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[2] A. T. Winfree, The Geometry of Biological Time. New York: Springer, 2001. [3] H. Wünsche, S. Bauer, J. Kreissl, O. Ushakov, N. Korneyev, F. Henneberger, E. Wille, H. Erzgrässer, M. Peil, W. Elsässer, and I. Fischer, “Synchronization of delay-coupled oscillators: A study of semiconductor lasers,” Phys. Rev. Lett., vol. 94, p. 163901, 2005. [4] L. Herrgen, S. Ares, L. G. Morelli, C. Schröter, F. Jülicher, and A. C. Oates, “Intercellular coupling regulates the period of the segmentation clock,” Curr. Biol., vol. 20, pp. 1244–1253, 2010. [5] W. O. Friesen and G. D. Block, “What is a biological oscillator?,” Am. J. Physiol., vol. 246, pp. 847–853, 1984. [6] , C. Fall, E. Marland, J. Wagner, and J. J. Tyson, Eds., Computational Cell Biology. New York: Springer, 2005. [7] M. Yeung and S. Strogatz, “Time delay in the Kuramoto model of coupled oscillators,” Phys. Rev. Lett., vol. 82, pp. 648–651, 1999. [8] M. G. Earl and S. Strogatz, “Synchronization in oscillator networks with delayed coupling: A stability criterion,” Phys. Rev. E, vol. 67, p. 036204, 2003. [9] I. J. Domian, A. Reisenauer, and L. Shapiro, “Feedback control of a master bacterial cell-cycle regulator,” Proc. Nat. Acad. Sci. USA, vol. 96, pp. 6648–6653, 1999. [10] J. Lewis, “Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator,” Curr. Biol., vol. 13, pp. 1398–1408, 2003. [11] J. C. Dunlap, “Molecular bases for circadian clocks,” Cell, vol. 96, pp. 271–290, 1999. [12] J. Kim, D. Shin, S. Jung, P. Heslop-Harrison, and K. Cho, “A design principle underlying the synchronization of oscillations in cellular systems,” J. Cell Sci., vol. 123, pp. 1537–1543, 2010. [13] B. Doiron, M. J. Chacron, L. Maler, A. Longtin, and J. Bastian, “Inhibitory feedback required for network oscillatory responses to communication but not prey stimuli,” Nature, vol. 421, pp. 539–543, 2003. [14] T. Bal, D. Debay, and A. Destexhe, “Cortical feedback controls the frequency and synchrony of oscillations in the visual thalamus,” J. Neurosci., vol. 20, pp. 7478–7488, 2000. [15] R. Bertram, L. Satin, M. Zhang, P. Smolen, and A. Sherman, “Calcium and glycolysis mediate multiple brsting modes in pancreatic islets,” Biophys. J., vol. 87, pp. 3074–3087, 2004. [16] P. Rapp, “Mathematical techniques for the study of oscillations in biochemical control loops,” Bull. Inst. Math. Appl., vol. 12, pp. 11–20, 1976. [17] Y. Hori, M. Takada, and S. Hara, “Biochemical oscillations in delayed negative cyclic feedback: Existence and profiles,” Automatica, vol. 9, no. 9, pp. 2581–2591, 2013. [18] T. Iwasaki, “Multivariable harmonic balance for central pattern generators,” Automatica, vol. 44, pp. 3061–3069, 2008. [19] Z. Chen, M. Zheng, W. Friesen, and T. Iwasaki, “Multivariable harmonic balance analysis of neuronal oscillator for leech swimming,” J. Comput. Neurosci., vol. 25, pp. 583–606, 2008. [20] Z. Chen and T. Iwasaki, “Matrix perturbation analysis for weakly coupled oscillators,” Syst. Control Lett., vol. 58, pp. 148–154, 2009. [21] N. A. M. Monk, “Oscillatory expression of Hes1, p53, and NFB driven by transcriptional time delays,” Curr. Biol., vol. 13, pp. 1409–1413, 2003. [22] D. A. Goodenough, J. A. Goliger, and D. L. Paul, “Connexins, connexons, and intercellular communication,” Annu. Rev. Biochem., vol. 65, pp. 475–502, 1996. [23] J. Kim, J. Kim, Y. Kwon, H. Lee, P. Heslop-Harrison, and K. Cho, “Reduction of complex singnaling networks to a representative kernel,” Sci. Signal., vol. 4, p. ra35, 2011. [24] R. Kageyama, Y. Niwa, H. Shimojo, T. Kobayashi, and T. Ohtsuka, “Ultradian oscillations in notch signaling regulate dynamic biological events,” Curr. Top. Dev. Biol., vol. 92, pp. 311–331, 2010. [25] Y. Q. Wang, Y. Hori, S. Hara, and F. J. Doyle, III, “The collective oscillation period of inter-coupled Goodwin oscillators,” in Proc. 51st IEEE Conf. Decision Control, Maui, FL, 2012, pp. 1627–1632. [26] H. K. Khalil, Nonlinear Systems. Upper Saddle River, NJ: Prentice Hall, 2002. [27] R. Courant and D. Hilbert, Methods of Mathematical Physics, Volumn 1. New York: Interscience Publishers, 1953. [28] H. G. Schuster and P. Wagner, “Mutual entrainment of two limit cycle oscillators with time delayed coupling,” Prog. Theor. Phys., vol. 81, pp. 939–945, 1989.
[29] S. Wang, L. Chandrasekaran, F. Fernandez, J. White, and C. Canavier, “Short conduction delays cause inhibition rather than excitation to favor synchrony in hybrid neuronal networks of the entorhinal cortex,” PLoS Comput. Biol., vol. 8, p. e1002306, 2012. [30] G. Schmid, X. Ao, and P. Haenggi, In-Phase and Anti-Phase Synchronization in Noisy Hodgkin-Huxley Neurons submitted [Online]. Available: http://arxiv.org/abs/1206.4841 [31] F. Giudicelli, E. M. Özbudak, G. J. Wright, and J. Lewis, “Setting the tempo in development: An investigation of the zebrafish somite clock mechanism,” PLoS Biol., vol. 5, pp. 1309–1323, 2007. [32] Y. Hori, T. Kim, and S. Hara, “Existence criteria of periodic oscillations in cyclic gene regulatory networks,” Automatica, vol. 47, pp. 1203–1209, 2011. [33] H. Hirata, Y. Bessho, H. Kokubu, Y. Masamizu, S. Yamada, J. Lewis, and R. Kageyama, “Instability of Hes7 protein is crucial for the somite segmentation clock,” Nature Genet., vol. 36, pp. 750–754, 2004.
Improved Feed-Forward Command Governor Strategies for Constrained Discrete-Time Linear Systems Alessandro Casavola, Emanuele Garone, and Francesco Tedesco
Abstract—This technical note presents an improved version of the FeedForward Command Governor (FF-CG) strategy recently proposed in [1] for the supervision of input/state constrained discrete-time linear systems subject to bounded disturbances, whose main feature is not to require any explicit on-line measure (or estimation) of the state for its implementation. Although effective also in the case of bounded disturbances, the performance of earlier FF-CG schemes [1] was mainly limited by the fact that such strategies were required to maintain constant their actions for a prescribed number of sampling steps. Here such a restriction is removed and the proposed FF-CG solution is allowed to update its action at each sampling step. Numerical simulations on a physical plant have been undertaken and comparisons with other strategies have been reported in order to show the effectiveness of the proposed approach. Index Terms—Command governor, linear systems, sensorless supervision.
I. INTRODUCTION The Command Governor (CG) is a nonlinear device which is added to a compensated plant, whose primal controller has been typically designed without considering the presence of input and state-related constraints, so as to ensure stability and good tracking performance when the constraints are supposedly not active (small signal regimes). The CG main objective is that of modifying the reference signal supplied to such a pre-compensated system when its direct application would produce constraints violation. This modification is typically achieved by solving on-line a constrained Quadratic Programming optimization Manuscript received May 20, 2011; revised August 09, 2012 and August 14, 2012; accepted February 08, 2013. Date of publication June 19, 2013; date of current version December 19, 2013. Recommended by Associate Editor C. Prieur. A. Casavola and F. Tedesco are with the Department of Informatics, Modeling, Electronics and Systems Engineering (DIMES), University of Calabria, Calabria 87036, Italy (e-mail:
[email protected];
[email protected]). E. Garone is with Université Libre de Bruxelles, B-1050 Brussels, Belgium (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2013.2270037
0018-9286 © 2013 IEEE
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problem, where the prescribed constraints are enforced along the future plant state predictions starting from the currently measured or estimated state, according to a receding horizon philosophy. Many results and a variety of schemes have been presented in the literature: see e.g. [3]–[6]. In particular, CG schemes dealing with disturbances and model uncertainties are considered in [5], with partial state information in [6]. See [8] for recent results on hybrid piecewise-affine systems. Different perspectives on CG are reported in [7]. For recent applications refer to [9], [10]. In [1], it has been shown that CG solutions which do not explicitly use the plant state for their action computation are possible at the price of some additional conservativeness. The idea behind such approach is that, if sufficiently slow transitions in the set-point modifications are acted by the CG unit, one can have a high confidence on the expected value of the state, even in the absence of an explicit measurement of it, because of the asymptotical stability of the system at hand. This feature may be of interest in all applications where either the measurement or the estimation of the state may be difficult or unsuitable. For example, it could be desirable in designing multi-agent distributed supervisory schemes. In fact, unlike approaches based on distributed MPC ideas, they would not require the knowledge of the entire aggregate state (or part of it) at each time instant, the latter being costly or even unrealistic in some large-scale applications. See e.g. [2] for distributed constrained supervision strategies for networked large-scale systems based on the earlier FF-CG ideas of [1]. Work is in progress to extend the scheme proposed in [2] to the novel FF-CG approach here proposed. Clearly, the standard CG approach, which explicitly makes use of the system state, usually enjoys better tracking performance and robustness to model uncertainty w.r.t. the FF-CG one. Nevertheless, the level of performance and robustness degradation is in many cases modest and justifies the use of FF-CG schemes in situations where the measure of the state is problematic. Notice also that in reference management problems the use of feedback is not a strictly necessary requirement and many classical solutions, e.g., filtering the reference signal to avoid high frequencies, are open-loop. The earlier FF-CG scheme proposed in [1], although quite effective, was shown to exhibit suboptimal tracking performance with respect to traditional state-based CG methods. The reason is that in such a scheme, unlike the traditional state-based CG approach, the FF-CG sampling steps and kept constant beaction is computed every tween two subsequent computations. Indeed, this has a direct consequence on the tracking performance of the algorithm, especially when references with fast variations are considered. Such a drawback is here overcome by means of a novel class of enhanced FF-CG strategies whose action is computable and applicable at each sampling time. The proposed solution is achieved by observing that, under the same assumptions of [1], the uncertainty about the state evolution arising from the absence of measurements, involves only the dynamics related to the initial conditions, which can be bounded by a time-valued monotonically vanishing function. The resulting FF-CG strategy is then formulated as a standard state-feedback CG generating a suitable signal that has to fulfill a restricted set of constraints. Moreover, in the disturbance-free case it is proved that the performance of this particular FF-CG scheme tends asymptotically (when the effect of initial conditions becomes negligible) to the one pertaining to standard state-based CG strategies.
II. NOTATIONS AND DEFINITIONS , and denote respectively the real, non-negative real and non-negative integer numbers. The Euclidean norm of a vector is denoted by whereas ,
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. For given sets , is the Pontryagin Set Difference and the Pontryagin-Minkowski Set be a matrix. Then, its entries will be indicated , .
denotes the quadratic form
Sum. Let , as
III. SYSTEM DESCRIPTION AND PROBLEM FORMULATION Consider the following closed-loop system
(1)
where: , is the state vector (which includes the conthe manipulable reftroller states under dynamic regulation), erence vector which, if no constraints (and no CG) were present, would and the output coincide with a desired reference vector which is required to track . The vector is a disturbance signal assumed to belong to the closed ball . Finally, represents the constrained vector which has to fulfill the set-membership constraint (2) being a polyhedral set. It is further assumed that (A1) the system (1) is asymptotically stable, i.e. is a Schur matrix. Such an assumption follows from the fact that the system is pre-compensated. Classical solutions to the CG design problem (see [4], [5]) have been achieved by computing, at each time , a CG action such that is the best approximation of under is a nonlinear memoryless function. the condition (2) and Here we will focus on a slight different approach to the CG design problem in which no measurement of the state vector is assumed avail. In order to better introduce the key ideas, able for determining case and let us consider temporarily the disturbance-free adopt the following notation for the steady-state solutions of (1) to a constant command (3) and will be used In the forthcoming analysis, the notation to indicate the above steady-state solutions corresponding to a constant . command of value The idea explicitly employed in early FF-CG schemes is that, if the were generated “changing slowly manipulable reference signal enough” w.r.t. system dynamics, then, because of system stability (see could always be maintained within a A1), the constrained vector from the closed-loop certain known (and “small”) distance steady-state equilibrium -vector (4) denotes hereafter the ball of radius centered at (in this where is centered at the origin). This approach has been used in case [1], where strategies of the form have been determined with computed every sampling steps and constantly applied between two successive CG action computations. In this technical note, a novel less conservative approach is proposed. To this end, observe that, because of linearity and the fact that univocally depends on and , the latter may be proven to be a and of the history of the commands function of its initial condition , viz. from time 0 up to time (5)
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separately. Then, it is possible to conceive CG schemes where, instead , decisions of considering the dependence on the measured state and of the past values of , that is are taken on the basis of (6) As it will be clear soon, we do not have to store in memory the entire se. In fact, a suitable aggregate expression can be found which quence for the computation of (6). is equivalent to the knowledge of A. The Proposed Improved FF-CG Approach In order to make precise statements, consider the constrained closedloop system (1), (2) satisfying assumption A1. Moreover, to simplify the developments, let us exploit the linearity of the system to separate the effects of initial conditions and commands from those of disturbances, i.e., (7) is the disturbance-free component of the state where and commands) (depending only on the initial state condition depends only on the disturbances (starting from whereas zero initial conditions). Next, consider the following set recursion (8) and characterize the tube of future constrained -vector predictions reachable from zero under the effect of dis. turbances only. From the definition (8) it follows that Furthermore, it has be shown [11] that the sets , if non-empty, are convex because of the convexity of . Let us now introduce The sets
(9) the set-valued future predictions (virtual evolutions) of the -vector for all possible disturbance sequence realizations along the virtual predicted from the time under a constant virtual command (at virtual time ). In particular, the set-valued initial state can be rewritten as the sum of three terms, vector , where i.e. represents the nominal (disturbances-free) evolution of the -variable along the and from virtual time under a constant virtual command the initial state , the vector represents the free evolution and of the disturbance-dependent component of the state
(10) denotes the set of virtual evolutions of the -variable due to all possible disturbance sequence. It is possible to prove (see [1] for details) that, in spite of state unavailability, the following implication holds:
Thus, constraints fulfillment can be ensured by only considering the disturbance-free evolutions of system (1) and adopting a “worst-case” approach. To this end, let us introduce, for a given sufficiently small scalar , the sets , where , which we will assume non-empty, is the set of all constant commands whose corresponding disturbance-free equilibrium points satisfy the constraints with margin . Notice that emptiness of would mean that no admissible equilibrium states exist. From the is closed and foregoing definitions and assumptions, it follows that as convex. If we rewrite the virtual evolutions (12) it results that is the steady-state component whereas is the transient evolution. Like in the standard CG solution, we , i.e. will restrict our attention to virtual commands within the set . This ensures that the steady-state component of the virtual evolutions does not violate constraints and will always belong to . Moreover, in order to satisfy the constraints also during the transients we need to guarantee (13) Then, the key idea used here for the construction of an effective FF-CG algorithm is as follows: let us assume that at time a command has been applied to the system, that the transient compo, result confined into a ball of known nents of around and that the constraints are not violated, viz. radius . Thus, it can be shown that the transient part of the predictions can be bounded according to the following expression: (14) We may note that, if we were waiting for a sufficient long time after the application of a new FF-CG command, the transient contribution would decrease and could be bounded within a certain percentage of its initial bound . In order to exactly quantify the contraction rate along the system evolutions the following of the initial bound definitions are in order: Definition (Guaranteed Contraction Sequence): The sequence is a Guaranteed Contraction Sequence for at time if , the pair implies that , holds true , with the real denoting any for each . upper-bound to Definition (Maximal Guaranteed Contraction Sequence): The seis a Maximal Guaranteed Contraction quence at time if: i) is a Guaranteed Sequence for the pair at time , ii) Contraction Sequence for the pair , for all Guaranteed Contraction Sequences for the at . pair A direct consequence of the above definitions is that if the command computed at time were constantly applied for the sub, then, given a sequent steps, i.e. , , the disMaximal Guaranteed Contraction Sequence turbance-free -transient could be bounded as follows: (15) because of the following equalities
(16) (11)
holding true.
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In [1] the latter idea has been exploited to build up a FF-CG scheme is modified only every steps. Here where the command signal we will overcome such a limitation. Consider at time the disturbancefree -transient evolution along the virtual horizon assuming that a has been applied from generic sequence of inputs time
(17) The latter, by introducing the translated command may be rewritten as
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Finally, we can denote
(25) as the set of all admissible FF-CG commands for a given sequence . On-line, at each time , this sequence is instantiated as . Moreover, convexity and compactness of are inherbecause of the properties of the Pontryagin-Differited by ence (see [11] for details). As a consequence, because the predictions is convex and compact as well. in (25) are linear, the set , we Then, by using the quadratic selection index may formulate the FF-CG algorithm as follows: The FF-CG Algorithm
(18)
repeat at each time 1.1 solve (26)
(19) By recalling (15), (16), the term depending from the initial conditions in (19) may be embedded as follows
(20) represents an upper-bound to the effect where the quantity of the initial conditions on the dynamics at time . Equality (19) and expression (20) suggest that, in order to guarantee , one can enforce the following sufficient condition:
(21) In fact, by summing up both left and right members in (20) and (21), it can be ensured that (22) Moreover, from system stability and by definition of Maximal Guaranteed Contraction Sequence, it follows that there exists a scalar such that for all such that and for all such that . As a consequence when . Conditions (20), (21) are crucial because they allow one to guarantee constraints fulfilment by exploiting only the knowledge of the sequence in (5). However, the storage of such a of past applied commands sequence can be avoided by introducing the translated state that satisfies
1.2 1.3 update (23) For the forthcoming analysis, we need to introduce the following property holding true for linear and asymptotically stable systems (1): . Then, there exists a real P1. Let such that . IV. MAIN RESULTS Lemma 1: Consider the set . Then, for all there exists a such that . . Being a Proof: Let us focus on the sequence Maximal Guaranteed Contraction Sequence, it will asymptotically conthere would exist a finite time verge to 0. Hence, for each such that . By looking at the definition (25), this latter consideration entails that , , and the statement of Lemma 1 directly follows. be an equilibrium solution Lemma 2: Let for system (23). Then, for any perturbation satisfying , , there exists a scalar such that . Proof: First it is useful to rephrase the set as where . Because
, the , or equivalently are sufficient to ensure . Such a condition can be further
condition
enforced in the following way:
(23) under the assumption membering that the sufficient condition (21) as
. By using such a definition and re, one may rewrite (24)
(27) where value of
and denotes the maximum singular . By recalling the triangular property of the norms
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, simple algebraic manipulations change (27) into the following stronger condition: (28) Since enforced by satisfying the equation
, the latter can be
In order to prove the statement, it is enough to note that for . Therefore, because of (33), when , and, consequently, the set in (25) takes the following form
. 4) See [4, Lemma 2]. 5) Thanks to Lemma 1, we can approximate, after a certain time , the FF-CG problem as instant (34)
(29) Next, let such that
. Then (29) implies that for all there exists a ball centered at with radius entirely included in
. Proposition 1: Let assumption A1 be fulfilled. Consider system (1) along with the FF-CG selection rule (26) and let an admissible comsuch that be applied mand signal where is a known scalar such that at time . Then: 1) At each decision time , the minimizer in (26) uniquely exists and can be obtained by solving a convex constrained optimization problem; 2) The system supervised by the FF-CG never violates the confor all regardless of any possible straints, i.e. ; admissible disturbance realization denote the standard 3) Let CG solution (details in [4], [5]) for the disturbance-free CG design where problem
Next, let be the FF-CG solution at time of (34) with . Because still admissible but not necessarily optimal at time , the sequence of costs is non-increasing, . Then, . i.e. In fact, , because of strict convexity of , would for a certain imply . Since A1 holds true, then for , and, because of Lemma 2, it exists such that . In such a case, the FF-CG solution to the better one and would change from would not be a convergence point. Next, let^{\prime}s define . By taking into account property (P1) for system (23) and by noticing that , we can state that there exists a finite time such which implies that , or equivalently . The latter implies that the predictions for along virtual time , starting from , will satisfy (35)
(30) is the state-dependent admissible region. Then, the time-varying regions of admissible commands , achieved by at each time instant, asymptotapplying the FF-CG actions , reically converge to (30) and ; gardless of the reference sequence , is bounded for any arbitrary 4) The sequence of . bounded reference sequence , with a constant set-point, the sequence of 5) Whenever converges in finite time either to or to its best admissible steady-state approximation : (31) Moreover, if the disturbance is identically zero has
one (32)
Proof: 1) See Proposition 1 in [5] and Theorem 1 in [4]. 2) See Theorem 1 in [4] and [5]. 3) In the absence of disturbances, by combining (18), (19), (20) and (24), we have that
(33) where and the tryagin-Minkowski sum, i.e.
operator denotes the Pon.
, if we add to the right side of the latter Because , we obtain that becomes . Then, because , the FF-CG solution will be for sure at the finite time . Finally, conditions (32) simply follows by assumption A1.
V. COMPUTATION A. Maximal Guaranteed Contraction Sequence In principle, one should determine any possible sequence for every . However, interesting enough, the following recursive property holds true: (36) and only has to be computed in practice. Moreover, one can implement an easy procedure for the computation of by inheriting some technicalities introduced in [1] for the computation of the Generalized Settling Time, hereafter briefly recalled is said to Definition (Generalized Settling Time): The integer , for be a Generalized Settling Time with parameter , with , if implies the pair for each , with the denoting any upper-bound to real Then, the computation of may be undertaken as follows: • ; if is not a Generalized Settling Time with param• ; eter
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•
if is a Generalized Settling Time with parameter and is the minimum amongst all parameters associate to the Generalized Settling Time (see [1], [12] for computational details). Observe also that the computations should be done for any . However, in the proof of the FF-CG properties it is shown that any approximating such that Guaranteed Contraction Sequence may be used in the place of without affecting its feasibility properties, at the price of introducing some additional conservativeness in the start-up phase of the algorithm. A practicable procedure is then that of computing offline and storing only the first samples of and approximating the tail with an exponentially the sequence , , with and decreasing sequence reals computed as detailed in [12]. B. Computation of Let a sequence be given. Being a polyhedral [11]. Under set, it can be represented as the above assumptions, the following structures for the sets and straightforwardly follow: and . Finally, if provided with a a time , one as can characterize the set where , referred to as the Constraint Horizon [3], is computed off-line according to an algorithm presented in [11]. VI. ILLUSTRATIVE EXAMPLE: AN EIGHT-TANK WATER DISTRIBUTION SYSTEM Consider the water tank network depicted in Fig. 1. The system consists of the interconnection of four cascaded two-tank models. Each cascaded subsystem is described by the following nonlinear equations (see [13]):
where is the water flow supplied by the pump whose command is . Moreover, for each , 2, the voltage , , are the tank cross sectional area, , the sections of pipes conthe water levels in the tanks, and necting the tanks, and the gravity constant and the water density respectively. denotes the set of subsystems which provide water to the downstream tank of the -th subsystem; in our case , , and . Finally, parameters , model the splitting water flows between upper and lower tanks. A simple static equation is used to model the relationship between the input voltage and the incoming mass of water
The following local and global constraints are to be enforced at each time instant:
Fig. 1. Four cascaded two-tank water system.
Each subsystem is regulated by a simple proportional controller that on the basis of the tracking error on water adapts the pump action levels in the tanks. Since the FF-CG algorithm requires a discrete-time linear model of the plant, the closed-loop system is linearized around the equilibrium , , and discretized with a . sampling step Three strategies have been contrasted in the simulations. In particular, we have analyzed the behavior of the system under the action of i) the FF-CG strategy described in Section II, ii) the Standard CG strategy of [4], iii) the FF-CG strategy presented in [1], hereafter referred to as FF-CG(Fixed). All the above methods use the same running cost to be minimized. Concerning , we selected the value . A horizon , the set in (25), has been chosen. needed for the computation of was Finally, for the FF-CG(Fixed) implementation, the choice . undertaken which implies A. Simulation Scenarios and Results The simulations, which have been carried out by applying the designed CG and FF-CG schemes to the nonlinear model of the tank network, aim at investigating the capability of the supervisor scheme to enforce the prescribed constraints when the desired set-points to the water levels of the downstream tanks have the profiles depicted in Fig. 2 (red dashed line). Notice that such profiles are not admissible for the system and constraint violations would occur if they were tracked without modifications. At the beginning, the desired references , correspond to an admissible equilibrium. At time , the reference related to the downstream tank of subsystem 1 is changed from 32 [cm] to 42 [cm]. At the same time, also the reference is modified from 32 [cm] to 34 [cm]. These values are when they are changed kept constant until time instant back to their initial values. Simultaneously, the desired references and change their values at time from 32 [cm] to 27.85 [cm] and, respectively, 28.5 [cm]. After that, these new values are kept , when they are brought back to the constant up to time previous values. In Figs. 3, some components of the constrained vector response can be observed. It is important to note that such a vector violates the constraints at several time instants when no CG unit is used. The same figure depicts that, on the contrary, when a CG/FF-CG unit is operating
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Fig. 4. CG actions for Subsystem 1 when a square-wave set-point with increasing frequency is considered.
Fig. 2. Water levels in the downstream tanks.
TABLE I (LEFT) RESIDUAL COST, (RIGHT) CPU TIME PER STEP
TABLE II VALUES OF THE RESIDUAL COST IN THE UNCERTAIN SCENARIO. MISSING DATA CORRESPONDS TO CASES OF CONSTRAINTS VIOLATION
Fig. 3. Relative level constraint and . constraints on actuators
and input saturation
the constraints happen to be fulfilled. Fig. 2 shows the system output. In this figure, the performance of the FF-CG, CG and FF-CG(Fixed) strategies can be compared. One can observe that the level of conservativeness introduced by the proposed FF-CG version is negligible after 250 [sec] when contrasted with the standard state-feedback CG approach. Moreover, FF-CG(Fixed) introduces a certain level of delay in the system response which is not present on the contrary in FF-CG. In order to evaluate the possible performance degradation of FF-CG w.r.t. CG under fast changing reference signals we have repeated the simulations with a set-point which is a swept sinusoidal wave for the first 1500 [sec] and a swept square-wave for the rest of the time (see Fig. 4). For space reasons, we show in Fig. 4 only the results related to subsystem 1. For this second scenario, in Table I are reported the values for of the residual cost each method. In Fig. 4 and Table I, a performance degradation is evident for FF-CG(Fixed) while FF-CG is capable of guaranteeing similar performance of CG. In the same Table I, the on-line computational burdens for all schemes are reported.
A further simulation scenario has been considered in order to investigate the robustness of FF-CG w.r.t CG. In this case, both schemes have been designed on the basis of the following model:
where the matrices and represent the uncertainty existing over the and nominal matrices and . In particular, where and are random scalars selected according to a continuous uniform distribution defined on . The robustness of the schemes can be increased by assuming the fictitious presence of persistent disturbances of suitable magnitude acting on the system. In the simulations, an uniformly distributed random noise has been considered with defined in such a way that represents a disturfor bance on the upstream water flows. Table II reports the value of the over the same simulation horizon for sevaverage cost . The same reference signal of Fig. 4 has been eral values of and used in the experiments. The lack of data in some entries of Table II corresponds to cases in which the constraints were violated due to the model uncertainty. As expected, the FF-CG scheme is slightly less robust than the CG one in the presence of model uncertainty. Anyway, it is worth pointing out that the use of a swept square-wave set point
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represents a worst case scenario for the robustness analysis and the difference between the two schemes is usually smaller in more realistic situations.
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Stochastic Stability of Jump Discrete-Time Linear Systems With Markov Chain in a General Borel Space O. L. V. Costa, Senior Member, IEEE, and D. Z. Figueiredo
VII. CONCLUSIONS In this technical note, a novel FF-CG scheme is proposed which does not make use of any measurement of the state to govern the set-point manipulations. The main idea under its development was to limit the set-point variations in order to always maintain the state trajectory “not too far” from the region of the steady-state admissible equilibria. The properties of the proposed algorithm have been carefully analyzed and the improvements and differences with earlier FF-CG approaches pointed out. Performance and robustness analyses and comparisons with classical CG and previously proposed FF-CG solutions have been provided and discussed in the final example.
Abstract—Necessary and sufficient conditions for stochastic stability (SS) of discrete-time linear systems subject to Markov jumps in the parameters are considered, assuming that the Markov chain takes values in a general Borel space . It is shown that SS is equivalent to the spectrum radius of a bounded linear operator in a Banach space being less than 1, or to the existence of a solution of a Lyapunov type equation. These results generalize several previous results in the literature, which considered only the case of the Markov chain taking values in a finite or infinite countable space. Index Terms—General Borel space, Lyapunov equation, Markov jump linear systems, stochastic stability.
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I. INTRODUCTION There has been lately an intensive interest on dynamic linear systems which are subject to abrupt changes in their structures. Among these models one that has received a great deal of attention is the so-called linear systems with Markov jump parameters (MJLS). Regarding the mean square stability theory for MJLS, we can mention, for instance, [3], [9], [10], [13], [14], [17], [19], [20], as a sample of works on this subject, which has by now a fairly complete body of results (see, for instance, [6] for further references on this topic). In these works the state space of the Markov chain was assumed to take values in a finite state space. For the case in which the state space of the Markov chain is countably infinite it is shown in [4] that mean square and stochastic stability ( -stability) are no longer equivalent. Regarding other issues such as almost sure stability, robust stability, stabilizability and detectability, the readers are referred, for instance, to [1], [7], [8], [18], [22] and [24]. In particular in [16] it was considered MJLS with the Markov chain taking values in a general Borel space, and the main goal was to derive conditions for the uniform exponentially almost sure stability (UEAS-stability). Assuming that the Markov chain is a positive Harris chain, the authors showed in Theorem 4.3 that UEAS is equivalent to a contractivity condition being satisfied. In this paper we deal with a different stability criterion, the so-called stochastic stability (SS). We consider a discrete-time MJLS with the jumps being modeled by a time-homogeneous taking values in a general Borel space Markov chain having a density with and with transition probability kernel respect to a -finite mesure on . No positive Harris assumptions are required, and the necessary and sufficient conditions are based on a Lyapunov type equation. As far as the authors are aware of, this is the first time that the SS of MJLS with the Markov chain taking values in a general Borel space is considered in the literature. The paper is organized as follows. In Section II we define the notation and some basic concepts. The problem statement is presented in Section III and some auxiliary results are addressed in Section IV. In Section V we present the necessary and sufficient conditions for SS of discrete-time MJLS as well as some easy-to-check sufficient conditions Manuscript received December 17, 2012; revised April 08, 2013, May 23, 2013; accepted June 04, 2013. Date of publication June 20, 2013; date of current version December 19, 2013. This work was supported in part by the Brazilian, National Research Council-CNPq, under Grant 301067/09-0 and USP project MaCLinC. Recommended by Associate Editor M. L. Corradini. The authors are with the Departamento de Engenharia de Telecomunicações e Controle, Escola Politécnica da Universidade de São Paulo, CEP: 05508 900-São Paulo, Brazil (e-mail:
[email protected];
[email protected]). Digital Object Identifier 10.1109/TAC.2013.2270031
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