a perturbation method can be used to analyze the (local) stability of the limit cycle. It can be shown that for the limit cycle to be stable a sufficient condition is
∂ R(E , ω) ∂ I (E , ω) ∂ R(E , ω) ∂ I (E , ω) − >0 (17) ∂E ∂ω ∂ω ∂E at (E0 , ω0 ) = (0.15, 1), where R(E , ω) = Re(1 + FOREδ (E , ω)P (jω)) and I (E , ω) = Im(1 + FOREδ (E , ω)P (jω)). In this case, condition (17) can be checked numerically and thus the limit cycle is stable. 4.2. CIδ with a second order plant Consider again the feedback system of Fig. 1, in this case with a Clegg integrator CIδ as reset compensator with δ = 0.25, 1 and a second order plant P (s) = s+ . A simulation has been run first to analyze the existence of limit cycles, the result is s2 that there exist a limit cycle as shown in Fig. 9, with amplitude E = 0.5 and frequency ω = 0.92 rad/s. Let us consider now the use of the describing function method. Different critical loci −1/CIδ/E (1, ω), for ω = 0.1, 0.3, 0.5, 0.92 rad/s, are shown in Fig. 10. In this case, again critical loci are circle arcs where centers and radii depend on the frequency, and are given by (14)–(16) for a = 0, that is xc (ω) = − K2ωπ , yc = − ω , and rc (ω) = K2ωπ . After a numerical K evaluation of the different crossings, the result is that there exist a limit cycle for any value of E such as δ/E = 0.82, and
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A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
0.4
-2
-1.6
-1.2
-0.8
-0.4
0
-0.4
0.4 ω = 0.1 rad/s
0.8
ω = 0.3 rad/s ω = 0.5 rad/s
-0.8 δ/E = 0.82 ω = 0.92 rad/s ω = 0.92 rad/s δ/E
-1.2
-1.6
Fig. 10. Crossings of critical locus −1/CIδ/E (1, ω) with Nyquist plot of the second order plant.
ω = 0.92 rad/s. For this example, the amplitude is E = δ/0.82 = 0.3. Again, the prediction of the limit cycle is very accurate. The deviation of the amplitude is consistent with the fact that signal v in Fig. 9 may have significant higher order harmonics. In addition, condition (17) is satisfied for (E0 , ω0 ) = (0.3, 0.92), and thus the limit cycle is also stable. 5. Justification of the describing function As is well known, in spite of the fact that the describing function technique gives only approximate results it has been found useful in many practical applications. Although a formal justification is clearly desirable, it turns out that the several analytical justifications that can be found in the literature are of limited use. The first work in this direction is [14], where it is shown that a limit cycle exist, using Fig. 1, if C is bounded and P is able to filter enough high order harmonics (low pass condition). Also, the works [15,16] develop formal results for nonlinear systems that are sector-bounded or exhibit a passivity condition, respectively. These results are not applicable to reset control systems, and in general an analytical treatment seems to be a hard open problem. In the following, the low pass condition will be explored for some particular types of reset control systems with reset band. In addition, the harmonics structure of the sinusoidal response of FOREδ will be explored (CIδ is a particular case). In general, the response of FOREδ to a sinusoidal input e(t ) = E sin(ω(t − tδ )) is periodic with fundamental period 2π /ω (Fig. 4). In addition, it is also symmetric and can be expressed as
KE (ω cos(ωtδ ) + a sin(ωtδ )) −at e + M (ω)E sin(ω(t − tδ ) + Φ (ω)) a2 + ω 2 v(t ) = π −y t − ω
π [ ω ] π 2π t ∈ , ω ω
t ∈ 0,
(18)
K | (see Fig. 11). where M (ω) and Φ (ω) are the magnitude and phase of | jω+ a
Now, computation of high order harmonics is relatively simple in this case since by symmetry only odd harmonics are different to zero. Therefore, by using the first n odd harmonics of FOREδ response as an approximation, a measure of the approximation can be defined as the L2 [0, 2ωπ ]-norm of the residual term. Then, v(t ) as given by (18) can be expressed by
v(t ) = vn (t ) + vˆ n (t ) where vn (t ) = shown that the ratio
‖ˆvn ‖ = ‖v‖
‖ˆvn ‖ ‖v‖
k=−n
a2k−1 ej(2k−1)ωt , and vˆ n (t ) = v(t ) − vn (t ). After some computation it may be
is given by
Im Ψ n +
Im Ψ
∑n
1 2
1 2
21 + i 2aω a
+ i 2ω
(19)
A. Baños et al. / Nonlinear Analysis: Hybrid Systems 5 (2011) 163–173
171
y
0
0.25π
0.5π
0.75π
π
1.25π
1.5π
1.75π
2π
Fig. 11. (black) Sinusoidal response of FOREδ for ω = 1 rad/s, K = 1, and a = −1. (red) First plus third harmonics. (blue) Approximation with 20 harmonics. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
1
2
3
4
5 6 7 8 Number of odd harmonics (n)
9
10
‖ˆv ‖
n ) vs. n (number of odds harmonics). (blue) | ωa | = 1, (red) | ωa | = 0.5, (green) | ωa | = 0.25. (For interpretation of the references to Fig. 12. Ratio (1 − ‖v‖ colour in this figure legend, the reader is referred to the web version of this article.)
where Ψ is the digamma function. Fig. 12 shows a measure of how low-order harmonics give a good approximation of the ‖ˆvn ‖ sinusoidal FOREδ response. Note that in general the ratio ‖v‖ only depends on the parameters a and ω trough the quotient
| ωa |.
For the example of FOREδ plus a first order plant, the low-pass condition can be now evaluated by computing how the plant filter the harmonics of the FOREδ sinusoidal response. In Fig. 12, it is shown how for different values of |a/ω| the relative weight of the first harmonic is much more significative and it is a good approximation of the filtered FOREδ response. For the example of CIδ with a second order plant, the results are analogous (Figs. 13–14). Note that the harmonics structure of the FOREδ or the CIδ response is always given as shown in Figs. 12–14, then the lowpass condition can be evaluated for a given plant by computing how the harmonics are filtered, and then evaluating how the plant modifies the harmonics structure. For example, Fig. 12 shows how the relative error norm of the first harmonic
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y
π
0.5π
0
1.5π
2π
2.5π
3π
3.5π
4π
4.5π
5π
5.5π
6π
x
Fig. 13. (black) Sinusoidal response of CIδ for ω = 0.92 rad/s. (green) First harmonic approximation. (red) First plus third harmonics. (blue) Approximation with 20 harmonics. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
1
0.8
0.6
0.4
0.2
0
1
2
3
4
5 6 7 8 Number of odd harmonics (n)
9
10
‖ˆv ‖
n Fig. 14. Ratio (1 − ‖v‖ ) vs. n (number of odds harmonics). (blue) | ωa | = 1, (red) | ωa | = 0.5, (green) | ωa | = 0.25. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
approximating the FOREδ response is around 10%, and that the relative error considering both the first and third harmonic is around 5%, thus it may be expected that the describing function method works properly. The above analysis is general for feedback systems with FOREδ or CIδ compensators and arbitrary order plants with a given frequency response function (including systems with time delays). Although the procedure can be directly extended for higher order reset compensators that can be built as the parallel connection of several FOREδ and also LTI compensators (and thus partial reset compensators are allowed), in general computation of higher order harmonics it needs to be done for every specific case in order to evaluate the low-pass condition. 6. Conclusions The problem of existence and stability of limit cycles in reset control systems with reset band has been approached. It has been solved by using standard methods based on describing function analysis. After computing closed-form expressions
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for the describing function of general reset compensators with reset band, and in particular for FORE and CI compensators, the method has been applied in two particular cases. In general, it may be concluded that the method gives a very accurate prediction of existence and stability of limit cycles, that can be computed for values of the rate δ/E. Appendix. Describing function Cδ (E , ω) of a reset compensator with reset band A derivation of expression (10) will be sketched here. Assume that the base compensator Cbase (s) has a state-space realization given by (Ar , Br , Cr , 0), then to calculate the fundamental component of the Cδ (E , ω) sinousidal response it is sufficient to obtain its values for the interval [−tδ , πω − tδ ] (see Fig. 4), where ωtδ = sin−1 (δ/E ). In that interval, the reset compensator behaves exactly as its base system, then the response can be computed by integrating the base system equations. Equivalently, any interval of length π can be used. By simplicity, the interval [0, π] is used and also the phase Ψ = ωt is used as an independent variable, the result is that compensator output is given by:
v(Ψ ) =
Ψ E A 1 A Ψ Cr (−e r ω F (Ψ1 )e r ω + F (−Ψ ))Br , 2j E
2j
C r ( e− A r
Ψ1 ω
F (Ψ1 )eAr
Ψ ω
if Ψ ∈ [0, π − Ψ1 ) (A.1)
+ F (−Ψ ))Br ,
if Ψ ∈ [π − Ψ1 , π )
where F (Ψ ) = (jωI − Ar )−1 e−jΨ + (jωI + Ar )−1 ejΨ , and Ψ1 = sin−1 (δ/E ). Now, the fundamental component of v(Ψ ) is computed, and since the Fourier transform of the input is E /2j, the describing function is given by Cδ (E , ω) =
1
π
Cr
+ Cr e
Ψ1 eA r ω
F ( Ψ1 )
∫
π−Ψ1
eA r
Ψ ω
e− j Ψ d Ψ
0
−Ar
π −Ψ1 ω
F ( Ψ1 )
∫
π
π −Ψ1
Ψ eA r ω
e
−jΨ
dΨ + +Cr
π
∫
F (−Ψ )e
−j Ψ
dΨ
Br .
(A.2)
0
Finally, after calculating the integrals and simplificating the resulting expression (10) is obtained. Details are omitted by brevity. References [1] D.D. Bainov, P.S. Simeonov, Systems with impulse effect: stability, theory and applications, E. Horwood Ltd, 1989. [2] W.M. Haddad, V. Chellaboina, S.G. Nersesov, Impulsive and Hybrid Dynamical Systems: Stability, Dissipativity, and Control, Princeton University Press, 2006. [3] J.C. Clegg, A nonlinear integrator for servomechanisms, Trans. A. I. E. E. m, Part II 77 (1958) 41–42. [4] I.M. Horowitz, P. Rosenbaum, Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty, Internat. J. Control 24 (6) (1975) 977–1001. [5] K.R. Krishman, I.M. Horowitz, Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances, Internat. J. Control 19 (4) (1976) 689–706. [6] O. Beker, C.V. Hollot, Y. Chait, H. Han, Fundamental properties of reset control systems, Automatica 40 (2004) 905–915. [7] A. Baños, J. Carrasco, A. Barreiro, Reset times dependent stability of reset systems, in: European Control Conference, Greece, 2007, pp. 4792–4798. [8] A. Baños, A. Barreiro, Delay-independent stability of reset systems, IEEE Trans. Automat. Control 54 (2) (2009) 337–341. [9] D. Nešić, L. Zaccarian, A.R. Teel, Stability properties of reset systems, Automatica 44 (8) (2008) 2019–2026. [10] T. Loquen, S. Tarbouriech, C. Prieur, Stability of reset control systems with nonzero reference, in: IEEE Conf. on Decision and Control, Cancun, Mexico, 2008, pp. 3386–3391. [11] W.H.T.M. Aangenent, G. Witvoet, W.P.M.H. Heemels, M.J.G. van der Molengraft, M. Steinbuch, Performance analysis of reset control systems, Internat. J. Robust Nonlinear Control (2009) doi:10.1002/rnc.1502. [12] A. Vidal, A. Baños, QFT-based design of PI + CI compensators: application in process control, IEEE Mediterr. Conference Control Automation (2008) 806–811. [13] M. Vidyasagar, Nonlinear Systems Stability, Prentice-Hall, London, 1993. [14] A.R. Bergen, R.L. Franks, Justification of the describing function method, SIAM J. Control 9 (1971) 568–589. [15] A. Mees, A. Bergen, Describing functions revisited, IEEE Trans. Automat. Control 20 (4) (1975) 473–478. [16] S.R. Sanders, On limit cycles and the describing function method in periodically switched circuits, IEEE Trans. Circuits Syst.-I 20 (4) (1975) 473–478.