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) forfirst-order linear systems under relay feedback. The plant can have a time delay, and the relaycan be asymmetric with hysteresis. Necessary and sufficient conditions are presented.

1. IntroductionSome efforts have been devoted to the analysis of

relay feedback systems. See the work of Astrom,1Goncalves et al.,2 and Johansson et al.3 and referencestherein. Most analysis work has to be based on theassumption that a limit cycle does exist because of thedifficulty of determining if it is really the case. Astrom1

gave formulas for computing periods of limit cycles (ifany). These types of oscillations were further studiedby Varigonda and Georgious.4

First-order systems under relay feedback are alsoimportant, and the results are in demand from relayapplications such as identification and autotuning.5-7

So far, less exact analysis has been reported, and onlypartial work has been presented by Majhi and Ather-ton.5-7 This paper aims to provide a complete analysisfor first-order systems under relay feedback, includingthe uniqueness of solutions, existence and stability oflimit cycles, and limit cycle periods and amplitudes.

2. Problem Formulation and ResultsGiven a transfer function G(s) ) Ke-τs/(s - a), consider

its first-order state-space realization described by

where x(t), y(t), u(t) ∈ R are the state, output, and controlinput, respectively; a, b, and c are constants with cb )K * 0; and τ g 0 stands for the time delay. The systemis under relay feedback:

where d+, d- ∈ R, with d- e d+ indicating hysteresis;u-, u+ ∈ R, and u- * u+. The initial function u(t̃) for t̃ ∈[-L, 0] is

We call eqs 1-3 a relay feedback system and denote itby Στ. Here, an absolutely continuous function x(t) iscalled a solution to system Στ if it satisfies eqs 1-3almost everywhere.8 Note that in this work we do notconsider chattering solutions, which may occur in thecase of d+ ) d-.

Define the switching planes (indeed, the switchingpoints) as

Suppose a limit cycle of system Στ exists. Then, it iscalled regular if the relay switches twice a period andthe trajectory of the limit cycle leaves S+ or S- after eachintersecting instant. For a solution x(t) to system Στ, wesay that it is locally stable if there exists a neighborhoodaround the initial condition x(t0) such that any trajectorystarting in it converges to x(t). The solution x(t) is called(globally) stable if the trajectory starting from any initialcondition converges to x(t).

Our purpose is to give a complete analysis for theuniqueness of solutions and the existence and stabilityof 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. Theresults for the periods and amplitudes of limit cycles (ifany) are given in proposition 2.4. The proofs for thepropositions 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- ed+ 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, thelimit cycle is regular and unique.

(iii) If a limit cycle exists, then the limit cycle isglobally stable. Moreover, the limit cycle is the commontrajectory after the first switch, independent of theinitial conditions.

* To whom correspondence should be addressed. Tel.: 65-68742282. Fax: 65-67791103. E-mail: elewqg@nus.edu.sg.

x̆(t) ) ax(t) + bu(t - τ)y(t) ) cx(t) (1)

u(t) ) {u+, if y(t) > d+ or y(t) g d- andlimεf0+

u(t - ε) ) u+

u-, if y(t) < d- or y(t) e d+ andlimεf0+

u(t - ε) ) u-

(2)

u(t̃) ≡ {u+, if y(0) > d-u-, if y(0) e d-

(3)

S+ :) {ê ∈ R: cê ) d+} ) {d+/c} (4)

S- :) {ê ∈ R: cê ) d-} ) {d-/c} (5)

8400 Ind. Eng. Chem. Res. 2004, 43, 8400-8402

10.1021/ie034043a CCC: $27.50 © 2004 American Chemical SocietyPublished on Web 11/25/2004

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- gd+ g d- g -Ka-1u+.

(ii) A limit cycle exists if and only if -Ka-1u- < d- ed+ < -Ka-1u+ and

If this is the case, the limit cycle is regular and unique.(iii) If a limit cycle exists, then the limit cycle is locally

stable, and the stability range is -Ka-1u- < cx(0) <-Ka-1u+. Moreover, the limit cycle is the commontrajectory after the first switch, independent of theinitial conditions in the stability range.

Proposition 2.3. 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 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 regularand unique.

(iii) If a limit cycle exists, then the limit cycle isglobally stable. Moreover, the limit cycle is the commontrajectory after the first switch, independent of theinitial conditions.

Remark 2.1. First-order unstable relay feedback sys-tems have been studied before by Majhi and Atherton.5-7

It is shown therein (under the assumption that a limitcycle does exist) that if there is no hysteresis and u- )-u+ ) 1, it holds that τ < a-1 ln 2, which is a partialresult compared with proposition 2.2(ii). Our results alsoprovide a complete analysis for the existence of solutionsand the existence and stability of limit cycles.

Next, we consider the amplitudes and period for alimit cycle (if any). Note that the relay under a limitcycle switches at the instant when the trajectory of y(t)reaches the peak values A+ and A-; see Figure 1. Let

the two states corresponding to A+ and A- be xA+ andxA-, respectively; let the two intersecting points be x+and x- and the time taken for the limit cycle to go fromxA- (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 andperiod are given below.

Proposition 2.4. If a limit cycle exists in system Στ,then

3. Proofs

We omit the proof of proposition 2.1 because it issimilar to (but a bit simpler than) the proof of proposi-tion 2.2.

Proof of Proposition 2.2. We first show point i. Forτ > 0, it is easy to show that there exists a uniquesolution for any given initial condition. We now concen-trate on τ ) 0. Without loss of generality, let the initialcondition x0 satisfy cx0 > d- and thus the relay startsat u+. Then the trajectory of x(t) will be governed by

Because a > 0, it is easy to see that if d- < -Ka-1u+,then, for cx0 g -Ka-1u+, the relay will remain u+ forall t g 0 and, for cx0 < -Ka-1u+, x(t) will intersect S-at some instant t1 > 0. However, if d- g -Ka-1u- alsoholds, after t ) t1, the trajectory x(t) cannot evolve.Otherwise, for t > 0, we have

which contradicts the control law (2). If d- < -Ka-1u+and d- < -Ka-1u-, after the instant t ) t1, the trajectorywill 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-1u-also holds, a unique solution exists for any initialcondition. Under d- g -Ka-1u+, if d+ > -Ka-1u-, thena similar analysis leads to a unique solution for anyinitial condition if d+ > -Ka-1u+ also holds. So far, pointi is proved.

Figure 1. Limit cycles for system Στ.

0 < τ <

min {a-1 lnKa-1(u+ - u-)

d- + Ka-1u+

, a-1 lnKa-1(u- - u+)

d+ + Ka-1u-}

(i) for a * 0

A- ) eaτ(d- + Ka-1u+) - Ka-1u+ (6)

A+ ) eaτ(d+ + Ka-1u-) - Ka-1u- (7)

T- ) a-1 lneaτ(d+ + Ka-1u-)

eaτ(d- + Ka-1u+) + Ka-1(u- - u+)(8)

T+ ) a-1 lneaτ(d- + Ka-1u+)

eaτ(d+ + Ka-1u-) + Ka-1(u+ - u-)(9)

(ii) for a ) 0A- ) Ku+τ + d- (10)

A+ ) Ku-τ + d+ (11)

T- )Kτ(u- - u+) + d+ - d-

Ku-(12)

T+ )Kτ(u+ - u-) + d- - d+

Ku+(13)

x(t) ) eat(x0 + ba-1u+) - ba-1u+ (14)

y(t1 + t) ) cx(t1 + t) )

{eat(d- + Ka-1u-) - Ka-1u- g d-, for u ) u-

eat(d- + Ka-1u+) - Ka-1u+ < d-, for u ) u+

Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004 8401

Next we show points ii and iii. It is seen from theabove that, for τ ) 0, there is no limit cycle because thesolution, if any, tends to +∞ or -∞. We now concentrateon the case of τ > 0. Without loss of generality, assumethat cx0 > d+. It is easy to see (like the case τ ) 0) thatif cx0 g -Ka-1u+, then the trajectory x(t) starting fromx0 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 14until, for some time t1 > 0, it satisfies cx(t1) ) d-. Aftert ) t1, because of the time delay τ > 0, the trajectorywill satisfy

before the switch occurs at t ) τ. It is easy to check thatcx(t1 + t) < d- holds for all t ∈ (0, τ]. After time t1 + τ,the trajectory of x(t) will be governed by

Similarly, the switch will occur if and only if

With some simple manipulations, eq 16 is equivalentto

Under condition (17), for some time t2 > 0, the trajectoryin eq 15 satisfies cx(t1 + τ + t2) ) d+. After time t1 + τ+ t2, because of τ > 0, the trajectory will satisfy

before the switch occurs at t1 + τ + t2 + τ. Again, thenext switch will occur if and only if

Also, with simple manipulations, eq 18 holds if andonly if

Hence, when eqs 17 and 19 are combined, points ii andiii are proved by noting that, after time t1, the trajectoryx(t) will be a limit cycle with two switchings per period.Moreover, any trajectory x(t) starting from the range

-Ka-1u- < cx(0) < -Ka-1u+ will traverse S- and S+ attwo points d-/c and d+/c, respectively. 0

Proof of Proposition 2.3. Note that in this case thetrajectory of x(t) will be governed by x(t) ) but + x0. Theproof is similar to but simpler than that for the case ofa * 0 and thus is omitted here. 0

Proof of Proposition 2.4. (i) From eq 1 and Figure1a, we can see that

which gives eq 6 because of cb ) K. Similarly, we canobtain eq 7. Expression (8) follows from

and cb ) K, while eq 9 follows from similar equations.(ii) It is obvious from Figure 1b that eqs 10 and 11

hold. Also, from

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. Globalstability of relay feedback systems. IEEE Trans. Autom. Control2001, 46 (4), 550-562.

(3) Johansson, K. H.; Rantzer, A.; Astrom, K. J. Fast switchesin relay feedback systems. Automatica 1999, 35 (4), 539-552.

(4) Varigonda, S.; Georgious, T. T. Dynamics of relay relaxationoscillators. IEEE Trans. Autom. Control 2001, 46 (1), 65-77.

(5) Atherton, D. P.; Majhi, S. Plant parameter identificationunder relay control. Proc. 37th IEEE CDC 1998, 2, 1272-1277.

(6) Majhi, S.; Atherton, D. P. Autotuning and controller designfor unstable time delay processes. UKACC Int. Conf. Control 1998,769-774.

(7) Majhi, S.; Atherton, D. P. Autotuning and controller designfor processes with small time delays. IEE Proc. Control TheoryAppl. 1999, 146 (5), 415-425.

(8) Filippov, A. F. Differential Equations with DiscontinuousRighthand Sides; Kluwer Academic Publishers: Dordrecht, TheNetherlands, 1988.

Received for review August 5, 2003Revised manuscript received February 19, 2004

Accepted February 20, 2004

IE034043A

x(t1 + t) ) eat[x(t1) + ba-1u+] - ba-1u+, 0 e t e τ

x(t1 + τ + t) ) eat[x(t1 + τ) + ba-1u-] - ba-1u- (15)

cx(t1 + τ) + cba-1u- > 0 (16)

0 < τ < a-1 lnKa-1(u+ - u-)

d- + Ka-1u+

(17)

x(t1 + τ + t2 + t) )

eat[x(t1 + τ + t2) + ba-1u-] - ba-1u-, 0 e t e τ

cx(t1 + τ + t2 + τ) + cba-1u+ < 0 (18)

0 < τ < a-1 lnKa-1(u- - u+)

d+ + Ka-1u-

(19)

A- ) cxA-

xA-) eaτ(x- + a-1bu+) - a-1bu+

cx- ) d-

xA+) eaT-(xA-

+ a-1bu-) - a-1bu-

cxA+) A+

cxA-) A-

A+ ) Ku-T- + A-, A- ) Ku+T+ + A+

8402 Ind. Eng. Chem. Res., Vol. 43, No. 26, 2004