Stability Analysis of reset control systems with reset band

Stability Analysis of reset control systems with reset band ? Alfonso Ba˜ nos ∗ Sebasti´ an Dormido ∗∗ Antonio Barreiro ∗∗∗ ∗

Dpt. Inform´ atica y Sistemas, Univ. Murcia, 30071 Murcia, Spain (e-mail: abanos@ um.es). ∗∗ Dpt. Inform´ atica y Autom´ atica, U.N.E.D., 28040 Madrid, Spain (e-mail: sdormido@dia. uned.es) ∗∗∗ Dpt. Ingenier´ıa de Sistemas y Autom´ atica, Univ. Vigo, 36200 Vigo, Spain, (e-mail: [email protected]) Abstract: 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 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 funtion of arbitrary order full reset compensators with and without reset band. Keywords: Hybrid systems, Stability of hybrid systems, impulsive systems, reset control systems 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 Bainov and Simeonov (1989) or the most recent work Haddad et al. (2006), 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 Clegg (1958), and was followed by the works Horowitz and Rosenbaum (1975) and Krishman and Horowitz (1976), 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 Beker et al. (2004). 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 been approach the stability problem. See for example Beker et al. (2004), Ba˜ nos et al. (2007), and Ba˜ nos and Barreiro (2009), and references therein. An important practical issue of reset control is that compensator implementation is usually done by using a reset band. In addition, it has been noted that use of a reset band may improve stability and performance in systems ? This work has been supported by Ministerio de Ciencia e Innovaci´ on (Spanish Government) under projects DPI2007-66455-C02

with time-delays (Vidal and Ba˜ nos (2008)), due to the phase lead characteristic that is common to reset compensators. Phase lead can be even improved by using a 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 Vidyasagar (1993)). Although it is well-known that the describing function may fails in some cases, it has been proved to be an efficient method and has been formally justified in many practical cases (Bergen and Franks (1971); Mees and Bergen (1975)), including some types of switched systems (Sanders (1993)). 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 singleoutput system 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 finally approach the existence and stability of limit cycles by using a describing function-based analysis. 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.

e

C

v

u

P

y

Fig. 1. Reset compensator C and LTI plant P 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  ˙ ∈ / Bδ  x˙ r (t) = Ar xr (t) + Br e(t), (e(t), e(t)) C : xr (t+ ) = Aρ xr (t), (e(t), e(t)) ˙ ∈ Bδ (1)  v(t) = Cr xr (t) + Dr e(t)

3.1 FORE FORE is s simple reset compensator with a first oder base K compensator given by F OREbase (s) = s+a . As it is well known (see for example Vidyasagar (1993)), 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 F ORE (with δ = 0 in (1)), to the sinusoidal input e(t) = Esin(ωt) is given in the s-domain simply by K Eω V (s) = (2) 2 s + a s + ω2 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.

where the reset band surface Bδ is given by Bδ = {(x, y) ∈ R2 |(x = −δ ∧ y > 0) ∨ (x = δ ∧ y < 0)}, being δ some nonnegative 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 Bδ+ and Bδ− in the plane, as shown in Fig. 2.

Fig. 3. FORE sinusoidal response

Fig. 2. Reset band 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 . 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 F OREδ compensator is considered at a first instance. In addition, since DF computation will be based on FORE DF, this DF is considered first.

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, 2π ω ]. By symmetry (see Fig. 3), it will be computed by using the interval [0, ωπ ], where vt (t) = The result is R 1 π/ω vt (t)e−jωt ωdt π 0 E 2j

=

KEω −at e a2 + ω 2

(3)

π j2Kω 2 (1 + e−a ω )(4) π(a + jω)(a2 + ω 2 )

and finally adding to the frequency response of the base system, the describing function of FORE results in F ORE(ω) =

π j2Kω 2 K (1 + e−a ω ) + (5) 2 2 π(a + jω)(a + ω ) a + jω

3.2 F OREδ To obtain the describing function with reset band, again the sinusoidal response needs to be calculated. In this

case, note that F OREδ response to e(t) = Esin(ωt) over the period [−tδ , 2π ω − tδ ] is the same (except a traslation) that its response to e(t − tδ ) over the period [0, 2π ω ] (see Fig. 4). Thus, F OREδ DF can be obtained as the contribution of the transitory terms in the response to e(t − tδ ) = Ecos(ωtδ )sin(ωt) − Esin(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 Ecos(ωtδ )sin(ωt), and a second term vt2 with the response to −Esin(ωtδ )cos(ωt).

Fig. 5. F OREδ (E, ω) Phase for different values of δ/E Note that the F OREδ introduces extra phase lead for frequencies over a1 , in comparison with F ORE 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.

Fig. 4. FORE with reset band sinusoidal response By using (3), the term vt1 is directly given by vt1 = KEcos(ωtδ ) −at e , and thus using again symmetry arguments a2 +ω 2 its harmonic balance (corresponding to the fundamental component) is given by R π KEω 2 cos(ωtδ ) −a ω 1 π/ω ) vt1 (t)e−jωt ωdt π(a+jω)(a2 +ω 2 ) (1 + e π 0 (6) = R 2π/ω E −jωt δ 1 e(t − tδ )e−jωt ωdt 2j e 2π

0

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 F OREδ for K = 1 and a = 0. The result is q   1 − ( Eδ )2 ) j4( −1 δ 1  ejsin ( E )  (9) CIδ (E, ω) = 1+ jω π

On the other hand, the second transitory term is vt2 = δ ) −at − KEasin(ωt e , and its harmonic balance is now a2 +ω 2 R π KEaωsin(ωtδ ) −a ω 1 π/ω Again, note that the reset band introduces extra phase ) vt2 (t)e−jωt ωdt π(a+jω)(a2 +ω 2 ) (1 + e π 0 = (7) lead for values 0 < δ/E < 0.8, and at every frequency (Fig R E −jωtδ 2π/ω 1 e(t − tδ )e−jωt ωdt 2j e 6). 2π 0 Finally, the F OREδ describing function is obtained by adding to (6) and (7) the base frequency response. After some simple manipulations the result is: F OREδ (E, ω) =1+ F OREbase (jω) r π −1 δ j2ω(1 + e−a ω ) δ δ (ω 1 − ( )2 + a )ejsin ( E ) π(a2 + ω 2 ) E E

(8)

K where F OREbase (jω) = jω+a , and the identities sin(ωtδ ) = q 2 δ 1 − Eδ has been used. Note that E and cos(ωtδ ) = in contrast to (5), F OREδ (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.

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 F OREb 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. On the other hand, although the derivation is somehow involved (see Appendix A), 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 δ π δ 1 + Cr e ω Ar sin−1( E ) G( , ω)(I + e−Ar ω ) · π E

(10)

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 give approximate results, and that works well only if the plant P (s) filters higher order harmonics of the periodic signal v in Fig. 1. Alternatively, the above procedure is equivalent to find the crossings of the critical locus −1/Cδ (E, ω) with the plant Nyquist plot P (jω), and then obtain 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 F OREδ and CIδ , respectively. Note that since, in general, their describing functions depends 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 exist 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.

Fig. 6. CId (E, ω) Phase for different values of δ/E ·e(

Ar ω

δ −jI)(π−sin−1 ( E ))

Br

where G is given by −1 −jsin−1 (x)

G(x, ω) = {(jωI − Ar ) −1 jsin−1 (x)

+(jωI + Ar )

e

e

+

} · 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 + Cr e ω Ar π F (0, ω)(I + e−Ar ω ) · π Ar ·e( ω −jI)(π−π) Br

(12)

and after simplification results in C(jω) = Cbase (jω) + ω + Cr {(jωI − Ar )−1 + (jωI + Ar )−1 } · π π ·(jωI − Ar )−1 (eAr ω + I)Br

(13)

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 should also be noted that these expressions appears to be new, and thus by itself they are a important contribution to the reset control field. Obviously, expression for F OREδ as given in (8) is a particular case of (10) for K Cbase (jω) = jω+a , Ar = −a, Br = 1, Cr = K. In addition, CIδ FD expression in (9) is a particular case for K = 1 and a = 0. 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

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 segments in the Nyquist plane as δ/E goes from 0 to 1. 4.1 F OREδ with a first order plant A simple analysis shows that there are not limit cycles in the case corresponding to a feedback control system (Fig. 1) with a F OREδ 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 closedloop system will always correspond to a zero compensator output, and a alternating value of the error δ, −δ, . . .. 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 posible if the base feedback system is unstable. In this example, a F OREδ compensator, with an unstable 2 base compensator F OREbase (s) = s−1 and δ = 0.1, and a 1 first order plant P (s) = s+0.5 is considered. A simulation of the feedback system reveals (see Fig. 7) that a limit cycle exist with amplitude E = 0.20 and frequency ω = 1 rad/s. In Fig. 8 crossings of critical loci −1/F OREδ/E (1, ω), for ω = 0.5, 1, 1.5 rad/s, with P (jω) are shown. 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, Loeb method can be used to analyze the stability of the limit cycle . . .

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. After a numerical 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 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, Loeb method can be used to analyze stability . . .

Fig. 7. Time simulation of feedback system with F OREδ and first order plant (note that e = −y)

Fig. 10. Crossings of critical locus −1/CIb (E, ω) with Nyquist plot of the second order plant Fig. 8. Crossings of critical loci −1/F OREδ/E (1, ω), for ω = 0.5, 1, 1.5 rad/s, with Nyquist plot of the first order plant 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, and a second order plant P (s) = s+1 s2 . A simulation has been made first to analyze the existence of limit cycles, the result is that there exist a limit cycle as shown in Fig. 9, with amplitude E = 0.5 and frequency ω = 0.92 rad/s.

5. 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 for the describing function of general reset compensators with reset band, and in particular for FORE and CI compensator, 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. REFERENCES

Fig. 9. Crossings of critical locus −1/CIb (E, ω) with Nyquist plot of the plant

A. Ba˜ nos, J. Carrasco, and A. Barreiro. Reset times dependent stability of reset systems. European Control Conference, Greece, pp. 4792-4798, 2007. A. Ba˜ nos and A. Barreiro. Delay-independent stability of reset systems. IEEE Transactions on Automatic Control, 54, 2, pp. 337-341, 2009. D. D. Bainov and P. S. Simeonov. Systems with impulse effect: stability, theory and applications, E. Horwood Lted., 1989. O. Beker, C.V. Hollot, Y. Chait, and H. Han. Fundamental properties of reset control systems. Automatica, 40, pp. 905-915, 2004. A. R. Bergen and R. L. Franks. Justification of the Describing Function Method. SIAM J. Contro, 9, pp. 568-589, 1971. J. C. Clegg. A nonlinear integrator for servomechnisms. Transactions A.I.E.E.m, Part II, 77, pp. 41-42, 1958.

W. M. Haddad, V. Chellaboina, and S. G. Nersesov. Impulsive and hybrid dynamical systems: stability, dissipativity, and control, Princeton University Press, 2006. I.M Horowitz, and P. Rosenbaum. Nonlinear design for cost of feedback reduction in systems with large parameter uncertainty. International Journal of Control, 24, 6, pp. 977-1001, 1975. K. R. Krishman and I. M. Horowitz. Synthesis of a nonlinear feedback system with significant plant-ignorance for prescribed system tolerances. International Journal of Control, 19, 4, pp. 689-706, 1976. A. Mees and A. Bergen. Describing functions revisited. IEEE Trans. Automatic Control, 20, 4, pp. 473-478, 1975. S. R. Sanders. On limit cycles and the describing function method in periodically switched circuits. IEEE Trans. Circuits and Systems-I, 20, 4, pp. 473-478, 1975. A. Vidal and A. Ba˜ nos. QFT-based design of PI+CI compensators: application in process control. IEEE Mediterranean Conference on Control and Automation, pages 806–811, 2008. M. Vidyasagar. Nonlinear systems stability, Prentice-Hall, London, 1993.

Finally, after calculating the integrals and simplificating the resulting expression (10) is obtained. Details are omitted by brevity.

Appendix A. DESCRIBING FUNCTION Cδ (E, ω) OF A RESET COMPENSATOR WITH RESET BAND Derivation of expression (10) will be sketched here. Assume that the base compensator Cbase (s) has a statespace 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 , ωπ − tδ ] (see Figure 4), where ωtδ = sin−1 (δ/E). In that interval, the reset compensator exactly behaves 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 i used as independent variable, the result is that compensator output is given by:  Ψ1 Ψ E   Cr (−eAr ω F (Ψ1 )eAr ω + F (−Ψ))Br , if Ψ ∈ [0, π − Ψ1 ) 2j v(Ψ) = (A.1) Ψ Ψ E   Cr (e−Ar ω1 F (Ψ1 )eAr ω + F (−Ψ))Br , if Ψ ∈ [π − Ψ1 , π)) 2j 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 Ψ1 1 Cδ (E, ω) = {Cr eAr ω F (Ψ1 ) π

π−Ψ Z 1

Ψ

eAr ω e−jΨ dΨ +

0

+Cr e−Ar

π−Ψ1 ω

Zπ F (Ψ1 )

Ψ

eAr ω e−jΨ dΨ +

π−Ψ1

Zπ +Cr 0

F (−Ψ)e−jΨ dΨ}Br

(A.2)