Outline. Introduction to Time-Delay Systems. lecture no. 2. What is changed for delay systems? Nyquist stability criterion

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1 Outline ntroduction to Time-Delay Systems Nyquist criterion for time-delay systems e sh Roots of quasi-polynomials: general observations lecture no. Delay sweeping (direct method) Leonid Mirkin Commensurate delays Faculty of Mechanical Engineering, Technion srael nstitute of Technology Department of Automatic Control, Lund University Lyapunov s methods Nyquist stability criterion What is changed for delay systems? y L.s/ e - r Nothing, except that we might no longer be interested in NC as stability region. The idea is to use plot of L.j!/ to count the number of closed-loop poles in NC. Namely, assume that L.s/ has no pole/zero cancellations in NC and denote: n ol number of poles of L.s/ in NC number of poles of CL.s/ in NC ~ number of clockwise encirclements of C j by Nyquist plot of L.j!/ as! runs from to n cl Then n cl D n ol C ~ Formal workaround: shift Nyquist contour a bit left or, equivalently, plot Nyquist plot of L. C j!/ for some <, yet by this intuition we have about frequency response gets lost. We still can use L.j!/ if we rule out situation with pole chain around j!-axis. f L.s/ D b.s/cb h.s/e sh a.s/ca h.s/e sh, then closed-loop system is C L.s/ D a.s/ C a h.s/e sh.a.s/ C b.s// C.a h.s/ C b h.s//e sh and we should rule out ˇˇ ah./cb h./ ˇ D by considering it unstable. a./cb./

2 Dead-time systems Examples of jl r./j D y L r.s/e sh e - r L.s/ D sc sc e s : L.s/ D s sc e s : Particularly simple analysis because of simple rules of plotting L r.j!/e j!h. Just remember that if jl r./j D, closed-loop systems unstable no matter how many times the critical point is encircled. maginary Axis maginary Axis infinite number of closed-loop poles in NC no closed-loop poles in NC Examples of jl r./j > Examples of jl r./j < L.s/ D sc sc e :5s : L.s/ D sc sc e 5s : L.s/ D :4 s C:sC : Nichols Chart maginary Axis maginary Axis maginary Axis Open Loop Gain (db) Open Loop Phase (deg) infinite number of closed-loop poles in NC infinite number of closed-loop poles in NC no closed-loop poles in NC

3 Examples of jl r./j < Examples of jl r./j < L.s/ D :4 s C:sC e s : L.s/ D :4 s C:sC e 5s : Nichols Chart Nichols Chart maginary Axis Open Loop Gain (db) 5 5 maginary Axis Open Loop Gain (db) Open Loop Phase (deg) Open Loop Phase (deg) two closed-loop poles in NC no closed-loop poles in NC Examples of jl r./j < Outline L.s/ D :4 s C:sC e s : Nichols Chart Nyquist criterion for time-delay systems maginary Axis Open Loop Gain (db) 5 5 Roots of quasi-polynomials: general observations Delay sweeping (direct method) Open Loop Phase (deg) Commensurate delays four closed-loop poles in NC Lyapunov s methods

4 Problem formulation Consider (characteristic) quasi-polynomial h.s/ D P.s/ C Q.s/e sh with P.s/ D s n C p n s n C C p s C p ; Q.s/ D q m s m C q m s m C C q s C q ; q m : Continuity of roots Define r;h sup Re s W h.s/ D Then, the following result holds: Theorem. r;h is continuous as a function of h for all h >. if, in addition, h.s/ is retarded, then r;h is continuous at h D as well The problem is to check whether h.s/ has all its roots in C n NC for some <. mportant consequence of this is that as h changes, roots of h.s/ may transit LHP--RHP (or vise versa) by crossing j!-axis only. Assumptions For P.s/ D s n C p n s n C C p s C p ; Q.s/ D q m s m C q m s m C C q s C q ; q m : we assume that A : m n, A : if m D n, then jq m j <, A : P.s/ and Q.s/ have no common roots in NC (i.e., coprime in NC ), A 4 : p C q. because A, guarantee ˇˇ Q./ P./ ˇ < (otherwise unstable for all h > ). if A does not hold, h.s/ unstable for all h. if A 4 does not hold, h./ D for all h. Outline Nyquist criterion for time-delay systems Roots of quasi-polynomials: general observations Delay sweeping (direct method) Commensurate delays Lyapunov s methods

5 dea To analyze stability h.s/ by continuous increase of h starting from h D. Namely, the analysis steps are as follows:. locate roots of.s/ D P.s/ C Q.s/;. increase h and check for j!-axis crossings of roots of h.s/: LHP to RHP crossings called are switches RHP to LHP crossings called are reversals Stability of h.s/ can then be verified by counting switches and reversals. As polynomial.s/ is finite dimensional, this step is trivial. We ll see below that this step can be efficiently performed. j! crossings f at some h roots of h.s/ cross j!-axis, we have (mind A ): P.j!/ C Q.j!/e j!h D Q.j!/ P.j!/ D ej!h : This, in turn, is equivalent to:. ˇˇ Q.j!/ ˇ D or jp.j!/j D jq.j!/j (magnitude relation), P.j!/.!h D arg Q.j!/ P.j!/ C k for some k Z (phase relation). Note that: for any! > satisfying, equality is always solvable for h; if! > is solution of, then so is!; if! D is solution of, equality cannot hold because of A 4. Conclusion: existence of j! roots of characteristic equation completely determined by magnitude equation and does not depend on delay. Positive solutions of ja.j!/j D jb.j!/j This equation can be rewritten as P.j!/P. j!/ Q.j!/Q. j!/ µ.!/ D ; which is polynomial equation in!. Thus, all frequencies! i at which rots of h.s/ cross j!-axis can be found from positive real roots of.s/. Example Let P.s/ D s C :s C and Q.s/ D q >. Then.!/ D.! C j:! C /.! j:! C / q D.! / C :! q D! 4 :995! C q D Three situations possible:. if < q < p :995 :99875, there are no real solutions;. if p :995 q <, there are two positive real solutions! D :995 C q :995 C q ;! D :995 q:995 C q. if q, there is one positive real solution! D :995 C q:995 C q :

6 Crossing directions ˇ Depend on.!/ sgn Re ds at (positive) crossing frequencies: ˇsDj! dh.! i / > roots migrate from LHP to RHP at! i > (switch).! i / < roots migrate from RHP to LHP at! i > (reversal) Some differential calculus Note that D d dh h.s/ D dp.s/ ds ds dh C dq.s/ ds ds dh e sh Q.s/e sh h ds dh C s : Thus, denoting by./ differentiation with respect to s, we have:.! i / D roots migration depends on higher derivatives or ds dh D sq.s/e sh P P.s/ C Q.s/e sh hq.s/e D s.s/ sh P.s/ Q.s/ Q.s/ C h ; The question now is how to compute sgn Re ds dh at jrc solutions of h.s/ D? where equality Q.s/e sh D P.s/ was used. Since sgn Re D sgn Re, P.j!/ Q.j!/.!/ D sgn Re j! P.j!/ Q.j!/ C h j P.j!/ Q.j!/ D sgn Re (as Re h! P.j!/ Q.j!/ j! D ) D sgn Re j ; (as! R) P.j!/ P.j!/ which does not depend on h. Q.j!/ Q.j!/ Some differential calculus (contd) Multiplying expression under sgn by P.j!/P. j!/ D Q.j!/Q. j!/ >, dp.j!/.!/ D sgn Re j d.j!/ P. j!/ dq.j!/ d.j!/ Q. dp.j!/ D sgn Re d! P. j!/ dq.j!/ d! Q. j!/ j!/ Then, since sgn Re D sgn. C /, dp.j!/.!/ D sgn Re d! P. j!/ dq.j!/ Q. j!/ d! C dp. j!/ dq. j!/ P.j!/ Q.j!/ d! d! D sgn d.!/ d! Crossing directions (contd) Thus, we have: sgn P.! i / > roots migrate LHP--RHP at! i > (switch) sgn P.! i / < roots migrate RHP--LHP at! i > (reversal) sgn P.! i / D roots migration depends on higher derivatives or Still, Walton and Marshall (987) showed that if.!/ changes its sign from to C at! D! i, we have LHP--RHP migration, if from C to RHP--LHP migration, and if it doesn t change sign no migrations take place (touch point).

7 Example (contd) Return to example with P.s/ D s C :s C and Q.s/ D q >. Here.!/ D d d!.!4 :995! C q / D 4!.! :995/:. if q D p :995, then! D! D :995 and.! / D.! / D (in fact, in this case no j!-axis crossings take place);. if p :995 < q < (two different crossing frequencies), then Some qualitative observations about crossing frequencies.!/ is even real polynomial in! with positive leading coeff. (mind A, ):.!/! 5! 4!!!!.! / D 4! q:995 C q > ;.! / D 4! q:995 C q < (so that! is always switch and! is always reversal);. if q (one crossing frequency), then (so that! is always switch)..! / D 4! q:995 C q > Crossing frequencies solve.!/ D and crossing directions determined by directions of.!/ at zero crossings (for increasing!). This means that the largest crossing frequency is always a switch, switch and reversal frequencies always interlace (with possible tangential points, at which no crossings occur, between them) Crossing delays t s time to make use of the phase relation at j!-crossing:!h D arg Q.j!/ P.j!/ C k; k Z and such that h : () For each crossing frequency! i this equation yields sequence of delays h i;k D h i; C! i k; k N; where h i; > is the smallest solution of (). Then: if! i is switch (.! i / > ), two poles move LHP--RHP at each h i;k ; if! i is reversal (.! i / < ), two poles move RHP--LHP at each h i;k. Thus, if we know poles of.s/, stability analysis of h.s/ needs. sorting all h i;k in increasing order,. counting all crossings up to h. Example (contd) Return to example with P.s/ D s C :s C and Q.s/ D q and let q D :4 n this case we have! :76 (switch) and! :78 (reversal) and h ;k :54 C 5:44 k D f:54; 5:598; :94; 6:86; : : :g; h ;k :778 C 8:6 k D f:778; :88; 9:898; 7:958; : : :g: This yields the following ordered list of crossing (switch and reversal) delays: fh ; ; h ; ; h ; ; h ; ; h ; ; h ; ; : : :g: Since.s/ D s C :s C :4 is stable, h.s/ is stable in h Œ; :54/, unstable in h Œ:54; :778 (two poles went to RHP at h D :54), stable in h.:778; 5:598/ (two poles returned to LHP at h D :778), unstable in h Œ5:598; / (two poles went to RHP at h D 5:598, then another two at h D :94, before first two returned at h D :88). Smallest delay at which first crossing occurs need not correspond to largest frequency.

8 Example (contd) Thus, quasi-polynomial h.s/ D s C :s C C :4e sh is stable in h Œ; :54/ [.:778; 5:598/: And now compare with Nichols charts (doesn t it ring a bell?): h D Nichols charts of :4 s C:sC e sh Some qualitative observations about crossings The following facts are worth remembering: Delay distance between two crossings at the same frequency! i is! i. Hence, shortest distance is between two switches, which means that at some point there is always RHP poles. Consequently, there always exists a delay, say hmax, such that h.s/ unstable 8h h max. This h max is calculable. Open Loop Gain (db) 5 5 h D 5 f there are no crossing frequencies, then stability / instability properties are delay independent. h D h D Open Loop Phase (deg) Frequency domain vs. modal analysis There is some nice interplay between these methods. For example: roots crossing frequencies are crossover frequencies of loop frequency response, root crossing delays correspond to delay margins,... Other modal analysis methods bilinear (Rekasius) transformation replace e sh with s, which covers the same area in C as grows from to Cs D representation replace e sh with and analyze stability with respect to both s jr and T Arguably ( m rather opinionated on this:-), frequency-response analysis provides more insight (cleaner thinking), modal analysis easier leads to reliable computations. What is certain is that one must not confine oneself to either one of these approaches, interplay between them yields better understanding (and is also fun).

9 Outline Nyquist criterion for time-delay systems Roots of quasi-polynomials: general observations Delay sweeping (direct method) Commensurate delays Lyapunov s methods Example Let h;.s/ D s C e sh C e sh : Like in single-delay case, we look for j!-crossings of roots. As h;.s/ real, its j!-axis roots coincide with those of h;. s/, i.e., they satisfy both s C e sh C e sh D and s C e sh C e sh D From the first equation, e sh D s e sh, then from the second equation: D C e sh se sh D C e sh C s.s C e sh / D C s C. C s/e sh This is a single-delay quasi-polynomial with.!/ D.! /. C! / D!.! / from which crossing! D p (switch) and crossing h ; D As ;.s/ stable, h;.s/ stable iff h Œ; p /. p Cj arg p D p. More general observations Let h;.s/ D P.s/ C Q.s/e sh C Q.s/e sh Clearly, s c D j! c is root of h;.s/ iff it is root of e sh h;. s/ D Q. s/ C Q. s/e sh C P. s/e sh More general observations (contd) f s c D j! c is crossing point of h;.s/ such that jp.j! c /j jq.j! c /j; then it also crossing point of h;.s/. Hence, this s c D j! c is also root of h;.s/ P. s/ h;.s/ Q.s/e sh h;. s/ D P.s/P. s/ Q.s/Q. s/ C Q.s/P. s/ Q.s/Q. s/ e sh which is single-delay quasi-polynomial. Yet converse not necessarily true: h;.s/ might have more j!-roots 4 than h;.s/, which complicates the analysis. Moreover, crosing directions at h;.j! c / and h;.j! c / coincide iff jp.j! c /j > jq.j! c /j opposite iff jp.j! c /j < jq.j! c /j 4 n fact, j!-roots of h;.s/ are roots of either h;.s/ or P.s/P. s/ Q.s/Q. s/.

10 Outline Nyquist criterion for time-delay systems Roots of quasi-polynomials: general observations Delay sweeping (direct method) Commensurate delays Lyapunov s methods Lyapunov s method for finite-dimensional systems Consider LT system Px.t/ D Ax.t/; x./ D x and assume there is (differentiable) V.x/ W R n 7! R C such that 5. V./ D. V.x/ > for all dx. derivative along trajectory dt called Lyapunov function. Then system is stable (in the sense of Lyapunov). f replaced with. PV.x/ < system is asymptotically stable. Yet another way to end up with asymptotic stability is via LaSalle nvariance Principle. PV.x/ and PV.x/ implies x.t/. 5 n principle, statements like V.x/ > should be more specific, yet we proceed with this sloppiness to simplify exposition. Lyapunov s method for finite-dimensional systems (contd) Let s choose for some P >. Then V.x/ D x.t/p x.t/ PV.x/ D Px.t/P x.t/ C x.t/p Px.t/ D x.t/a P x.t/ C x.t/pax.t/ D x.t/.a P C PA/x.t/ f we can choose P > satisfying A P C PA D C C for some C, PV.x/ D x.t/c Cx.t/ ; which implies stability. For asymptotic stability we then need observability of.c; A/ as in that case Cx.t/ implies x.t/. n fact and P D 9P > such that A P C PA < A Hurwitz Z e A C C e A d > is observability Gramian of.c; A/. Adding delays: developing intuition via discrete systems Consider NxŒk C D A NxŒk C A NxŒk As state vector here Nx a D Nx Œk Nx Œk Nx Œk h, quadratic Lyapunov function should look like ƒ PN PN PN h NV. Nx a / D Nx Œk Nx Œk Nx Œk h N P PN PN h 6 4 : : : :: 7 6 : 5 4 PN h PN h PN hh hx hx D Nx Œk i PN ij NxŒk j id j D h > NxŒk NxŒk : NxŒk 7 5 h

11 Adding delays: Lyapunov-Krasovskiĭ functional Consider Px.t/ D A x.t/ C A x.t h/ Quadratic Lyapunov function (actually, functional fœ h; 7! R n g 7! R C ) for this system could be of the form V.x / D Z h Z h x.t /P.; /x.t /dd for some function P.; /, where ; h, such that P.; / D P.; / and R h R h./p.; /./dd > for all./ 6. This functional called Lyapunov-Krasovskiĭ functional. Alternative expression: V.x / D Z t t h Z t t x./p.t ; t /x./dd h Adding delays: Lyapunov-Krasovskiĭ functional (contd) Choice regarded sufficiently general is P.; / D P ı./ı./ C P./ı./ C P./ı./ C P./ı. / C P. / for some matrix P, matrix functions P./ and P./ defined in Œ; h, and matrix function P.t/ defined in t Œ h; h. n this case V.x / D x.t/p x.t/ C C x.t/ Z h Z h Z h P./x.t the derivative of which is a mess 6... x.t /P. /x.t /dd /d C Z h x.t /P./x.t /d 6 t is possible to choose P.; / of this form by reverse engineering: choose observable measurement operator for state vector and construct P.; / as observability Gramian. Delay-independent conditions via LK approach Consider Px.t/ D A x.t/ C A h x.t h/ and choose Z t V.x / D x.t/p x.t/ C x./p x./d t h for some P > and P h >. Then, using Leibniz integral rule, PV.x / D Px.t/P x.t/ C x.t/p Px.t/ C x.t/p x.t/ x.t h/p x.t h/ D x.t/ x.t h/ A P C P A C P P A h x.t/ A h P P x.t h/ Thus, if there are P > and P > such that A P C P A C P P A h A h P < ; P PV.x / and system asymptotically stable (mind LaSalle). This is Linear Matrix nequality (LM), which can be efficiently solved. Adding delays: Razumikhin approach Consider again Px.t/ D A x.t/ C A x.t h/ Lyapunov function for it, V.x /, doesn t have to be quadratic. We may take V.x / D max Œ h; QV.x.t C // for some Lyapunov function QV.x/. n this case P QV.x/ may be even positive at points where QV.x/ < V.x /. This, relaxed, condition reads as follows: Theorem (Razumikhin) System is asymptotically stable if there is V.x/ W R n 7! R C such that. V./ D. V.x/ > for all x. PV.x/ < whenever V.x.t// V.x.t C //, Œ h;, for some >

12 Delay-independent conditions via Razumikhin approach Consider Px.t/ D A x.t/ C A h x.t h/ and choose V.x/ D x.t/p x.t/ for some P >. For every > and > we can define function.t/ PV.x/ C V.x.t// V.x.t h// D Px.t/P x.t/ C x.t/p Px.t/ C x.t/p x.t/ x.t h/p x.t h/ D x.t/ x.t h/ A P C PA C P PA h A h P P x.t/ x.t h/ At x.t/ W V.x.t C // V.x.t// we have that PV. Thus, if <, the system is stable by Razumikhin arguments. Hence, if there is matrix P > and scalar > such that A P C PA C P PA h A h P P < ; Lyapunov-Krasovskiĭ vs. Razumikhin (not generic) f there are P > and > such that A P C PA C P A h P then PA h P < ; A P C P A C P P A h A h P < for P P D P and P D P : Thus, the LK condition holds whenever so does the Razumikhin condition, but not necessarily vise versa. Hence, in this case Razumikhin approach is potentially more conservative (solvability of LK LM does not necessarily mean that P h D P ). then the system asymptotically stable. Delay-independent stability in scalar case Consider Px.t/ D a x.t/ C a h x.t h/ for x R. Delay-independent stability would require stability at h D and no positive crossing frequencies. These read a C a h < and! C a D a h or, equivalently, a a h and a < (the latter can be interpreted as stability under h! ). has no positive real solutions Scalar case: Lyapunov-Krasovskiĭ and Razumikhin LK solvability LM becomes 9p > ; p > such that a p C p a h p < : a h p p Taking Schur complement of the.; / term, the last condition equivalent to a p C p C a h p =p < p C a p p C a h p < : The latter reads a < and a p a h p a a h ; i.e., we recover exact conditions (LK conservative in general). n scalar case a p C p a h p a p C p a < h p < a h p p a h p p and Razumikhin result coincides with Lyapunov-Krasovskiĭ one.