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Ind. Eng. Chem. Res. 2004, 43, 8400-8402
RESEARCH NOTES Relay Feedback: A Complete Analysis for First-Order Systems Chong Lin, Qing-Guo Wang,* and Tong Heng Lee Department of Electrical and Computer Engineering, National University of Singapore, Singapore 119260, Singapore
This paper is concerned with the existence and stability of solutions (including limit cycles) for first-order linear systems under relay feedback. The plant can have a time delay, and the relay can be asymmetric with hysteresis. Necessary and sufficient conditions are presented. 1. Introduction Some efforts have been devoted to the analysis of relay feedback systems. See the work of Astrom,1 Goncalves et al.,2 and Johansson et al.3 and references therein. Most analysis work has to be based on the assumption that a limit cycle does exist because of the difficulty of determining if it is really the case. Astrom1 gave formulas for computing periods of limit cycles (if any). These types of oscillations were further studied by Varigonda and Georgious.4 First-order systems under relay feedback are also important, and the results are in demand from relay applications such as identification and autotuning.5-7 So far, less exact analysis has been reported, and only partial work has been presented by Majhi and Atherton.5-7 This paper aims to provide a complete analysis for first-order systems under relay feedback, including the uniqueness of solutions, existence and stability of limit cycles, and limit cycle periods and amplitudes. 2. Problem Formulation and Results Given a transfer function G(s) ) Ke-τs/(s - a), consider its first-order state-space realization described by
x˘ (t) ) ax(t) + bu(t - τ) y(t) ) cx(t)
(1)
where x(t), y(t), u(t) ∈ R are the state, output, and control input, respectively; a, b, and c are constants with cb ) K * 0; and τ g 0 stands for the time delay. The system is under relay feedback:
{
u+, if y(t) > d+ or y(t) g d- and lim u(t - ) ) u+
u(t) )
f0+
u-, if y(t) < d- or y(t) e d+ and lim u(t - ) ) u-
(2)
f0+
where d+, d- ∈ R, with d- e d+ indicating hysteresis; u-, u+ ∈ R, and u- * u+. The initial function u(t˜) for ˜t ∈ [-L, 0] is * To whom correspondence should be addressed. Tel.: 6568742282. Fax: 65-67791103. E-mail:
[email protected].
u(t˜) ≡
{
u+, if y(0) > du-, if y(0) e d-
(3)
We call eqs 1-3 a relay feedback system and denote it by Στ. Here, an absolutely continuous function x(t) is called a solution to system Στ if it satisfies eqs 1-3 almost everywhere.8 Note that in this work we do not consider chattering solutions, which may occur in the case of d+ ) d-. Define the switching planes (indeed, the switching points) as
S+ :) {ξ ∈ R: cξ ) d+} ) {d+/c}
(4)
S- :) {ξ ∈ R: cξ ) d-} ) {d-/c}
(5)
Suppose a limit cycle of system Στ exists. Then, it is called regular if the relay switches twice a period and the trajectory of the limit cycle leaves S+ or S- after each intersecting instant. For a solution x(t) to system Στ, we say that it is locally stable if there exists a neighborhood around the initial condition x(t0) such that any trajectory starting in it converges to x(t). The solution x(t) is called (globally) stable if the trajectory starting from any initial condition converges to x(t). Our purpose is to give a complete analysis for the uniqueness of solutions and the existence and stability of limit cycles for system Στ. The results for stable plants (a < 0), unstable plants (a > 0), and integral plants (a ) 0) are given in propositions 2.1-2.3, respectively. The results for the periods and amplitudes of limit cycles (if any) are given in proposition 2.4. The proofs for the propositions are given in section 3. Proposition 2.1. Consider a system Στ with a < 0. (i) A unique solution exists for any initial condition if and only if any of the following hold: (a) τ > 0; (b) τ ) 0 and d- > max {-Ka-1u+, -Ka-1u-}; (c) τ ) 0 and d+ < min {-Ka-1u+, -Ka-1u-}; (d) τ ) 0 and -Ka-1u- e d+ and -Ka-1u+ g d-. (ii) A limit cycle exists if and only if τ > 0 and -Ka-1u- > d+ g d- > -Ka-1u+. If this is the case, the limit cycle is regular and unique. (iii) If a limit cycle exists, then the limit cycle is globally stable. Moreover, the limit cycle is the common trajectory after the first switch, independent of the initial conditions.
10.1021/ie034043a CCC: $27.50 © 2004 American Chemical Society Published on Web 11/25/2004
Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 8401
the two states corresponding to A+ and A- be xA+ and xA-, respectively; let the two intersecting points be x+ and x- and the time taken for the limit cycle to go from xA- (respectively xA+) to xA+ (respectively xA-) be T(respectively T+). Then, T- + T+ is the limit cycle period. The expressions for the limit cycle amplitudes and period are given below. Proposition 2.4. If a limit cycle exists in system Στ, then
(i) for a * 0 A- ) eaτ(d- + Ka-1u+) - Ka-1u+
(6)
A+ ) eaτ(d+ + Ka-1u-) - Ka-1u-
(7)
aτ
T- ) a-1 ln -1
T+ ) a Figure 1. Limit cycles for system Στ.
Proposition 2.2. Consider a system Στ with a > 0. (i) A unique solution exists for any initial condition if and only if any of the following hold: (a) τ > 0; (b) τ ) 0 and d- < min {-Ka-1u+, -Ka-1u-}; (c) τ ) 0 and d+ > max {-Ka-1u+, -Ka-1u-}; (d) τ ) 0 and -Ka-1u- g d+ g d- g -Ka-1u+. (ii) A limit cycle exists if and only if -Ka-1u- < d- e d+ < -Ka-1u+ and
0 max {Ku+, Ku-}; (c) τ ) 0 and 0 < min {Ku+, Ku-}; (d) τ ) 0 and Ku+ g 0 g Ku-. (ii) A limit cycle exists if and only if τ > 0 and Ku- > 0 > Ku+. If this is the case, the limit cycle is regular and unique. (iii) If a limit cycle exists, then the limit cycle is globally stable. Moreover, the limit cycle is the common trajectory after the first switch, independent of the initial conditions. Remark 2.1. First-order unstable relay feedback systems have been studied before by Majhi and Atherton.5-7 It is shown therein (under the assumption that a limit cycle does exist) that if there is no hysteresis and u- ) -u+ ) 1, it holds that τ < a-1 ln 2, which is a partial result compared with proposition 2.2(ii). Our results also provide a complete analysis for the existence of solutions and the existence and stability of limit cycles. Next, we consider the amplitudes and period for a limit cycle (if any). Note that the relay under a limit cycle switches at the instant when the trajectory of y(t) reaches the peak values A+ and A-; see Figure 1. Let
ln
-1
e (d+ + Ka u-) e (d- + Ka-1u+) + Ka-1(u- - u+) aτ
eaτ(d- + Ka-1u+) eaτ(d+ + Ka-1u-) + Ka-1(u+ - u-)
(8)
(9)
(ii) for a ) 0 A- ) Ku+τ + d-
(10)
A+ ) Ku-τ + d+
(11)
T- )
Kτ(u- - u+) + d+ - dKu-
(12)
T+ )
Kτ(u+ - u-) + d- - d+ Ku+
(13)
3. Proofs We omit the proof of proposition 2.1 because it is similar to (but a bit simpler than) the proof of proposition 2.2. Proof of Proposition 2.2. We first show point i. For τ > 0, it is easy to show that there exists a unique solution for any given initial condition. We now concentrate on τ ) 0. Without loss of generality, let the initial condition x0 satisfy cx0 > d- and thus the relay starts at u+. Then the trajectory of x(t) will be governed by
x(t) ) eat(x0 + ba-1u+) - ba-1u+
(14)
Because a > 0, it is easy to see that if d- < -Ka-1u+, then, for cx0 g -Ka-1u+, the relay will remain u+ for all t g 0 and, for cx0 < -Ka-1u+, x(t) will intersect Sat some instant t1 > 0. However, if d- g -Ka-1u- also holds, after t ) t1, the trajectory x(t) cannot evolve. Otherwise, for t > 0, we have
y(t1 + t) ) cx(t1 + t) )
{
eat(d- + Ka-1u-) - Ka-1u- g d-, for u ) ueat(d- + Ka-1u+) - Ka-1u+ < d-, for u ) u+
which contradicts the control law (2). If d- < -Ka-1u+ and d- < -Ka-1u-, after the instant t ) t1, the trajectory will be governed by x(t) ) eat[x(t1) + ba-1u-] - ba-1u-. Next, if d- g -Ka-1u+, we check that if d+ e -Ka-1ualso holds, a unique solution exists for any initial condition. Under d- g -Ka-1u+, if d+ > -Ka-1u-, then a similar analysis leads to a unique solution for any initial condition if d+ > -Ka-1u+ also holds. So far, point i is proved.
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Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004
Next we show points ii and iii. It is seen from the above that, for τ ) 0, there is no limit cycle because the solution, if any, tends to +∞ or -∞. We now concentrate on the case of τ > 0. Without loss of generality, assume that cx0 > d+. It is easy to see (like the case τ ) 0) that if cx0 g -Ka-1u+, then the trajectory x(t) starting from x0 evolves for all t g 0 while the relay remains u(t) ≡ u+. Let the initial state x0 satisfy -Ka-1u+ > cx0 > d+. Then the trajectory of x(t) will be governed by eq 14 until, for some time t1 > 0, it satisfies cx(t1) ) d-. After t ) t1, because of the time delay τ > 0, the trajectory will satisfy
x(t1 + t) ) eat[x(t1) + ba-1u+] - ba-1u+, 0 e t e τ before the switch occurs at t ) τ. It is easy to check that cx(t1 + t) < d- holds for all t ∈ (0, τ]. After time t1 + τ, the trajectory of x(t) will be governed by
x(t1 + τ + t) ) eat[x(t1 + τ) + ba-1u-] - ba-1u- (15) Similarly, the switch will occur if and only if -1
cx(t1 + τ) + cba u- > 0
Ka-1(u+ - u-) d- + Ka-1u+
(16)
x(t1 + τ + t2 + t) ) eat[x(t1 + τ + t2) + ba-1u-] - ba-1u-, 0 e t e τ before the switch occurs at t1 + τ + t2 + τ. Again, the next switch will occur if and only if
(18)
Also, with simple manipulations, eq 18 holds if and only if -1
0 0, the trajectory will satisfy
cx(t1 + τ + t2 + τ) + cba-1u+ < 0
A- ) cxA-
cxA- ) A-
With some simple manipulations, eq 16 is equivalent to
0 < τ < a-1 ln
-Ka-1u- < cx(0) < -Ka-1u+ will traverse S- and S+ at two points d-/c and d+/c, respectively. 0 Proof of Proposition 2.3. Note that in this case the trajectory of x(t) will be governed by x(t) ) but + x0. The proof is similar to but simpler than that for the case of a * 0 and thus is omitted here. 0 Proof of Proposition 2.4. (i) From eq 1 and Figure 1a, we can see that
(19)
Hence, when eqs 17 and 19 are combined, points ii and iii are proved by noting that, after time t1, the trajectory x(t) will be a limit cycle with two switchings per period. Moreover, any trajectory x(t) starting from the range
we obtain eqs 12 and 13.0 Literature Cited (1) Astrom, K. J. Oscillations in systems with relay feedback. IMA Vol. Math. Its Appl. 1995, 74, 1-25. (2) Goncalves, J. M.; Megretski, A.; Dahleh, M. A. Global stability of relay feedback systems. IEEE Trans. Autom. Control 2001, 46 (4), 550-562. (3) Johansson, K. H.; Rantzer, A.; Astrom, K. J. Fast switches in relay feedback systems. Automatica 1999, 35 (4), 539-552. (4) Varigonda, S.; Georgious, T. T. Dynamics of relay relaxation oscillators. IEEE Trans. Autom. Control 2001, 46 (1), 65-77. (5) Atherton, D. P.; Majhi, S. Plant parameter identification under relay control. Proc. 37th IEEE CDC 1998, 2, 1272-1277. (6) Majhi, S.; Atherton, D. P. Autotuning and controller design for unstable time delay processes. UKACC Int. Conf. Control 1998, 769-774. (7) Majhi, S.; Atherton, D. P. Autotuning and controller design for processes with small time delays. IEE Proc. Control Theory Appl. 1999, 146 (5), 415-425. (8) Filippov, A. F. Differential Equations with Discontinuous Righthand Sides; Kluwer Academic Publishers: Dordrecht, The Netherlands, 1988.
Received for review August 5, 2003 Revised manuscript received February 19, 2004 Accepted February 20, 2004 IE034043A
