Feedback stabilization of Networked Control Systems

Transcription

1 Feedbak stabilization of Networked Control Systems Bitsoris, G., Athanasopoulos, N., & Dritsas, L. (29). Feedbak stabilization of Networked Control Systems. In Proeedings of the IEEE Conferene on Control Appliations (CCA) & Intelligent Control (ISIC) IEEE. Published in: Proeedings of the IEEE Conferene on Control Appliations (CCA) & Intelligent Control (ISIC) Doument Version: Peer reviewed version Queen's University Belfast - Researh Portal: Link to publiation reord in Queen's University Belfast Researh Portal General rights Copyright for the publiations made aessible via the Queen's University Belfast Researh Portal is retained by the author(s) and / or other opyright owners and it is a ondition of aessing these publiations that users reognise and abide by the legal requirements assoiated with these rights. Take down poliy The Researh Portal is Queen's institutional repository that provides aess to Queen's researh output. Every effort has been made to ensure that ontent in the Researh Portal does not infringe any person's rights, or appliable UK laws. If you disover ontent in the Researh Portal that you believe breahes opyright or violates any law, please ontat openaess@qub.a.uk. Download date:6. Mar. 29

2 Feedbak Stabilization of Networked Control Systems George Bitsoris, Nikolaos Athanasopoulos and Leonidas Dritsas Abstrat In this paper the stability analysis and ontrol synthesis problems for Networked Control Systems (NCS) with bounded transmission delays (onstant and unknown or timevarying) are investigated. First, stability onditions for NCS desribed by ARMA models are established and a method for the determination of admissible delay range is developed. Then, a linear programming method for the design of linear state-feedbak ontrollers guaranteeing the stability of the system for any delay belonging to a prespeified range is developed. Contrary to the usual approahes based on the use of quadrati Lyapunov funtions, a polyhedral Lyapunov approah is adopted for both analysis and synthesis. A ontrol synthesis numerial example is given to illustrate the redution of onservatism of the tolerable delay range when ompared to former results. I. INTRODUCTION It is well known that the major ontrol hallenge in analysis and synthesis of Networked Controlled System (NCS) is to fae the problems due to the presene of unertain network-indued delays stemming from the very fat of utilizing a ommon ommuniation hannel for losing the loop [. These delays stem from the information flow between: a) the sensor and the ontroller, and b) the ontroller and the atuator. They have in general different harateristis depending primarily on the utilized network protool, the sheduling methods and the ommuniation overhead (paket ollisions/retransmissions/losses) used in NCS, while their presene imposes strit limitations on the ahievable feedbak performane [, [2, [3, [4. For the ase of a disrete stati feedbak implemented with a period h, these delays an be lumped into a single term τ k, where k refers to the sampling instant kh [2. Signifiant effort has reently been invested in developing ontrol methodologies to handle the network delay effet in NCSs (see surveys [, [5). In [3, LMIs are used for robust stability analysis and ontroller synthesis for networked systems subjet to unertain time-varying delays upper bounded by a sampling period; the ase of NCS with delays longer than one sampling period is presented in [4, [6. By treating the unertain NCS delay as a time-varying parameter unertainty, suffiient onditions, expressed as LMIs, for the existene of a stati stabilizing state feedbak ontroller appear in [7, [8, [9. Swithed system approahes that expliitly take into aount both the delay unertainty and This work was supported in part by the Greek State Sholarships Foundation. G. Bitsoris, N. Athanassopoulos and L. Dritsas are with the Control Systems Laboratory, University of Patras, Patras 265, Greee. e mail: bitsoris@ee.upatras.gr, nathanas@ee.upatras.gr, ldri@otenet.gr the ontroller asynhroniity have reently developed [, [. Most of these ontrol approahes use as analysis and synthesis tools either Lyapunov-Krasovskii funtionals or quadrati Lyapunov funtions. In the best of the authors knowledge, the use of polyhedral Lyapunov funtions has not yet been investigated in the ontext of NCS, although it has been shown to be a powerful tool in many interesting ontrol problems (inluding the ase of robust and onstrained ontrol) where it yields generi and less onservative results ompared to quadrati Lyapunov approahes [2, [3, [4, [5. In this paper NCSs are desribed by ARMA models. Thus the results onerning the positive invariane of polyhedral sets for ARMA models, established in [2 are used. The paper is organized as follows: Setion II refers to the modelling aspets of NCS with varying transmission delays. In Setion III, stability onditions for NCS systems desribed by ARMA models are established. These onditions lead to the development of a method for the determination of the admissible delay range for a systems ontrolled by linear state-feedbak. In Setion IV the design problem of linear state-feedbak ontrollers guaranteeing the stability of the system for any delay belonging to a prespeified range is investigated. An illustrative ontrol synthesis numerial example is given in Setion V. II. NCS-DYNAMICS Throughout this paper, apital letters denote real matries and lower ase letters denote olumn vetors or salars. For two real vetors x [ T x x 2... x n and y [ T y y 2... y n x < y (x y) is equivalent to x i < y i (x i y i ) for i,2,...,n. Similar notation holds for matries. Given a real matrix H (h ij ), H denotes the matrix H ( h ij ). Finally, O n denotes the n n null matrix. The dynamis of the NCS under investigation is desribed by the ombination of a ontinuous time linear time invariant plant with a disrete time ontroller and its onfiguration is shown in Figure. This onfiguration orresponds to the ase of a remote ontroller, non-olloated with the sensor and atuator [6, [7, [8. The sampling period h is assumed to be onstant and known, whereas both ontroller and atuator (inluding the zero-order-hold ZOH) are event-driven devies in the sense that they update their outputs as soon as they reeive a new sample. The state vetor x is sampled periodially, transmitted through the network, fed to the disrete time ontroller whih omputes the ontrol ation and transmits

3 it to the atuator after an unertain delay. The plant reeives this ommand after an unertain delay τ k. Inhere, the ase Delay t k ZOH Fig.. u(t) PLANT x (t) Ax (t)+bu(t) y(t)c x (t) DISCRETE TIME CONTROLLER u k-ksf x(kh) NCS strutural framework Periodi Sampler h x(kh) of SISO systems with less than one sampling period delay, (τ k < h), is examined. For the ontrol arhiteture shown in Figure, the system dynamis is desribed below, for t [ kh + τ k,kh + h + τ k+) : ẋ(t) A x(t) + B û(t), y(t) C x(t), () { [ u(k ), t kh h + τ û(t) k, kh + τ k) u(k), t [ kh + τ k, kh + h + τ k+). (2) The total delay within the kth sampling period, that is the time from the instant when the sampling node samples sensor data from the plant to the instant when atuators exert a ontrol ation (whose omputation was based on this sample) to the plant is denoted by τ k τs k + τa. k Moreover this total delay is assumed upper bounded as τ min < τ k τ max h and in general it is a time varying and unertain quantity, refleting the nature of the network involved, the network load, et. In (2), û(t) is the most reent ontrol ation presented to the event driven atuator at the time instane t within a sampling period (i.e. within the time interval [kh, kh + h)), and an take either one of the two values u(k ) or u(k). Certain part of the material in this setion an be traed in reent publiations [6, [7, [8 hene the presentation will be brief. The important modeling issue arising from (2) is that the atuation time instanes are not equidistant beause the pieewise onstant ontrol ation û(t) experienes a jump at the unertain time instane kh+τ k when the ontrol ation oming out of the event driven ZOH devie is updated from value u(k ) into u(k). Hene, unless τ k is onstant, it is not in general possible to treat the ensuing NCS in a standard sampled-data or time delayed setting and a hybrid setup should be used [, [6,[9, [2. Despite the jump nature of û(t), the disretization of (2) between onseutive sampling instanes is straightforward and the ensuing exat disretization is given by [7, [8: x(k + ) Φx(k) + Γ (τ k )û(k) + Γ (τ k )u(k ) (3) where Φ exp(a h) and Γ (τ k ) h τ k Γ (τ k ) Γ (τ k ) + exp(a λ)b dλ, h exp(a λ)b dλ (4) The unertain delay an always be deomposed as τ k τ + τ k with τ denoting the seleted nominal value, τ o [τ min,τ max. In this paper the nominal value τ of the unertain delay is hosen to be τ τ min. System variables with ( ) as supersript will denote the orresponding nominal value. The matries Γ (τ k ), Γ (τ k ) an then be deomposed into onstant and known nominal parts Γ (τ ), Γ (τ ) and unertain though bounded parts Γ, Γ, that is where Γ i (τ k ) Γ i (τ ) + Γ i (τ k,τ ) i, Γ (τ k,τ ) Γ (τ ) Γ (τ ) h τ h exp(a λ)b dλ h τ exp(a λ)b dλ h τ k exp(a τ)b dτ Γ (τ k,τ ) (5) System (3) an thus be equivalently written in the form x(k + ) Φx(k) + (Γ (τ ) + Γ (τ k,τ ))u(k) + +(Γ (τ ) + Γ (τ k,τ ))u(k ) (6) Using a disrete time linear state feedbak law u(k) K sf x(k), the losed loop dynamis beomes x(k+) [ Φ + Γ (τ k )K sf x(k)+ [ Γ (τ k )K sf x(k ). (7) III. STABILITY ANALYSIS OF NCS System (3) an be written in the form A (q )x(k) (8) where q is the bakward shift operator and A (q ) is a real polynomial matrix of the form A (q ) I n + A q + A 2q 2 The stability of this of lass systems via polyhedral Lyapynov funtions has been investigated in [2. The authors of this paper have established neessary and suffiient onditions for a salar funtion { } v(x) (Gx)i max i n w i

4 to be a Lyapunov funtion for system (8). These onditions are stated in the following theorem: Theorem :[2 If there exist matries G R p p,p n, rankg n, H R p p,h R p n, a vetor w R p with positive omponents and a ε > suh that GA i H i G, i,2 ( H + H 2 )w εw by setting w [ T and seleting matrix G as follows: For eah value of K of, G is omposed of the left eigenvetors of matrix (Φ + B d K sf ) where B d exp(a λ)b h dλ. Comparing to [8, where the stability of the same NCS has been studied, it an be learly seen that using Theorem 2 larger delay bounds are omputed even when matrix G is hosen randomly. ε < then the equilibrium x of system (8) is asymptotially stable. By applying this result to system (3) we establish onditions guaranteeing the asymptoti stability of NCS with fixed delay τ k : Theorem 2:If there exist matries G R p n,p n, rankg n, H (τ k ) R p p,h (τ k ) R p p, a vetor w R p with positive omponents and a ε(τ k ) > suh that G[Φ + Γ (τ k )K sf H (τ k )G (9) Network delay range τ k /h GΓ (τ k )K sf H (τ k )G () ( H (τ k ) + H (τ k ) )w ε(τ k )w () ε(τ k ) < (2) then the equilibrium x of system (3) is asymptotially stable. It an be easily seen that if for a fixed matrix G R p n,p n, rankg n and a vetor w R p with positive omponents onditions (9)-(2) of Theorem 2 are satisfied for all τ k belonging to a delay range [τ min,τ max, the equilibrium x of system (3) is asymptotially stable for any time varying delay τ k [τ min,τ max. Conditions (9)-(2) in Theorem 2 an be used to determine the range of admissible delay time for whih the stability of the losedloop NCS is guaranteed. This is illustrated in the following example. We onsider the open-loop stable ontinuous time linear system () with [ [ A, B C [ The sampling period is h.333 se while the unertain input delay an vary between zero and one full sampling period, i.e. τ k [,h). We assume a state-feedbak gain matrix K sf of the form K sf [ K of whih in fat orresponds to an output feedbak ontrol u(k) K of y(k). In Fig. 2, the stability margins are drawn for two different hoies of matrix G: The smaller margins are omputed when the nonsingular matrix G R 2 2 and the positive vetor w are hosen randomly, while the larger delay bounds for whih stability is preserved are omputed..5.5 Ouput feedbak gain K range of Fig. 2. Stability margins for the losed-loop system when output feedbak gain varies from - to. IV. A DESIGN APPROACH FOR FIXED AND UNCERTAIN DELAYS The design problem is formulated as follows: Given the ontinuous-time system (), the bounds τ min and τ max of the unertain input delay τ k and a sampling period h, determine a state-feedbak ontrol law u(k) K sf x(k) suh that the resulting losed-loop NCS is asymptotially stable for any time varying delay τ k [τ min,τ max. In order to establish an approah to this problem we onsider the perturbed desription of the NCS: x(k + ) Φ + Γ (τ )K sf + Γ (τ k,τ )K sf )x(k) + +(Γ (τ ) + Γ (τ k,τ )K sf )x(k ) (3) Theorem 3: If for a nonsingular matrix G R n n,there exist n n matries, H,H, H(τ k ),H + (τ k ),H (τ k ),H + (τ k ),H (τ k ), a vetor w R n with positive omponents and a positive salar ε < suh that G[Φ + Γ (τ )K sf H G (4) GΓ (τ )K sf H G (5) G Γ (τ k )K sf H(τ k )G (6) H H(τ k ) H + (τ k ) H (τ k ) (7) H + H(τ k ) H + (τ k ) H (τ k ) (8)

5 (H + (τ k ) + H (τ k ) + H + (τ k ) + H (τ k ))w εw (9) H + (τ k ) O n,h (τ k ) O n H + (τ k ) O n,h (τ k ) O n for all τ k [τ min,τ max then the equilibrium x of system (3) is asymptotially stable for any time varying delay time τ k [τ min,τ max. Proof: Taking into aount that Γ (τ k ) Γ (τ k ), from (4)- (9) it follows that G[Φ + Γ (τ k )K sf G[Φ + Γ (τ )K sf + Γ (τ k )K sf (H H(τ k ))G G[Γ (τ k )K sf G[Γ (τ )K sf + Γ (τ k )K sf (H + H(τ k ))G ( H H(τ k ) + H + H(τ k ) )w ( H + (τ k ) H (τ k ) + H + (τ k ) H (τ k ) )w ( H + (τ k ) + H (τk ) + H + (τ k ) + H (τ k ) w (H + (τ k ) + H (τ k ) + H + (τ k ) + H (τ k ))w εw beause H + (τ k ) O n, H (τ k ) O n, H + (τ k ) O n, H (τ k ) O n. Thus, setting H (τ k ) H H(τ k ) H (τ k ) H + H(τ k ) we onlude that all hypotheses of Theorem 2 are satisfied. Therefore the equilibrium x of system (3) is asymptotially stable for any time varying delay time τ k [τ min,τ max. A diret appliation of this result to the design of statefeedbak ontrollers is not possible beause unknown matries H + (τ k ),H (τ k ),H + (τ k ),H (τ k ) depend on τ k.in order to overome these diffiulties we next establish stability onditions independent of τ k.due to spae limitations the analysis is restrited to systems with real open-loop eigenvalues. The exponential exp(a τ) of a matrix an always be written in the form exp(a τ) a (τ)z + a 2 (τ)z a n (τ)z n (2) where Z i R n n i,2,...,n are real matries (onstituent matries) and a i (τ) i,2,...,n are real funtions of the form a i (τ) τ q e λiτ. Therefore, Γ (τ k,τ ) i h τ k a i (τ)dτz i B h τ k exp(a τ)b dτ i (τ k )Z i B (2) i where i (τ k ) are integrals of the form i (τ k ) h τ k e λiτ dτ in the ase where Z i orresponds to a simple real eigenvalue or i (τ k ) h τ k τ q e λiτ dτ in the ase where Z i orresponds to a multiple real eigenvalue. Let us define with + 2 max n max i max max i (τ k ) τ min τ k τ max In the ase where τ τ min, i (τ k ) are positive for any τ k [τ min,τ max, it follows that i max i (τ max ). We an now establish the following result: Theorem 4: If for a nonsingular matrix G R n n, there exist n n matries, H,H, H Z,H2 Z,...,Hr Z, a vetor w R n with positive omponents and a ε > suh that G[Φ + Γ (τ )K sf H G (22) GΓ (τ )K sf H G (23) ( H + H )w εw (24) GZ i B K sf H j ZG j,2,...,n (25) ( H H j Z + H + H j Z )w εw j,2,...,n (26) ε < (27) then the equilibrium x of system (3) is asymptotially stable for any time varying delay τ k [τ min,τ max. Proof: From (2) it follows that G Γ (τ k )K sf (τ k )GZ B K sf + 2 (τ k )GZ 2 B K sf r (τ k )GZ n B K sf for any τ k [τ min,τ max.and,by virtue of (25), G Γ (τ k )K sf (τ k ) H ZG (τ k ) H max ZG n ( max n ) i (τ k ) HZ i G H(τ k )G(28) where i H(τ k ) i i (τ k ) H i Z Taking into aount that Γ (τ k ) Γ (τ k ), from (22), (23) and (29) it follows that G[Φ + Γ (τ k )K sf G[Φ + Γ (τ )K sf + Γ (τ k )K sf (H H(τ k ))G

6 G[Γ (τ k )K sf G[Γ (τ )K sf + Γ (τ k )K sf (H + H(τ k ))G Thus onditions (9) and () of Theorem 2 are satisfied with Furthermore, + ( i H H H(τ k ) H H + H(τ k ) i i ) i (τ k ) H i (τ k ) w + H + H i max Z w i i max i (τ k ) ( H + H )w+ + ( i i (τ k ) ( H H i Z + H + H i Z )w i i (τ k ) ) εw + i i (τ k ) εw εw beause n i(τ k ) i. Therefore onditions () and (2) of Theorem 2 are also satisfied for any delay τ k belonging to the time interval [τ min,τ max Consequently the equilibrium x of system (3) is asymptotially stable for any time varying delay τ k [τ min,τ max. Aording to this result, a solution to the mentioned problem is obtained by first seleting a pair (G, w) and then by solving relations (22)-(27) with respet to the unknown matries K sf,h,h, H(τ k ),HZ i i,2,...,n and parameter ε. Sine these onditions imply the positive invariane of the polyhedral set R(G,w) {x R n : Gx w} with respet to the system desribed by the equation x(k + ) Φx(k) + Γ (τ )u(k),a pair (G,w) whih is (Φ,Γ (τ )) positively invariant [22 must be seleted. These relations an be formulated as linear algebrai equalities and inequalities by setting H H j Z Hj+ Z H j+ Z with H j+ Z H + H j Z Hj+ Z H j+ Z,H j Z,H j+ Z i,2,...,n,h j Z i,2,...,n. A solution of relations (22)-(27) an be obtained by defining an optimization problem having these relations as linear onstraints. Thus, a state-feedbak ontrol law u(k) K sf x(k) that stabilizes the NCS for any delay time τ k [τ min,τ max τ k an be determined by solving the linear programming problem ( H (τ k ) + H (τ k ) )w min {ε} (29) K sf,h,h,h j Z,Hj+ Z,Hj Z,Hj+ Z,Hj Z,ε ( H H(τ k ) + H + H(τ k ) )w under linear onstraints ( ( n ) ( i (τ k ) n ) ) H HZ i i max + i (τ k ) H + HZ i G[Φ + Γ (τ )K sf H G (3) w i max GΓ (τ )K sf H G (3) i (τ k ) H i (τ k ) + (H H i i max Z) i max w+ ( H + H )w εw (32) GZ i B K sf H j ZG j,2,...,n (33) i (τ k ) + H i (τ k ) + (H + H i i max Z) i max w H H j Z Hj+ Z H j Z i,2,...,n (34) H + H j Z i (τ k ) Hj+ Z H j+ Z i,2,...,n (35) H i (τ k ) w + H H i Z w+ (H j+ Z + H j+ Z + H j+ Z + H j+ Z)w εw j,2,...,n (36) max H j+ Z,H j Z,H j+ Z,H j Z j,2,...,n (37) If the optimal value of parameter ε satisfies inequality ε < the orresponding ontrol law u(k) K sf x(k) is a solution to the problem under onsideration. It should be emphasized that minimization of parameter ε results to improved transient behavior, beause parameter ε is a measure of the exponential onvergene of the state to the equilibrium of the delayed system. Indeed, it an be proven that under onditions (29)-(36) the positive definite funtion { } v(x) (Gx)i max i n is a Lyapunov funtion for system (3) whih satisfies inequality v(x(k + )) εv(x(k)) with ǫ <. w i V. NUMERICAL EXAMPLE We onsider an unstable ontinuous-time linear system () with matries [ [ A, B The sampling period is h. seond, the bounds of the unertain input delay are τ min, τ max.7 and the nominal disrete-time dynamis are omputed for τ. The feedbak gain K sf [ was omputed by solving the linear programming problem (29)- (36) setting G [ [, w

7 The optimal value of ε is.98 <. In Fig.3 and Fig.4 the state response of the disrete-time system and the ontrol effort for initial state x [ T are shown respetively. state variables x, x x x time (se) Fig. 3. State response of the losed-loop system for initial state x [ T. ontrol effort u time (se) Fig. 4. Control strategy for initial state x [ T. VI. CONCLUSIONS. A novel approah for both the stability analysis and state feedbak ontroller design for linear Networked Control Systems has been presented. The NCS dynamis is desribed by ARMA models. Contrary to ommon quadrati Lyapunov funtions or Lyapunov-Krasovskii funtionals used in previous papers, asymptoti stability is proved using polyhedral Lyapunov funtions. A benefit of this approah is the redution of onservativeness in the stability analysis when ompared to other existing results. 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