The Nyquist criterion for LTI Time-Delay Systems

Transcription

1 Recent Researches in Autoatic ontrol The Nyquist criterion for LTI Tie-Delay Systes LIBOR PEKAŘ, RADEK MATŠŮ, PETR DOSTÁLEK, JAN DOLINAY Departent of Autoation and ontrol Engineering, Faculty of Applied Inforatics Toas Bata niversity in Zlin na. T.. Masaryka 5555, 761 Zlin ZEH REPBLI Abstract: - This paper extends results about stability and stabilization of a retarded quasipolynoial obtained using the Mikhaylov criterion earlier. Retarded quasipolynoials appear as nuerators and denoinators of linear tie-invariant tie-delay systes (LTI-TDS. A LTI-TDS syste of retarded type (destitute of distributed delays is said to be stable if all roots of its characteristic quasipolynoial are located in the open left-half coplex plane. The contribution transfors the forulation of spectru assignent of a characteristic quasipolynoial into the language of the Nyquist criterion for the open loop of a control syste. Again, the arguent principle is utilized to derive generalized Nyquist criterion for LTI-TDS. Stability easures related to the criterion are discussed with the specifications for LTI-TDS. An illustrative exaple is presented to illuinate the results. Key-Words: - Tie-delay systes, Nyquist criterion, stability, stabilization, quasipolynoial, gain argin 1 Introduction Asyptotic stability, spectru analysis and stabilization of linear tie-invariant tie-delay systes (LTI-TDS have been challenging tasks in control theory during last decades. The probles are nontrivial even for these siple-odeled systes due to their infinite diensional nature. A vast bulk of various significant results was obtained and reported; without any attept to be exhaustive, see for instance [1] [7]. LTI-TDS can be represented in the input-output description by the Laplace transfer function as a ratio of so-called quasipolynoials in one coplex variable, instead of polynoials as for delay-free systes. Delay in the feedback can significantly deteriorate the quality of control perforance, naely stability and periodicity. For luped delays, the denoinator quasipolynoial decides about the control syste asyptotic stability because of the fact that its zeros are syste poles with the sae eaning as for polynoials; however, the spectru is infinite due to a quasipolynoial transcendental for. Analysis of asyptotic properties of the characteristic quasipolynoial and its spectru is one of a possible ways how to handle LTI-TDS stability. The presented paper extends results obtained for a retarded quasipolynoial with two delays in [8] and for the control feedback with a first order LTI- TDS in [9]. The findings in these papers were obtained via the arguent principle (or via the Mikhaylov stability criterion. Applying the arguent principle for the control feedback along with the knowledge the open loop frequency response results in the use of the well known Nyquist criterion. The notorious precept about the nuber of open loop unstable poles, however, is not easy to utilize in the case of LTI-TDS due to the infinite spectru which is of an effort to be calculated [1]-[11]. Hence, in this paper we siply derive the generalized Nyquist criterion for LTI- TDS and we turn over soe stability easures, such as gain and phase argins and the iniu of the sensitivity function. There are soe specific features about the easures LTI-TDS which are discussed in the paper. An exaple supported by siulations in Matlab- Siulink is presented to clarify the results and stateents. 2 Preliinaries Recall basic results about the arguent (increent principle for retarded quasipolynoials and findings introduced in [8] and [9]. Theore 1 (Arguent increent principle [12]. onsider a retarded quasipolynoial of the for n ( 1 hi n i s s + s exp( s ϑ (1 i j 1 ij ij ISBN:

2 Recent Researches in Autoatic ontrol If for any iaginary s jω, ω, function has no zero in the closed right half s- plane iff the arguent of reaches the increent ( arg s ω [, nπ 2 (2 Definition 1 (Retarded quasipolynoial stability. Retarded quasipolynoial of the for (1 is said to be asyptotically stable if it has no root in the closed right half s-plane, i.e. if there is no σ such that ( σ,re{ σ}. Proposition 1 (Nuber of unstable roots [13]. onsider a retarded quasipolynoial (1. Then the nuber N of poles of located in the closed right half s-plane (i.e. unstable ones is n 1 N arg (3 2 π ω [, The following results have been derived for siple quasipolynoials with n 1 and h 1 and h 2, respectively. Theore 2 [9]. onsider the quasipolynoial s+ a ( ϑs kq exp + (4 where a ; k,ϑ > are fixed, whereas q is selectable. Then, if a ϑ 1 (5 the quasipolynoial (4 is asyptotically stable iff a q> (6 k ontrariwise, if a ϑ>1 (7 the quasipolynoial (4 is asyptotically stable iff ( a cosϑω > (8 k q where the crossover frequency ω is the iniu nonzero eleent of the set { : ω>,i{ ( jω } } Ω : ω (9 Definition 2. onsider quasipolynoial s+ a ( ϑs + kq exp( τs exp (1 with a ; k, ϑ, τ >. Here, the set of crossover frequencies is defined as { ω : ω>,i{ ( jω } Re{ ( jω } } Ω1 : The critical frequency ω is defined as ω : in ω : ω Ω1, arg s, arg (11 π ω [, ω ω [ ω, 2 for a particular critical gain q given by ( ϑω ( τω (12 ω asin q k sin (13 Reark 1 [8]. Eleents ω 1 Ω1 are calculated as all solutions of the transcendental equation ( τω ( (( ϑ τ ω ω1 cos 1 a sin 1 (14 Theore 3 [8]. If sin ( τω >, then quasipolynoial (1 is stable iff a ω a sin < q< k k sin ( ϑω ( τω (15 ontrariwise, if sin ( τω <, then quasipolynoial (1 is stable iff ω a sin q> k sin ( ϑω ( τω k where ω is the critical frequency. a (16 3 Main Result Nyquist riterion In this chapter the Nyquist criterion for retarded LTI-TDS based on the arguent principle is presented. As usual, the Nyquist criterion gives inforation about the closed-loop stability based on the knowledge of the overall phase shift (arguent increent of the open-loop transfer function around the critical point -1. onsider a siple control syste as in Fig. 1 and the corresponding closed loop transfer function ISBN:

3 Recent Researches in Autoatic ontrol WY where ( Y R (17 W 1 + s 1+ is the open-loop transfer function. Note that for LTI-TDS transfer functions are no longer rational but eroorphic. Express the transfer functions as b a, R q / p / R (18 where a, b, q, p are retarded quasipolynoials and is strictly proper and R is proper (the properness is defined as for delay-free systes using the highest s-power. Then q b p a a q b p a Y WY (19 W p + where the characteristic quasipolynoial reads p a q b + (2 We can forulate and prove the following theore. Theore 4 (The Nyqusit criterion for retarded quasipolynoials. Let the plant and the controller have transfer functions as in (18 and the control syste be in a siple for as in Fig. 1. Let a and p have no root on the iaginary axis, i.e. a, p for any iaginary s jω, ω. Then, if arg ω [, ( ( lπ / 2 p s a s The closed-loop syste is asyptotically stable if arg 1+ ω [, π ( ( n l 2 (21 (22 where n is the highest s-power in the closed-loop s as in (2. characteristic quasipolynoial ( Fig. 1 Siple feedback control loop Proof. Since the plant transfer function is strictly proper, the highest s-power n of s p s a s + q s b s p s a s. If ( ( ( ( ( equals that of ( ( arg jωω, [, ( nπ / 2 s (23 then the closed-loop syste is asyptotically stable according to Theore 1 (i.e. it characteristic quasipolynoial has all roots in the open left-half s- plane, and, siultaneously, arg ω [, ( /( a p nπ / 2 lπ / 2 s Fro (17 and (18 is, oreover, obvious that ( ( ( ( (24 ( arg s / a s p s arg 1+ ω [, ω [, (25 and the proof is finished. Thus, to test the closed-loop asyptotic stability, one can figure the Nyquist plot of and count its overall nuber of encircleents around the critical point -1, or ore precisely, the overall phase shift of the curve. Now, the natural question is, whether the notorious precept about the nuber of unstable poles of (as for delay-free systes can be used. The answer is the following odification of Theore 4. Theore 5 (The Nyqusit criterion for retarded quasipolynoials an alternative forulation. Let the plant and the controller have transfer functions as in (18 and the control syste be in a siple for as in Fig. 1. Let a and iaginary axis, i.e., p p have no root on the a for any iaginary s jω, ω. Then, the closed-loop syste is asyptotically stable if ω [, ( 1+ n π arg (26 where n is the nuber of poles of with positive real parts (i.e. unstable poles. Proof. Assue results fro Theore 4 and Proposition 1. If there in no pure coplex conjugate pair of poles of (i.e. roots of a p, all unstable poles have positive real parts. The nuber of unstable poles is given by (3. If notations (21 and (26 are taken into account, one can write ISBN:

4 Recent Researches in Autoatic ontrol n n 1 π l l ( n 2 (27 2 π 2 n Substitution (27 into (22 yields (26, finally. 4 Stability Measures sing the Nyquist plot, a factor of stability can be easured by the gain argin and the phase argin which characterize the distance of the plot fro the critical point. As alternate easure, one can take the iniu of the sensitivity function into account. All these easures can serve not only for the stability easureent but also for the controller tuning. The classic conception of soe of the easures is deficient in any cases. For exaple, the gain argin eans the aplification of so that the closed loop becoes unstable (or equivalently, reains stable. Traditionally, it is supposed that the Nyquist plot crosses the negative real axis in the coordinate grater than -1; in other words, the gain argin is greater than one. However, this is not true in general. Siilarly, the phase argin expresses the phase shift which akes the closed loop being on the stability border; however, there can be ore than one these frequencies. Thus, we suggest these definitions of the gain argin and the phase argin: Definition 3. onsider that the closed loop syste is stable. The lower gain argin defined as A ( 1/ Re{ ( ω }, in : ax j ω Ω P,in P,in The upper gain argin A, ax reads A P,in ( 1/ { Re ( ω }, ax : in j ω Ω P,ax P,ax P,ax, in A is (28 (29 where ω P, in and ω P, ax are the appropriate phase crossover frequencies (through the iaginary axis fro sets Ω Ω P,in P,ax : : { ω : I{ }, Re{ ( jω } (, 1 (, } { ω : I{ },Re{ ( jω } ( 1, } (3 Definition 4. onsider that the closed loop syste is stable. The phase argin ϕ is defined as ϕ : Ω A ω : in ( π + arg( jωa od( 2π Ω A { ω : ( jω 1} A (31 where ω A is the appropriate gain (aplitude crossing frequency. orollary 1. The closed loop syste reains stable iff the open loop gain varies within the interval ( A A (32, in,,ax and the absolute value of the open loop phase shift change is lower than ϕ. Soe authors [14], [15] suggest to use a single paraeter M ax, instead of both A,, ϕ, which is the axiu (peak of the sensitivity function. Definition 5. The axiu (peak of the sensitivity function is defined as M ax ( jω : sup WE ω (33 WE 1 WY The value of M ax corresponds directly with a duping ratio of a closed-loop syste; the lower M is, the ore duped the syste is. Regarding ax the Nyquist plot the inverse value of M ax eans the distance of a point on the Nyquist plot fro the critical point -1, i.e. ( j 1 M ax 1/inf ω + (34 ω 5 Deonstration Exaple Exaple 1. Let the LTI-TDS plant be described by the transfer function b a exp s 5exp ( 1.1s ( s (35 The controlled syste is unstable which is clear fro the Mikhaylov plot a ( jω displayed in Fig. 2 (a detailed zoo to the origin of the coplex plane is added since the overall phase shift (the arguent change is 5π / 2, i.e. l 5. In other words, the plant has three unstable poles because of Theore 1. onsider a proportional controller q q and the task is to find the appropriate range of q so that the closed loop is asyptotically stable. ISBN:

5 Recent Researches in Autoatic ontrol Fig. 2 Mikhaylov plot of a fro Exaple 1 (a and a detail of the vicinity of the origin (b. Let us use the Mikhaylov criterion first. Hence, the closed-loop characteristic quasipolynoial reads s 5exp( s + q exp( 1. s 1 (36 According to Reark 1, one can calculate the set of crossover frequencies as Ω 1 {.953,4.741,6.72,1.385,12.498,... } and easily verify that the critical frequency satisfying definition (12 is ω. 953 which gives rise to the critical gain q Since sin (.953 >, Theore 3 results in the stabilizing interval 5 < q < 5.83 (37 Because n 1, the closed loop is stable according to the Nyquist criterion (see Theore 4 and Theore 5 if ω [, ( 1+ 3π arg (38 Set e.g. q 5. 4 and display the Nyquist plot of the open loop, see Fig. 3. Indeed, the overall phase shift around the point -1 is 3 π. Fig. 3 Nyquist plot of fro Exaple 1 (a and a detail of the vicinity of the critical point -1 (b, for q In order to easure the stability, find phase crossing frequencies defined in (3. Aazingly, it holds that Ω Ω Ω (39 1 P,in P,ax which also eans that the phase crossing frequencies are independent of q. Re { ( j ω P,in } are 1.8, ,.781,.411,..., whereas the set of { } Re ω is {.931,.548, } { ( } j P,ax, which gives rise to A. 926, A , in, ax This result agrees with boundaries (37. ain crossing frequencies can be calculated as Ω A.634,3.354,5.756 yielding ϕ { } {.31,.988,.595}. 31 in rad. Obviously, the easures indicate a very low closed-loop stability safeness since the recoended values are approxiately A 3, ϕ π 3. ISBN:

6 Recent Researches in Autoatic ontrol 4 onclusion The paper was aied to present the Nyquist criterion odified to be valid also for LTI-TDS of the retarded type. It was i.a. verified that the obligatory stateent about the nuber of open-loop unstable poles holds for these systes as well. Stability easures related to the Nyquist criterion were put ore precisely for retarded LTI-TDS. This contribution extended authors previous results about the Mikhaylov criterion for first order LTI- TDSs, which is deonstrated on a siulation exaple. Acknowledgeent The work was perfored with financial support of research project MSM This support is very gratefully acknowledged. References: [1]. Stépán, Retarded Dynaical Systes: Stability and haracteristic Functions, Vol. 21 of Pitan Research Notes in Matheatics Series, Longan Scientific and Technical, New York, [2] W. Michiels, K. Engelborghs, P. Vansevenant, and D. Roose, ontinuous Pole Placeent for Delay Equations, Autoatica, Vol. 38, No. 5, 22, pp [3] W. Michiels and S.-I. Nicolescu, Stability and Stabilization of Tie-Delay Systes. An eigenvalue based approach, Vol. 12 of SIAM Advances in Design and ontrol, SIAM, Philadelphia, 27. [4] K. u, V.L. Kharitonov, and J. hen, Stability of Tie-Delay Systes, Birkhäuser, Boston, 23. [5] W. Michiels, T. Vyhlídal, P. Zítek, H. Nijeijer, D. Henrion, Strong Stability of Neutral Equations with an Arbitrary Delay Dependency Structure, SIAM Journal on ontrol and Optiization, Vol. 48, No. 2, 29, pp [6] I.I. Delice and R. Sipahi, ontroller Design for Delay-Independent Stability of Multiple Tie- Delay Systes via Déscart s Rule of Signs, in Proceedings of the 9 th IFA Workshop on Tie-Delay Systes, Prague, 21, [D- ROM]. [7] T. Hashioto, Feedback Stabilization of Abstract Delay Systes on Banach Lattices, International Journal of Matheatical Models and Methods in Applied Sciences, Vol. 4, No. 3, 21, pp [8] L. Pekař and R. Prokop, Arguent Principle Based Stability onditions of a Retarded Quasipolynoial with Two Delays, in Last Trends on Systes (Volue I, 14th WSEAS International onference on Systes, orfu Island, reece, 21, pp [9] L. Pekař and R. Prokop, Stabilization of a Delayed Syste by a Proportional ontroller, International Journal of Matheatical Models and Methods in Applied Sciences, Volue 4, Issue 4, 21, pp [1] T. Vyhlídal, Analysis and Synthesis of Tie Delay Syste Spectru, Ph.D. Thesis, Faculty of Mechanical Engineering, zech Technical niversity in Prague, 23. [11] K. Engelborghs, DDE-BIFTOOL: A Matlab Package for Bifurcation Analysis of Delay Differential Equations, TW Report 35, Departent of oputer Science, Katolieke niversiteit Leuven, Belgiu, 2. [12] P. Zítek, Anisochronic Modelling and Stability riterion of Hereditary Systes, Probles of ontrol and Inforation Theory, Vol. 15, No. 6, 1986, pp [13] H. órecki, S. Fuksa, P. rabowski, and A Korytowski, Analysis and Synthesis of Tie Delay Systes, John Wiley & Sons, New York, [14] S. Skogestad and I. Postlethwait, Multivariable Feedback ontrol-analysis and Design, John Wiley & Sons, hichester, 25. [15] M. Šebek and Z. Hurák, An Often Missed Detail: Forula Relating Peek Sensitivity with ain Margin Less than One, In Proceedings of the 17th International onference on Process ontrol 9, Štrbské Pleso, Slovakia, 29, pp ISBN: