Criteria for Global Stability of Coupled Systems with Application to Robust Output Feedback Design for Active Surge Control

Transcription

1 Criteria for Global Stability of Couple Systems with Application to Robust Output Feeback Design for Active Surge Control Shiriaev, Anton; Johansson, Rolf; Robertsson, Aners; Freiovich, Leoni 9 Link to publication Citation for publishe version APA): Shiriaev, A., Johansson, R., Robertsson, A., & Freiovich, L. 9). Criteria for Global Stability of Couple Systems with Application to Robust Output Feeback Design for Active Surge Control. Paper presente at IEEE Multi-conference on Systems an Control, 9, St. Petersburg, Russian Feeration. General rights Copyright an moral rights for the publications mae accessible in the public portal are retaine by the authors an/or other copyright owners an it is a conition of accessing publications that users recognise an abie by the legal requirements associate with these rights. Users may ownloa an print one copy of any publication from the public portal for the purpose of private stuy or research. You may not further istribute the material or use it for any profit-making activity or commercial gain You may freely istribute the URL ientifying the publication in the public portal Take own policy If you believe that this ocument breaches copyright please contact us proviing etails, an we will remove access to the work immeiately an investigate your claim. L UNDUNI VERS I TY PO Box7 L un +4646

2 8th IEEE International Conference on Control Applications Part of 9 IEEE Multi-conference on Systems an Control Saint Petersburg, Russia, July 8-, 9 Criteria for Global Stability of Couple Systems with Application to Robust Output Feeback Design for Active Surge Control Anton S. Shiriaev, Rolf Johansson, Aners Robertsson, Leoni B. Freiovich Abstract The well-known an commonly accepte finite imensional moel qualitatively escribing surge instabilities in centrifugal an axial) compressors is consiere. The problem of global output feeback stabilization for it is solve. The solution relies on two new criteria for global stability propose for a class of nonlinear systems exploiting quaratic constraints for infinite sector nonlinearities. The constructive steps in eveloping a family of output feeback controllers base on these stability tests are presente. Performance of the closeloop systems are illustrate by simulations. I. INTRODUCTION This paper is focuse on esigning globally stabilizing output feeback controllers for the following nonlinear system t φ = ψ + φ + + φ) ) t ψ = φ u) ) Here ψ an φ are state variables, u is a control input, an is a positive constant. The moel ) ) is known for more than two ecaes as the Greitzer moel an has been use for approximating the couple behavior of pressure an flow in ynamics of compressor systems 9,,,,, 4. Difficulties in eveloping feeback controllers are ue to the presence of the cubic nonlinearity in the equation ). The key for our evelopment is the fact that this nonlinearity satisfy certain quaratic constraints. Both variables φ ) an ψ ) have physical meaning being eviations of the average flow an pressure from their mean values. Somes, both of them can be assume as outputs of the system. However, on-line measurements of the flow require special instrumentation an are often not feasible. Here we will consier the most ifficult case when only ψ - variable is available y = ψ ) for feeback esign. Presence of the nonlinearity in ynamics of ) ) makes the search for an output feeback controller A. Shiriaev is with Department of Applie Physics an Electronics, Umeå University, SE-9 87 Umeå, Sween, an Department of Engineering Cybernetics, NTNU, NO-749 Tronheim, Norway. R. Johansson an A. Robertsson are with Department of Automatic Control, LTH, Lun University, P.O. Box 8, SE- Lun, Sween. L. Freiovich are with Department of Applie Physics an Electronics, Umeå University, SE-9 87 Umeå, Sween. The work has been partly supporte by the Sweish Research Council the grants: 5-48, 8-54) an the Kempe Founation. an an associate Lyapunov function for the close-loop system to be quite a nontrivial mathematical problem. Clearly, since the nonlinearity is not globally Lipschtz an epens on φ-variable, esigning a globally stabilizing output feeback controller, relying on measurements of y = ψ only, is a challenge. To the best of our knowlege, this problem has remaine open until now. A. The Key Structural Transformation To solve the poste problems, we consier a very particular class of output feeback controllers. The key assumption is that after an appropriate change of coorinates, the closeloop systems can be written as ẋ A A = x B + w ė A e Cx) + w B x,e) 4) where x an e are components of the new state, v = Cx is the scalar input to the nonlinearity of the first subsystem, A, A, A, B, B, an C are constant matrices of appropriate imensions, w ) an w ) are static nonlinearities constructe from the cubic nonlinearity present in the original ynamics ) ). The paper is organize as follows: In Section II we present two criteria to verify global stability of 4) expresse as conitions for stability of the x-an e-subsystems. We show in Section III how both statements can be use for esigning output feeback controllers to stabilize ) ). The results of numerical simulation are iscusse in Section IV, an concluing remarks are mae in Section V. This Section is continue by iscussion of important properties of ) ). B. Quaratic Constraints for the Nonlinearity in the Surge Dynamics )-) Let us iscuss useful properties of the nonlinearity of the ynamical system )-) wv) := + v) 5) Lemma : The static nonlinearity 5) satisfies the incremental quaratic constraint QC) wv ) wv ) v v ), v, v R 6) Proof: Substituting 5) into the left-han sie of 6) we obtain v + ) v + ) v v ) = v + ) + v + ) + v + )v + ) v v ) /9/$5. 9 IEEE

3 Consier now the general form of a ynamic output feeback control law u = Uz,y), ż = Fz,y) 7) where U ) an F ) are smooth. Lemma : Suppose,τ max ) is the maximal interval of existence of a solution Xt) = φt);ψt);zt) of the close-loop system ), ), 7), where τ max > can be finite or not. Let FX) be a smooth function. Then, there are following two possible cases. ) There exists a sequence {t k } k= of moments with lim k t k = τ max such that the integrals of the quaratic form G v,wv) = wv) v) 4 v 8) with w ) efine in 5), along this solution with vt) = F Xt)) are strictly positive, i.e. t k G vt),w vt) ) t >, k =,,... 9) ) Along this solution, either FXt)) or FXt)) Moreover, the integral in 9) of the quaratic form G v,w ientically equals zero for any t,τ max ). Proof: To check 9), observe that G v,wv) = wv) v) 4 v = v + ) v ) Clearly, if integrating this relation along a solution of the close-loop system over t k,τ max ) with vt) = FXt)) results in zero value for any t k,τ max ), then vt) equals either or on t k,τ max ) an it is just left to notice that we can shift the since the system is -invariant. II. STABILITY CRITERIA FOR DYNAMICAL SYSTEM 4) Let us postpone the iscussion on how to transform the close-loop system ), ), 7) into the form of 4) an search for conitions, uner which global stability of 4) follows from properties of the separate x an e-subsystems. Loosely speaking, such conitions coul be interprete as successful state feeback an reuce observer esign criteria. However, it is worth noting that neither x-nor e-subsystems are inepenent, an just assuming asymptotic stability of each of them will not necessary result in asymptotic stability of 4). Stronger properties will be requeste an features of the nonlinear functions w ) an w ) will be use. Namely, we will use the following. Assumption : the nonlinearity w ) satisfies the relations 9) an ) similar to w ) efine in 5); Assumption : the nonlinearity w ) satisfies an infinite sector quaratic constraint G e,w x,e) = e T Π e w x,e), x, e ) with Π e being a constant matrix, that is the relation similar to 6) efine for the original nonlinearity w ). The first stability conition will rely on quaratic stabilities of both x an e-subsystems. Theorem : Let Assumptions an hol. Suppose that: ) There exist matrices P = P T an Q = Q T > such that the following inequality is vali x T P A x+b w ) +G Cx, w < xt Q x ) for all x an w. Moreover, there exists a matrix K such that G Cx,K x, x, an A + B K ) is Hurwitz. ) There exist matrices P = P T an Q = Q T > such that e T A T P + P A )e < e T Q e, e T P B + Π e ) w = ) for all e an w. Moreover, the matrix A is Hurwitz. Then, the nonlinear system 4) is quaratically stable, i.e. there are matrices P = P T > an Q = Q T > such that along any nontrivial solution xt);et) of 4) we have T T xt) xt) xt) xt) t P < Q 4) et) et) et) et) Proof: It follows from the minimal stability conitions of Theorem, that the matrices P an P are positive efinite. Let us consier the positive efinite quaratic function T x Wx,e) = P e x e, P = P, 5) γ P where γ is a positive constant, an compute the erivative of W ) along a solution of 4). t Wxt),et)) = P xt A x + A e + B w Cx) +γ e T P A e + B w x,e) Since the inequalities ) an ) are vali along any solution of 4), we have t W < P xt A x + A e + B w Cx) +γ e T P A e + B w x,e) +G Cx,w Cx) + γ G e,w x,e) Taking into account the relations ), ), one can observe that the right han sie of the last inequality becomes less or equal to x T Q x + x T P A e γ e T Q e, which after completing the squares becomes R x + r e) e T γ Q r T r )e 6) Here R is such that Q = RR T, an r T = A T P R. If γ is chosen large enough, then the quaratic form 6), which serves as an upper boun for efinite. t Wxt),et)), is negative

4 The secon stability criteria is again base on quaratic stability of the e-subsystem, but requires weaker properties only asymptotic stability) of the x-subsystem. Theorem : Let Assumptions an hol. Suppose that: ) There exist a square matrix P = P T an a row matrix q such that the following inequality is vali x T P A x + B w ) +G Cx, w q x 7) for all x an w. Moreover, there exists a matrix K such that G Cx,K x, x, the matrix A + B K ) is Hurwitz, an the pair q,a + B K ) is observable. ) The system 4) has no nontrivial solution xt),et) along which et) an C xt) const {,}. ) There exist matrices P = P T an Q = Q T > such that ) is vali for all e an w. Moreover, the matrix A is Hurwitz. Then, the origin of nonlinear system 4) is locally exponentially stable an globally asymptotically stable. Proof is base on the following four claims. Claim : Uner the assumptions of Theorem the x- subsystem with et) is globally asymptotically stable. Inee, the inequality 7) is vali for any vectors x an w, hence it is vali when w = K x so that x T P A x + B K x ) q x This is a matrix Lyapunov inequality, an so the facts that A + B K ) is Hurwitz an q,a + B K ) is observable imply that P >. Let us consier the Lyapunov function caniate V x) = x T P x. Due to valiity of 7) an ), its erivative along a solution xt) of the x-subsystem with et) satisfies the non-strict inequality t V xt)) = xt) T P A xt) + B w t)) xt) T P A xt) + B w t)) + G Cxt),w t) q xt) 8) It implies existence of the solutions of the x-subsystem with et) on an infinite interval of 5, Theorem., Lyapunov stability 5, Theorem 4., an their bouneness. It is left to verify that xt) converges to the origin. Let us consier the integral form of 8) V xt k )) V xt k )) = < t k x T P A x + B w )t t k x T P A x + B w ) + G Cxt),w t) t q xt) t t k 9) where the sequence {t k } k= is from 9), which by assumption exists for any particular nontrivial solution of 4). Note that using these special intervals of allows us to make one of the inequalities strict. This is the key point of the proof. Hence, V xt)) c const as T. It follows from 8) that the ω -limit set of any solution is nonempty, compact, an invariant 5, Lemma 4.. Taking a solution x t) of x-subsystem with et) from an ω -limit set an applying 8) an 9) to it, we conclue that c = an so xt) as T. Claim : There are no solutions of 4) that escape to infinity in finite. The -erivative of the function W ) efine by 5) with γ = along any solution of 4) satisfies the inequality t Wxt),et)) xt)t q T q xt) + xt) T P A et) et) T Q et) ε Wxt),et)) ) for some ε >. Hence, solutions cannot grow faster than exponentially, see e.g. 5, Lemma.4. Claim : Along any even unboune) solution xt),et) of 4), et) exponentially converges to zero. This fact immeiately follows from ). Claim 4: All solutions of 4) are boune. The first inequality in ) can be rewritten as t Wxt),et)) ε Wxt),et)) t) with t) = et) an some ε >. Integrating the last inequality results in the following one T WxT),eT)) Wx),e)) ε t)t Exponential convergence of t) to zero implies that ) L,+ ). In turn, integrability of t) over the interval,+ ) implies the bouneness of W ) an so of the solution xt),et). To finish the proof of Theorem, one can observe that any solution xt),et) of 4) will have a non-empty ω -limit set, while on this set e-variable shoul be zero. That is, this set consists of solutions of x-subsystems with et), which are asymptotically stable. This implies that all solutions of 4) converge to the origin. Furthermore, it is reaily seen that with the conitions of Theorem the origin is locally exponentially stable by linearization. Hence, it is globally asymptotically stable. III. ROBUST OUTPUT FEEDBACK DESIGN FOR ), ) A. Output Feeback Controller Design: Example Consier the family of output feeback controllers u = λ ψ + λ z + α u + c ψ ψ + c z z ) ) ) ż = λ ψ + λ 4 z + + c ψ ψ + c z z Note that the QC in the form of sector conitions imply that for all the linear functions w ), w ) satisfying the quaratic constraints, i.e. those that appear after linearization, the ynamics of x an e-subsystems are quaraticaly stable.

5 where z is a scalar an eight constant parameters λ s, α s an c s are to be etermine. The close loop system ), ) with the controller ) has the form φ φ ψ = λ λ ψ + ż λ λ 4 z αu w ψ,z+ w φ, w φ = + φ) ) w ψ,z = + c ψ ψ + c z z) To meet the structure of the system 4), introuce vectors x an e as follows x = ψ; z, e = φ c ψ ψ c z z, ) an consier, first, e-subsystem of 4), which is now a scalar ifferential equation ė = A e + B w x,e), eπ e w x,e), x, e 4) Definition of e ) in ) an equations ) allow to procee with computing t e as ) ) ė = c ψ cψ λ φ + c z λ ψ+ ) ) cψ λ + c z λ 4 z + w cψ α φ + u c z w ψ,z To match the first term on the right han sie of 4), the coefficients shoul obey the ientities ) A = c ψ cψ λ ), A c ψ = c z λ ) cψ λ A c z = 5) c z λ 4 To meet the structure of nonlinearity infinite sector conition) in 4), we obtain one more equation c ψ α u c z = 6) Then, inee, we can efine B an w ) in 4) as an w ) = w φ ) w ψ,z ) respectively, so that the inequality 4) is achieve e w x,e) = φ c ψ ψ c z z) + c ψ ψ + c z z) + φ) with Π e =. To match the structure of ynamics of the x-subsystem in 4) with x efine as in ), the ifferential equation for ψ in ) shoul be rewritten in coorinates ψ, z an e instea of the original ones ψ, z an φ. Namely ) ψ = φ λ ψ λ z α u w ψ,z 7) = e + c ψ λ )ψ + c z λ )z α u w ψ,z Summing up the manipulations mae with the close-loop system ), we obtain the next claim. Lemma : Suppose coefficients λ s, α s an c s of the controller ) satisfy the relations 5), 6). Then ) The close-loop system ) can be equivalently written as 4), where x an e are efine by ), c ψ ), B =, w x) = w ψ,z, A = w w φ w ψ,z, an x,e) = c cψ λ z λ A = λ λ 4, A = α u, B = ) The nonlinear function w ) is efine by w x,e) = w φ w ψ,z an satisfies the infinite sector conition 4) with Π e = ; ) The nonlinearity w ) efine by w x) := w ψ,z an the linear fictitious output v = c ψ ψ c z z = C x 8) of the x-ynamics satisfies the QC 9) w x) v 4 v, ψ, z with the reefine state an the linear output. Once the coefficients of the controller ) are foun such that following a change of coorinates the close-loop system ) can be rewritten as 4), we can apply Theorem an search for a set of parameters corresponing to stabilizing controllers ). The result base on applying the Frequency Theorem to verify ) is formulate next. Proposition : Consier the close-loop system ). Suppose that the parameters λ s, α s, an c s are such that the relations 5), 6) are satisfie an the following conitions hol: ) The inequality { } Re Tjω) 4 Tjω) < 9) is vali for all ω, where Ts) = C si A ) B ) = s + czαzc ψ λ )+c zα uλ c ψ α uλ 4 c ψ c z λ ) { s s λ 4 + c ψ λ ) The matrix A + 4 B C = } c cψ λ z λ λ λ 4 + λ4c ψ λ ) λ c z λ ) + 4 α u c ψ c z is Hurwitz; ) ) The constant A = c ψ is negative. Then with any sets of these parameters the close loop system ), i.e. the surge subsystem ), ) with the efine by these parameters ynamic output feeback controller ), is quaratically stable. B. Output Feeback Controller Design: Example Consier the following moification of the family of output feeback controllers ) u = λ ψ + λ z + α u + cψ ψ + c z z) ) + ε u q ż = λ ψ + λ 4 z + + cψ ψ + c z z) ) + ε z q q = c ψ ψ + c z z) ) ; T 4

6 where λ λ 4, α u,, c ψ, c z, ε u an ε z are constant parameters to be etermine. The close-loop system ) ) with any of such controllers is then φ ψ ż q = λ λ ε u λ λ 4 ε z c ψ c z + α u w ψ,z + w φ φ ψ z q + w φ = + φ) ) w ψ,z = + c ψ ψ + c z z) To meet the structure of the system 4), introuce vectors x an e as follows x T = ψ, z, q T, e = φ c ψ ψ c z z. ) The e-ynamics are then t e = t φ c ψ t ψ c z t z = c ψ e + w φ w ψ,z }{{}}{{}}{{} = A = B = w x,e) 4) where the last equality hols provie that the coefficients λ s, c s, α s, ε s satisfy 5), 6) an new relation c ψ ε u = c z ε z 5) Rewriting ψ -ynamics in e instea of φ-variable, similar to 7), allows us to meet the structure of the system 4). Lemma 4: Suppose coefficients λ s, c s, α s, ε s of the controller ) satisfy the relations 5), 6), 5). Then ) The close-loop system ) can be equivalently written as 4), where x an e are efine by ), c ψ A = ), B =, w x) = w ψ,z, w x,e) = w φ w ψ,z, an c A = ψ λ c z λ ε u λ λ 4 ε z c ψ c z A =, B = α u ; 6) ) The nonlinear function w ) is efine by w x,e) := w φ w ψ,z an satisfies the infinite sector conition 4) with Π e = ; ) The nonlinear function w ) is efine by w x) := w ψ,z an the linear fictitious output v = c ψ ψ c z z = c ψ, c z, }{{} x 7) = C of the x-ynamics satisfies the QC 9) w x) v 4 v, ψ, z, q with the reefine state an the linear output. With coefficients of the controller as in Lemma 4, we can apply Theorem using the Frequency Theorem to verify 7), an escribe stabilizing controllers in the family ). Proposition : Consier the close-loop system ). Suppose that the parameters λ s, α s, c s, an ε s are such that the relations 5), 6), 5) are vali an the next conitions hol: ) The inequality Re { } Tjω) 4 Tjω) 8) is vali for any ω, where Ts)=C si A ) B = s + p s s + l s 9) + l s + l with l = ε u an cψ p = λ λ α u c z l = + λ c ψ λ c z l = λ + cψ c z c z λ ) The matrix A + 4 B C is Hurwitz, an the pair,,,a ) is observable; ) ) The constant A = c ψ is negative. Then, with any sets of these parameters the close-loop system ), i.e. the surge subsystem ), ) with the efine by these parameters ynamic output feeback controller ), is globally asymptotically stable. A. Example IV. COMPUTER SIMULATIONS The set of stabilizing controllers ) escribe in Proposition is not empty. For instance, if = then the following coefficients λ = 7, λ = 4, λ = 7, λ 4 = 47 α u =, = 9, c ψ = 5, c z = 4) satisfy all the requirements. In particular, the transfer function ) is s/ / Ts) = s + s/ + /. It is positive real, an hence satisfies 9). The eigenvalues of the matrix A + 4 B C are {,.875}. Fig. epicts the evolution of φ an ψ variables for one of a typical solutions of the close-loop system. 5

7 φt) φt) ψt) ψt) Fig.. The solution of the close-loop system with the ynamical controller ) when coefficients are chosen as in 4). Fig.. The solution of the close-loop system with the ynamical controller ) when coefficients are chosen as in 4), 4). B. Example The set of stabilizing controllers ) escribe in Proposition is not empty. For instance, if =, then the controllers with coefficients 4) an ε u = 4/5, ε z = 4 4) satisfy all the requirements. Fig. epicts the evolution of φ an ψ variables for the close-loop system with the same initial conitions as in Example above. V. CONCLUDING REMARKS We suggest here two new families of robust output ropin-pressure) feeback controllers that stabilize the wellknown finite imensional moel for surge instability of compressor systems. Theoretical results are rigorously prove using quaratic constraints. Controllers from the first family ensure global exponential stabilization. The ones from the secon family provie integral action but only ensure local exponential an global asymptotic stability. Performance is verifie by simulations. REFERENCES D. Fontaine, S. Liao, J.D. Pauano, & P.V. Kokotovic, Nonlinear control experiments on axial flow compressor, IEEE Trans. on Automatic Control, vol., pp , 4. J.T. Gravahl & O. Egelan, Compressor surge an rotating stall: Moeling an control, Lonon: Springer-Verlag, 999. E.M. Greitzer & F.K. Moore, A theory of post-stall transients in axial compressor systems: Part II-Application., J. Eng. Gas Turbines & Power, 8, pp. 9, J. van Helvoirt, Centrifugal compressor surge, moeling an ientification for control, Ph.D. thesis, Techn.Universiteit Eihoven, 7. 5 H. Khalil, Nonlinear systems r e.), Prentice Hall, Upper Sale River,. 6 M. Krstic, D. Fontaine, P.V. Kokotovic, & J.D. Pauano, Useful nonlinearities an global stabilization of bifurcations in a moel of jet engine surge an stall, IEEE Trans. on Automatic Control, vol. 4, pp , M. Krstic, I. Kanellakopoulos, & P.V. Kokotovic, Nonlinear an aaptive control esign, New York: Wiley, M. Maggiore & K.M. Passino, A separation principle for a class of non-uco systems, IEEE Trans. on Automatic Control, vol. 487), pp.,. 9 F.K. Moore, A theory of rotating stall of multistage axial compressors: Parts I-III, J. Eng. Gas Turbines & Power, 6, pp. 6, 984. F.K. Moore & E.M. Greitzer, A theory of post-stall transients in axial compressor systems: Part I-Development of equations, J. Eng. Gas Turbines & Power, 8, pp , 986. J.D. Pauano, L. Valavani, A. Epstein, E. Greitzer, & G.R. Guenette, Moeling for control of rotating stall, Automatica, vol. 9), pp. 57 7, 994. A. Shiriaev, R. Johansson, & A. Robertsson, Some comments on output feeback stabilization of Moore-Greitzer compressor moel, in Proc. of 4r Conf. on Decision & Control, pp , 4. V.A. Yakubovich, G.A. Leonov, & A.Kh. Gelig, Stability of stationary sets in control systems with iscontinuous nonlinearities, Singapore: Worl Scientific, 4. 6