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Robust Stability Analysis for Autonomous Min-Max Systems
European Journal of Control (2008)2:104–113 # 2008 EUCA DOI:10.3166/EJC.14.104–113
Robust Stability Analysis for Autonomous Min-Max Systems Qianchuan Zhao and Da-Zhong Zheng Center for Intelligent and Networked Systems, Department of Automation and TNLIST Lab, Tsinghua University, Beijing 100084, China
Min-max systems are natural extensions of the well known (max,plus) linear discrete event dynamic system (DEDS) models. They are motivated by recent research on digital circuits and communication networks. In this paper, we study the robust stability of a family of min-max systems whose parameters are perturbed in intervals. In the spirit of Kharitonov, we first reduce the robust stability test to the check of stability on the vertex set of parameter space. We then further reduce the test to a subset derived from the structure of the system family. Keywords: Min-max systems, Kharitonov-like criteria.
robust
stability,
1. Introduction Discrete event dynamic systems (DEDS’s) in which the operations min, max and plus appear simultaneously are known as min-max systems or min-maxplus systems. Some of the recent advances can be found in [3,5,15,19–21] and references therein. Good survey papers include [16] and [4]. A variety of problems arising in digital circuits, computer networks and automated manufacturing plants can be described using these models (see [6,14] and references therein for more detailed discussions on the applications of min-max system models). Theoretically, such systems are natural extensions of timed discrete event systems
Correspondence to: Q. Zhao Tel: 86-10-62783612; Fax: 86-1062786911; E-mail: [email protected]
which contain only maximum timing constraints (or only minimum timing constraints) and which can be studied as so called linear models in terms of max-plus algebra (see e.g., [1,9,10]). A given min-max system is called stable if it has an eigenvalue , or a uniform asymptotic growing rate of the system state vector, namely, limk!1 XðkÞ= k ¼ ð; . . . ; Þ0 . This notion of stability is introduced in [8] for max-plus systems. It is equivalent to the request that the system has an eigenvalue such that the system has periodic trajectory after a finite number of transient phase. This type of systems only needs finite buffer capacity for operation and all events fire at a constant speed in a long run. Note that such a stability concept is different from the usual stability concept used in continuous variable dynamic systems (CVDSs) or hybrid dynamic systems (including piecewise affine systems (e.g., [11]) which contains minmax systems as a special case) literature which requires the convergence of the state to the origin as time tends to infinity. It is also different from the stability concept referring to the boundedness of XðkÞ representing queuing length or waiting time (see e.g., [18]) in the study of DEDS containing queues. Compared with the origin stability or boundedness of state, stability studied here is a weaker one, in the sense that if state XðkÞ converges to zero or is bounded, then divided by k, the limit will automatically go to ð; . . . ; Þ0 with ¼ 0. This relaxation in stability is more suitable for min-max systems because in practice the state is usually used to describe the timing of
Received 10 January 2007; Accepted 2 February 2008 Recommended by J. Lunze, A.S. Morse
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repeated events and it is natural for some state variables to grow with at most linear speed as a function of ‘time’ (number of occurrence of the events). Stability of min-max systems plays an important role in solving engineering problems such as the clock schedule verification [7]. The policy iteration results in [7] or a direct calculation of cycle time vector based on the Duality Theorem [12] can be applied to check system stability, provided that we know the parameters exactly. However these results cannot be applied directly for the cases in which the parameters are not exactly known, for example due to uncertainties introduced in the modelling process, or perturbations experienced by plant parameters. Although there are extensive results on the study of robust stability in terms of plant parameters in CVDS (see e.g., [2,17,23,24], and references there in), and also some work in linear DEDS (see [25]), to our best knowledge, there is little work on robust stability analysis for minmax systems. Among existing results of min-max systems, the study of structural properties is most closely related. Balance condition [14,21], Strong connectivity [13], Inseparability [28] and Irreducibility [22]1 are all important examples of structural properties. Structural properties are sufficient to guarantee the stability of a system regardless of the specific values the parameters are taking. But they are not necessary when each parameter belongs to a specific interval because the loss of stability may happen only outside the interval of interest. In this paper, we study the robust stability of a family of min-max systems whose parameters are perturbed in intervals. In the spirit of Kharitonov [2,17], we first reduce the robust stability analysis to the check of stability on the vertex set of the parameter space. The structure of the family is then used to further restrict the scope of the vertex check.
2. Problem Formulation We follow the notations used in [7,14,28]. The operations a _ b and a ^ b are used to stand for maximum and minimum respectively: a _ b ¼ maxða; bÞ and a ^ b ¼ minða; bÞ. We use 0 to denote the transpose of a vector, that is, ðx1 ; . . . ; xn Þ0 is the column vector formed by transposing ðx1 ; . . . ; xn Þ. The upper case English letters X, and Y will be used in the rest of this paper to stand for vectors ðx1 ; . . . ; xn Þ0 and ðy1 ; . . . ; yn Þ0 . bðXÞ and tðXÞ will be used to represent mini fxi g and maxi fxi g respectively, and will be called the bottom and the top of X. 1
We proved that Inseparability is equivalent to Irreducibility in [27].
Definition 1: [7,14] A min-max function of type ðn; 1Þ is any function f : Rn ! R, which can be written as a term in the grammar: f :¼ x1 ; x2 ; . . . ; xn jf þ ajf ^ fjf _ f; ð1Þ where a 2 R will be understood as parameters. The notation used here is the Backus-Naur form wellknown in computer science [7]. The vertical bars separate the different ways in which terms can be recursively constructed. The simplest term is one of the n variables, xi . Given any term, a new one may be constructed by adding an a 2 R; given two terms, a new one may be constructed by applying a min or max function. Only these rules may be used to build terms. For example, fðx1 ; x2 ; x3 Þ ¼ ða þ ðx1 ^ x2 ÞÞ _ ðb þ x3 Þ is a min-max function of type ð3; 1Þ with two parameters (a and b). We will denote the set of parameters of a min-max function f as pf and denote the total number of parameters as mðfÞ. For the example, we have pf ¼ ða; bÞ and mðf Þ ¼ 2. Notice, expressions like 1 ^ x; 1 _ x should not be regarded as min-max functions according to Definition 1. Definition 2: A min-max function of type ðn; nÞ is any function F : Rn ! Rn , such that each component Fi is a min-max function of type ðn; 1Þ. The set of min-max functions of type (n, n) will be denoted by MM (n, n). Definition 3: An autonomous min-max system defined by a min-max function F 2 MMðn; nÞ is a dynamic system given by Xðk þ 1Þ ¼ FðXðkÞÞ
ð2Þ
where XðkÞ ¼ ðx1 ðkÞ; ; xn ðkÞÞ0 2 Rn is the it state vector. Note that (2) can be written in componentwise form 0 1 0 1 x1 ðk þ 1Þ F1 ðXðkÞÞ B C B C ... @ A ¼ @ ... A xn ðk þ 1Þ Fn ðXðkÞÞ 0 1 F1 ðx1 ðkÞ; . . . ; xn ðkÞÞ B C ... ¼@ A Fn ðx1 ðkÞ; . . . ; xn ðkÞÞ In order to study the robustness of a min-max system as defined in (2), we use a vector to denote all parameters appearing in the min-max functions F1 ; . . . ; Fn and use the notations FðX; Þ and ðÞ to indicate the dependency of F and on parameters respectively. Formally, is defined as ¼ ðpF1 ; . . . ; pFn Þ: P Its size is ni¼1 mðFn Þ.
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M1 Disassembly line
Assembly line M2
perturbed min-max systems whose parameters belong to pre-determined intervals. Definition 4: A family of min-max systems denoted by ðÞ is defined by a set of systems ðÞ ¼ fðÞj 2 g;
Fig. 1. A repair shop.
where the parameter space has the form Y þ ½ ¼ i;j ; i;j ; Example: Consider a simple repair shop having two machines M1 and M2. Each asset has two components A1 and A2. Assets need to be first disassembled, then their components are repaired and assembled as shown in Fig. 1. It is required that once disassembled, an asset’s component A1 must be processed immediately. To improve machine utilization, we also require that the asset’s component A2 must not wait for the same machine repairing the component A1 of the same asset. For simplicity, we assume that the disassembly time can be omitted. So, an asset is disassembled only when there is an idle machine. We assume that on both machines, the repair time of component A1 is a1 and the repair time of component A2 is a2 . Denote the starting time for the repairment of component A1 of the k-th asset and the starting time for component A2 as s1 ðkÞ and s2 ðkÞ respectively, for k ¼ 1; 2; . . .. Denote the complete time for the repairment of the k-th asset’s component A1 and the complete time for component A2 as x1 ðkÞ and x2 ðkÞ respectively. So we have x1 ðkÞ ¼ s1 ðkÞ þ a1 and x2 ðkÞ ¼ s2 ðkÞ þ a2 . When we assume that there is no starvation at the output of the disassembly line and there is no blocking at the assembly line, we have s1 ðkþ1Þ ¼ x1 ðkÞ^x2 ðkÞ and s2 ðkþ1Þ ¼ x1 ðkÞ_x2 ðkÞ. As a convention, we set x1 ð0Þ ¼ 0 and x2 ð0Þ ¼ 0 as the initial condition which means that both machines are ready at time t ¼ 0. In summary, the system’s dynamics can be modelled as a min-max system defined by the following min-max function. FðXÞ ¼
ðx1 ^ x2 Þ þ a1 ðx1 _ x2 Þ þ a2
ð3Þ
In this case, n ¼ 2. The parameter vector is ¼ ða1 ; a2 Þ; where pF1 ¼ ða1 Þ, pF2 ¼ ða2 Þ. Although the components of could Pnbe located using one index taking values from 1 to i¼1 mðFn Þ, we prefer to use two indices i and j to locate them. The first index i will be used to indicate which min-max function the parameter belongs to. The second index j elaborates the location of the parameter in the function once i is fixed. So, we define i;j ¼ pFi ðjÞ. As usual, we need to introduce a family of systems of interest. In this paper, we will study the class of
ð4Þ
ð5Þ
i;j
where a and aþ are lower and upper bounds of parameter a. The range of index i is f1; . . . ; ng and the range of index j is 1; . . . ; mðFi Þ. The robustness property we will study in this paper is stability. Let us introduce stability for a min-max system first and then introduce the robust stability for a family of min-max systems. Definition 5: A given system in form of (2) is stable, if there exists a real such that lim Fki ðXÞ=k ¼ ; i ¼ 1; . . . ; n:
k!1
The number is called the eigenvalue of the system. The limit limk!1 Fk ðXÞ=k is known as cycle time vector and will be denoted as ðÞ. It is independent of the starting point X 2 Rn (as a consequence of the non-expansive nature of min-max functions, see e.g., [16])2. So, the stability condition can also be written in vector form ðÞ ¼ ð; . . . ; Þ0 : with the convention that i ðÞ ¼ limk!1 Fki ðXÞ=k. Remark 1: For every min-max system, the limit ¼ ðlimk!1 Fki ðXÞ=k; i ¼ 1; . . . ; nÞ0 limk!1 XðkÞ=k always exists due to the Duality Theorem (see [12]). So, for an unstable min-max system, its cycle time vector must be a vector whose components are nonidentical. Note that not all min-max systems with finite initial state are automatically stable. In fact, even for an autonomous (max,plus) system (a system that only contains max and plus operation without min operation) with an initial state whose entries are all finite, it is not the case that the system is always stable. For example, the (max,plus) system maxðx1 ðkÞ þ 1; x2 ðkÞÞ x1 ðk þ 1Þ ¼ x2 ðk þ 1Þ x2 ðkÞ 2
This may not be true if we allow some components of X to be equal to
þ1 or 1.
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where x1 ð1Þ ¼ 1; x2 ð1Þ ¼ 0 has the state trajectory x1 ðkÞ ¼ k; x2 ðkÞ ¼ 0 for k ¼ 1; 2; . . .. The two state variables x1 and x2 have a different rate of increase: 6 limk!1 x2 ðkÞ=k ¼ 0. Thus, acclimk!1 x1 ðkÞ=k ¼ 1 ¼ ording to our definition, the system is not stable. Definition 6: A family of min-max systems ðÞ is said to have robust stability, if ðÞ is stable for all 2 .
3. Main Results We reduce the robust stability test problem for a family of min-max systems ðÞ to a finite test on the vertex set of . The vertex set of , denoted by VðÞ is defined as Y þ f VðÞ ¼ i;j ; i;j g: i;j
Theorem 1: A family of systems ðÞ defined by FðX; Þ in form of (2) has robust stability if and only if ðÞ is stable for every 2 VðÞ. This theorem is a Kharitonov-like result in the sense that it allows the robustness test for stability of the whole family to be reduced to the test on the vertex set. Proof: The ONLY IF part is obvious since VðÞ . We should focus on the IF part. We use proof by contradiction. The details are given in the Appendix. & To illustrate the application of our theorem, consider again our example. Example (cont’d): Consider the family of min-max systems given in form of (3), if the parameters ai belong þ to interval ½a i ; ai , i ¼ 1; 2, the parameter space can be written as þ þ ¼ ½a 1 ; a1 ½a2 ; a2
and the vertex set of its parameter space is þ þ VðÞ ¼ fa 1 ; a1 g fa2 ; a2 g which in general contains 22 elements. According to Theorem 1, to judge whether the family of systems has robust stability, we need only to check stability for these parameters. For the special cases where some of the intervals collapse to a single point, the size of the vertex set may be smaller. Consider for example, the special case where a1 ¼ 1 and a2 2 ½1; 2. We only need to check 2 vertices distinguished by the value of a2 : VðÞ ¼ fð1; 1Þ; ð1; 2Þg. Direct calculation shows that ðÞ is unstable for the cases a2 ¼ 2 and stable for a2 ¼ 1 (with a1 ¼ 1). In general, the check of stability on a vertex can be done by the policy iteration results in [7] or a direct calculation of cycle time vector based on the Duality Theorem [12].
Although the number of tests is finite according to Theorem 1, it is interesting to ask whether we can reduce the number of tests further since the size of the vertex set is exponential. The following theorem partially answers this question. For a given min-max function F, introduce a monotone Boolean function3 FðX; 0Þ by setting all parameters to 0 and regard ^ as ‘AND’ and _ as ‘OR’. Here we use 0 to represent a parameter vector whose elements are uniformly zero. Let Bn be the n-dimensional Boolean space. For the example in (3), the corresponding equation FðX; 0Þ ¼ X has the form x1 ^ x 2 x1 ¼ : ð6Þ x2 x1 _ x2 Obviously, the equation FðX; 0Þ ¼ X always has two trivial solutions 0 ¼ ð0; 0; . . . ; 0Þ0 and 1 ¼ ð1; 1; . . . ; 1Þ0 , the constant Boolean vector whose elements are 0 or 1. At the same time, for our example, the Boolean equation FðX; 0Þ ¼ X has one non-trivial solution, namely X ¼ ð0; 1Þ0 . In general, we will denote the set of nontrivial solutions to the Boolean equation FðX; 0Þ ¼ X by ðFÞ, that is, ðFÞ ¼ fXjFðX; 0Þ ¼ X; X 2 Bn ; X 6¼ 0; X 6¼ 1g: So, for our example in (3), we have ðFÞ ¼ fð0; 1Þ0 g. The importance of ðFÞ lies in the fact that these solutions can be used to narrow down the scope of parameter vertices we should check. The reduced vertex set VðFÞ ðÞ of the parameter space is defined as follows. [ fj8i s:t: zi ¼ 0; i;j VðFÞ ðÞ ¼ Z2ðFÞ þ ¼ i;j and 8i s:t: zi ¼ 1; i;j ¼ i;j g:
ð7Þ
In words, VðFÞ ðÞ are all the vertices in the parameter space such that whose parameters match one of the non-trivial solutions Z of the Boolean equation FðX; 0Þ ¼ X in the following way: for i ¼ 1; . . . ; n, the parameters (i;j ¼ pFi ðjÞ, j ¼ 1; . . . ; mðFi Þ) belonging to function Fi take their upper bounds þ i;j when zi ¼ 1 and take their lower bounds þ i;j when zi ¼ 0. Note that VðFÞ ðÞ only depends on the structure of F, not on the value of the parameters. For our example, the set VðFÞ can then be determined as VðFÞ ¼ f ¼ 0 þ ða 1 ; a2 Þg because there is only one element Z ¼ ð0; 1Þ in ðFÞ and it requires the parameters in F1 to be set to lower bounds and the parameters in F2 to be set to upper bounds. 3 Recall that monotone Boolean functions are Boolean functions with no negation operation.
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Theorem 2: A family of systems ðÞ defined by FðX; Þ in the form of (2) has robust stability if and only if ðÞ is stable for every 2 VðFÞ ðÞ. Proof: The ONLY IF part is obvious since VðFÞ ðÞ . We should focus on the IF part. Suppose ðÞ is stable for all 2 VðFÞ ðÞ but there is a 2 such that ð Þ is not stable. That is, for this , ¼ ðFðX; ÞÞ is not a constant vector. Define a vertex as (17) in the proof of Theorem 1 for all i 2 f1; . . . ; ng and j 2 f1; . . . ; mðFi Þg, ( þ i;j ; if i 2 S; ð8Þ i;j ¼ i;j ; if i 2 Sc : where S ¼ fsjs ¼ tðÞg and Sc ¼ fs0 js0 < tðÞg are two subsets of f1; . . . ; ng decided by the vector . is From the proof of Theorem 1, we know that ðÞ unstable. So it is sufficient to show that satisfies 2 VðFÞ ðÞ. This will lead to a contradiction with the assumption that ðÞ is stable for all 2 VðFÞ ðÞ which completes the proof. To establish the desired result, we need to cite a fact that has been used in [28] (Lemma 4 in [28]): For ¼ ðFðX; ÞÞ, it must be true that Fð; 0Þ ¼ . Now define a Boolean vector Z such that for i ¼ 1; . . . ; n, ( zi ¼
0;
if i < tðÞ;
1;
if i ¼ tðÞ:
ð9Þ
Following similar argument of Theorem 1 in [28], we can show that FðZ; 0Þ ¼ Z, that is Z 2 ðFÞ. It is clear that zi ¼ 0 if and only if i 2 Sc and zi ¼ 1 if and only if i 2 S. Hence, according to the construction of (17), it is also true that for all i, i;j ¼ i;j if and only if zi ¼ 0 if and only if z ¼ 1. Hence satisfies and i;j ¼ þ i i;j & 2 VðFÞ ðÞ. The proof is completed. Remark: Although the size of the vertex set VðÞ is exponential in terms of the number of parameters, Theorem 2 claims that we need only to check stability on a usually smaller vertex subset VðFÞ ðÞ. To determine the set ðFÞ is actually to solve Boolean equation which is an extensively studied topic (see e.g., [26]). Remark: There may be some vertices outside VðFÞ such that the system is unstable. But according to Theorem 2, we don’t need to check them. Example (cont’d): Consider the family of min-max systems given by the min-max function (3) with a1 ¼ 1 and a2 2 ½1; 2 again. According to Theorem 2, to test the robust stability of FðX; Þ, we only need to test vertices in VðFÞ . For this specific example, VðFÞ
þ contains a single vertex. VðFÞ ¼ f ¼ ða 1 ; a2 Þ ¼ ð1; 2Þg. And the test result, as above mentioned, is that þ ðða 1 ; a2 ÞÞ is unstable. This agrees with our result for the exhaustive checking of the whole vertex set VðÞ. Under the unstable configuration (a1 ¼ 1; a2 ¼ 2), suppose that M1 repairs component A1 of the first asset, then according to our operation rule, it will continue to process component A1 of all other assets since M1 always finishes earlier than M2. Component A2 of all assets will be processed by M2. The two machines in Fig. 1 have different processing speeds. M1 works faster than M2. Asset’s components will accumulate in the input buffer of M2 and also will accumulate in output buffer of M1. To avoid buffer overflow, control must be introduced to stabilise the system. One possible way to do so is to require M1 and M2 to wait for each other, that is, only after an asset’s both components finish processing on machines, M1 and M2 start to process next asset’s components.
4. Conclusions In this paper, a Kharitonov-like result is established for the robust stability test of a family of autonomous min-max systems whose parameters are given by intervals. It is sufficient to check only a subset of the vertex set of the parameter space to guarantee the robust stability of the family. This subset depends on the structure of the family. The study of robust stability under correlated perturbations and possible applications of these robust stability analysis results to synthesis robust controllers are future research topics.
Acknowledgments This work was supported by NSFC (Grant Nos. 60074012, 60274011 and 60574067, 60721003, 60736027) and National Key Project of China, Fundamental Research Funds from Tsinghua University and Chinese Scholarship Council, Ministry of Eduction of China. The authors would like to thank both anonymous reviewers for their suggestions to improve the quality of the manuscript and thank an anonymous reviewer of a different journal for pointing out a mistake in the original proof of Theorem 1.
References 1. Baccelli F, Cohen G, Olsder GJ, Quadrat J.-P. Synchronization and Linearity. John Wiley & Sons, Chichester, 1992
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2. Barmish BR. New Tools for Robustness of Linear Systems. Macmillan, New York, 1994 3. Chen W, Tao Y. Observability and reachability of nonlinear deds and its coloring graph. Chin Sci Bull, 2001; 46 (8): 62–64 4. Cheng Y. A survey of the theory of min-max systems. In: Huang G.-B, Huang DS, Zhang X.-P (eds), ICIC 2005, Part II, LNCS 3645, Springer-Verlag, Berlin, Heidelberg, 2005, pp. 616–625 5. Cheng Y, Zheng D.-Z. A cycle time computing algorithm and its application in the structural analysis of min-max. Discrete Event Dyn Syst Theory Appl, 2004; 14(1): 5–30 6. Cochet-Terrasson J, Gaubert S, Gunawardena J. Dynamics of min-max functions. Technical Report HPL-BRIMS97-13, HP Lab, 1997 7. Cochet-Terrasson J, Gaubert S, Gunawardena J. A constructive fixed point theorem for min-max functions. Dyn Stab Syst, 1999; 14: 407–433 8. Cohen G, Moller P, Quadrat J.-P, Viot M. Linear system theory for discrete event systems. In Proceedings of the 23rd Conference on Decision and Control, Las Vegas, NV, 1984; pp. 539–544 9. Cohen G, Moller P, Quadrat J.-P, Viot M. Algebraic tools for the performance evaluation of discrete event systems. Proc IEEE, 1989; 77(1): 39–58 10. Cuninghame-Green RA. Minimax Algebra. Lecture Notes in Economics and Mathematical Systems, Vol. 166. Springer-Verlag, Berlin, 1979 11. Ferrari-Trecate G, Cuzzola FA, Mignone D, Morari M. Analysis of discrete-time piecewise affine and hybrid systems. Automatica, 38:2139–2146, 2002. 12. Gaubert S, Gunawardena J. The duality theorem for min-max functions. Comptes Rendus Acad Sci, 1998; 326: 43–48 13. Gaubert S, Gunawardena J. Existence of eigenvectors for monotone homogeneous functions. Technical Report HPLBRIMS-99-08, HP Lab, 1999 14. Gunawardena J. Min-max functions. J DEDS; 1994, 4: 377–406 15. Gunawardena J (ed). Idempotency. Cambridge University Press, Cambridge, 1998 16. Gunawardena J. From max-plus algebra to non-expansive mappings: a nonlinear theory for discrete event systems. Theor Comput Sci, 2003; 293: 141–167 17. Kharitonov VL. Asymptotic stability of an equilibrium position of a family of systems of differential equations. Differential’nye Urauneniya, 1978; 11: 2086–2088 18. Perkins JR, Kumar PR. Stable, distributed, real-time scheduling of flexible manufac-turing/assembly/diassembly systems. IEEE Trans Autom Control, 1989; 34(2): 139–148 19. De Schutter B, and van den Boom T. Model predictive control for max-min-plus systems. In: Boel R, Stremersch G (eds), Discrete Event Systems: Analysis and Control, Kluwer Academic Publishers, Boston, 2000, pp. 201–208 20. Tao Y, Chen W. Cycle time assignment of min-max systems. Int J Control, 2003, 76(18): 1790–1799 21. Tao Y.-G, Chen W, Yin Y.-X. Balance properties and stabilization of min-max systems. Int J Autom Comput, 2006; 1: 76–83 22. van der Woude J, Subiono. Conditions for the structural existence of an eigenvalue of a bipartite (min,max, þ)system. Theor Comput Sci, 2003; 293(1):13–24 23. Wang L. Robust stability of a class of polynomial families under nonlinearly correlated perturbations. Syst Control Lett, 1997; 30:25–30
24. Wang L, Huang L. Finite verification for robustness of discrete-systems under struc-tured perturbations. Chin Sci Bull, 1994; 39(11): 886–891 25. Wang L, Zheng D.-Z. Robustness of linear discrete event dynamic systems. J Control Theory Appl (in Chinese), 1990; 7(2): 27–31 26. Wang Y, McCrosky C. Solving boolean equations using rosop forms. IEEE Trans Comput, 47(2):171–177, 1998. 27. Zhao Q. A remark on inseparability of min-max systems. IEEE Trans Autom Control, 2004; 49(6): 967–970 28. Zhao Q, Zheng D.-Z, Zhu X. Structure properties of minmax systems and existence of global cycle time. IEEE Trans Autom Control, 2001; 46(1):148–151
Appendix Before giving the proof, we need two lemmas. It has been established in [14] that every min-max function f can be written in conjunctive normal form (CNF) given by a minimum of a finite set of max-plus functions. Max-plus functions contains only ‘max’ and ‘ þ ’ operations. CNFs are not unique. For a minmax function f of type ðn; 1Þ, we apply the recursively the following five rules to construct a CNF for it based on the grammar generating the min-max function. If f is of form ða þ xi Þ or simply xi for some i, where a is a parameter, output f as a CNF for itself. (ii) If f is of form f ¼ f1 ^ . . . ^ fl , where each of fi is in form of (i), output f ¼ f1 ^ . . . ^ fl as a CNF for f. (iii) If f ¼ a þ g where g ¼ g1 ^ . . . ^ gl is in form of (ii), define fi ¼ a þ gi and output f ¼ f1 ^ . . . ^ fl as a CNF for f. (iv) If f is in form of ðg1 ^ g2 Þ, where g1 and g2 are already in CNF such that g1 ¼ g11 ^ . . . ^ g1l , and 2 ^ 0 g2l0 , then, let output f ¼ gV 2 ¼ g1 ^ . . .V l 1 ð i¼1 gi Þ ^ ð lj¼1 g2j Þ as a CNF for f. (v) If f is in form of ðg1 _ g2 Þ, where g1 and g2 are already in CNF such that g1 ¼ g11 ^ . . . ^ g1l , and . ^ g2l0 , then, define fij ¼ g1i _ g2j and g2 ¼ g21 ^ . .V output f ¼ i; j¼1 ðfij Þ as a CNF for f. The range of the index i is from 1 to l and j from 1 to l0 . (i)
Since with each rule (iii)-(v), one operation can be removed, the CNF construction procedure will terminate in finite steps for any min-max function. For example, for the min-max function f ¼ ½a þ ðx1 ^ ðb þ x2 ÞÞ _ ðc þ x3 Þ, we first apply rule (i) to decide that terms x1 , ðb þ x2 Þ and ðc þ x3 Þ are in CNF. According to rule (ii), x1 ^ ðb þ x2 Þ is in CNF. Applying rule (iii), we obtain ða þ x1 Þ ^ ða þ b þ x2 Þ as a CNF of ½a þ ðx1 ^ ða þ b þ x2 ÞÞ. Finally, applying rule (v), we obtain ½ða þ x1 Þ _ ðc þ x3 Þ^
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½ða þ b þ x2 Þ _ ðc þ x3 Þ as a CNF for f. It is important to note that constants added to xi in CNF do not need to be a single parameter. For example, the constant added to ða þ b þ x2 Þ is a þ b, a sum of two parameters. In general, let us assume after applying the five rules and collecting all terms sharing the same xi , a CNF of Fi is given in the following form, Fi ðX; Þ ¼
^
ð
_
i ð b;j ðÞ þ xj ÞÞ;
Lemma 4: For every real number h, a system given in form of 0
1 0 1 F1 ðXðkÞÞ x1 ðk þ 1Þ @ A ¼ @ ... A ... xn ðk þ 1Þ Fn ðXðkÞÞ
is stable iff the system h defined in (14) is stable. 0
ð10Þ
b¼1;...lðiÞ j2Bði;bÞ
where i 2 f1; . . . ; ng, X 2 Rn , 2 . Here we use i ðÞ to denote the constant added to xj . b;j Since we are studying min-max functions with parameters taking values within bounded intervals, it is easy to imagine that the constants added to the variables x1 ; . . . ; xn are also bounded. In fact we can give an upper bound for them. Let be ¼ 1 þ max max
mðF Xi Þ
2 i2f1;...;ng
ji;j j:
ð11Þ
j¼1
We have the following lemma on the size of the constants added to the variables x1 ; . . . ; xn in the CNF given in (10). i ðÞ in (10), it holds that Lemma 1: For every b;j i j b;j ðÞj < .
Proof: Let us observe the effects of rules (iii-v) in constructing CNFs. We can see that rule (iii) causes the adding of a parameter to the constant added to xi ; rule (iv) may cause two constants belonging to the same xi combine into the minimum of both; rule (v) may cause two constants belonging to the same xi combine into the maximum of both. In any case, the i ðÞ is absolute value of the resulting constant b;j always upper bounded by the sum of the absolute i ðÞj < : & value of all parameters, as a result j b;j Dually, we can construct disjunctive normal forms (DNFs) for min-max functions. Let Fi ðX; Þ ¼
_
ð
^
ð id;j ðÞ þ xj ÞÞ
ð12Þ
d¼1;...rðiÞ j2Dði;dÞ
be a DNF of Fi . Here we use id;j ðÞ to denote the constant added to xj . We have the following lemma. Lemma 2: For every id;j ðÞ in (12), it holds that j id;j ðÞj < . Lemma 3: [14] For every min-max function f of type ðn; 1Þ and real number h, it holds that fðx1 þ h; . . . ; xn þ hÞ ¼ fðx1 ; . . . ; xn Þ þ h.
ð13Þ
B Xh ðk þ 1Þ ¼ @ 0 B ¼@
xh1 ðk þ 1Þ ...
1 C A
xhn ðk þ 1Þ F1 ðXh ðkÞÞ þ h
1
ð14Þ
C ... A h Fn ðX ðkÞÞ þ h
where Xh ðkÞ is the state vector and the initial condition is xh1 ð1Þ ¼ x1 ð1Þ þ h; . . . ; xhn ð1Þ ¼ xn ð1Þ þ h. Proof: Due to Lemma 3, we can easily show by induction that the state vector of h is related to XðkÞ by Xh ðkÞ ¼ ðx1 ðkÞ þ kh; . . . ; xn ðkÞ þ khÞ. Thus the cycle time vector of h equals to ð1 ðÞ þ h; . . . ; n ðÞ þ hÞ0 : So whenever all components of ðÞ are the same, all components of ðh Þ are the same. & Proof of the sufficient part of Theorem 1: Suppose on the contrary, ðÞ is stable for all 2 VðÞ but there is a 2 such that ð Þ is not stable. That is, for this , ðð ÞÞ ¼ ðFðX; ÞÞ ¼ is not a vector whose entries are all equal. We shall show that there is also not stable, that exists a 2 VðÞ such that ðÞ is, ðFðX; ÞÞ is not a vector whose entries are all equal. The proof is divided into two cases: the case in which the maximum value of the components of (denoted as tðÞ) is positive and the case in which tðÞ is non-positive. Case a) The case in which tðÞ > 0. Based on the vector , we define two non-empty index sets S ¼ fsjs ¼ tðÞg and Sc ¼ fs0 js0 < tðÞg. It is clear that S \ Sc ¼ ; and S [ Sc ¼ f1; . . . ; ng. It is also clear that tðÞ maxs2Ic s > 0. Let " be a positive number such that tðÞ " > 0
ð15Þ
and s =3: " ½tðÞ max c s2I
ð16Þ
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Robust Stability of Min-Max Systems
Since both tðÞ > 0 and tðÞ maxs2Ic s > 0 such " always exists. It is clear that
Now let us establish that for k ¼ 1; 2; . . ., and for all s 2 S, s0 2 Sc
s þ "Þ > " > 0: ðtðÞ "Þ ðmax c s2I
Define a vertex ¼ ði;k Þ of as follows: for all i 2 f1; . . . ; ng and j 2 f1; . . . ; mðFi Þg, ( i;j ¼
þ i;j ;
if i 2 S;
i;j ;
if i 2 S : c
ð17Þ
ð18Þ
for all s 2 S and s0 2 Sc . Our proof is constructive. Define recursively Fk ðX; Þ; k ¼ 1; 2; . . . as Fk ðX; Þ ¼ FðFk1 ðX; Þ; Þ and F0 ðX; Þ ¼ X. According to the definitions of cycle time vector and sets S and Sc , for " > 0 chosen to satisfy (15) and (16), there exists a k0 > 0 such that for all k k0 , it holds that k0 " > ðtðÞ "Þ;
ð25Þ
zks0 < ðk0 þ q0 þ k 1Þðmaxc t þ "Þ
ð26Þ
t2S
Obviously these lead to the desired result that > s0 ðFðZ; ÞÞ since s ðFðZ; ÞÞ k ¼ lim zs s ðFðZ; ÞÞ k!1 k ðk0 þ q0 þ k 1ÞðtðÞ "Þ lim k!1 k ¼ ðtðÞ "Þ;
all parameters appearing in Fs , s 2 S are That is, in , set to upper bounds of their intervals, and all parameters appearing in Fs0 , s0 2 Sc are set to lower bounds of their intervals. is not stable by showing We shall show that ðÞ that > s0 ðFðX; ÞÞ; s ðFðX; ÞÞ
zks > ðk0 þ q0 þ k 1ÞðtðÞ "Þ
ð27Þ and k ¼ lim zs0 s0 ðFðZ; ÞÞ k!1 k ðk0 þ q0 þ k 1Þðmaxt2Sc t þ "Þ lim k!1 k ¼ ðmaxc t þ "Þ: t2S
ð28Þ
ð19Þ Note that we already have for k ¼ 1; 2; . . .
and Fks ðX; Þ > kðtðÞ "Þ;
ð20Þ
zks > ðk0 þ q0 þ k 1ÞðtðÞ "Þ
ð29Þ
Fks0 ðX; Þ < kðmaxc t þ "Þ;
ð21Þ
zks0 < ðk0 þ q0 þ k 1Þðmaxc t þ "Þ
ð30Þ
t2S
t2S
for a fixed initial state X and for all s 2 S and s0 2 Sc . Let Fk0 ðX; Þ ¼ Y. Let q0 ¼ dð="Þe, where dae is the least integer no less than a (e.g., d3:2e ¼ 4). Thus q0 " :
ð22Þ
Let Z ¼ F ðX; Þ. Start from the state Z, we define two sequence of vectors Zk and Zk , k ¼ 1; 2; . . . as follows.
since Zk ¼ Fk0 þq0 þk1 ðX; Þ. To prove (25) and (26), we use an induction method and prove a stronger conclusion, that is, for all k ¼ 1; 2; . . ., and s 2 S, s0 2 Sc , zks zks
ð31Þ
zks0 zks0
ð32Þ
k0 þq0
Zkþ1 ¼ Fk ðZ; Þ; kþ1
Z
¼ F ðZ; Þ; k
ð23Þ
1). For k ¼ 1, both (31) and (32) are true because z1s ¼ z1s ¼ zs ¼ Fks 0 þq0 ðX; Þ
ð33Þ
z1s0 ¼ z1s0 ¼ zs0 ¼ Fks00 þq0 ðX; Þ
ð34Þ
ð24Þ
for k ¼ 1; 2; . . . ; with Z1 ¼ Z1 ¼ Z. We have Zkþ1 ¼ It is straight forward to FðZk ; Þ and Zkþ1 ¼ FðZk ; Þ. see Zk ¼ Fk0 þq0 þk1 ðX; Þ since Z ¼ Fk0 þq0 ðX; Þ.
according to the definition of Z.
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Q. Zhao and D.-Z. Zheng
2). Assume that (31) and (32) are true for u k. We shall prove that they are also true for u ¼ k þ 1. For s 2 S, for Zskþ1 ¼ Fs ðZk ; Þ;
ð35Þ
all parameters appearing in Fs , we observe that in , are set to upper bounds of their intervals. As a result, we have Fs ðZk ; Þ Fs ðZk ; Þ
ð36Þ
The second inequality is due to the choice of " such that tðÞ " > 0. Furthermore, due to the induction assumption, j0 2 Sc implies from (30) that zkj0 ðk0 þ q0 þ k 1Þðmaxc t þ "Þ: t2S
As a result, we have from (16) and (22) that
sd0 ; j0 ðÞ ðk0 þ q0 þ k 1Þ½ðtðÞ "Þ ðmaxc t þ "Þ t2S
since s; j ¼ þ s; j s; j
ð38Þ
Now we use the DNF of Fs given in form of (12) _ ^ ð ð sd; j ðÞ þ xj ÞÞ ð39Þ Fs ðX; Þ ¼
for k > 1. This contradicts Lemma 2. Thus, we can conclude that for the vector Zk and an s 2 S, if d0 and are such that zskþ1 ¼ Fs ðZk ; Þ ¼ DV0 ðs; d0 Þ s k ð j2D0 ðs;d0 Þ ð d0 ;j ðÞ þ zj ÞÞ, it must hold that D0 ðs; d0 Þ S. Based on this fact, we can establish that ^ ð sd0 ;j ðÞ þ zkj ÞÞ zskþ1 ¼ Fs ðZk ; Þ ¼ ð
j2D0 ðs;d0 Þ
^
ð sd0 ;j ðÞ ð j2D0 ðs;d0 Þ
þ zki ÞÞ Fs ðZk ; Þ: ð46Þ
d¼1;...rðsÞ j2DðsÞ
to establish zskþ1 zskþ1 :
ð40Þ
Since the DNF of Fs is a maximum of rðsÞ items, for every X, there must be a corresponding d0 and D0 ðs; d0 Þ such V that Fs equals to one item, that is, Fs ðX; Þ ¼ ð j2D0 ðs;d0 Þ ð sd0 ;j ðÞ þ xj ÞÞ. We claim that for the vector Zk ,Vif d0 and D0 ðs; d0 Þ are such that zskþ1 ¼ Fs ðZk ; Þ ¼ ð j2D0 ðs;d0 Þ ð sb0 ;j ðÞþ zkj ÞÞ, it must hold that D0 ðs; d0 Þ S. We use proof by contradiction to show this. Suppose on the contrary, ^ ð sd0 ;j ðÞ þ zkj ÞÞ ð41Þ Fs ðZk ; Þ ¼ ð j2D0 ðs;d0 Þ
holds for a subset D0 ðs; d0 Þ not included in S. Then there would be a j0 2 D0 ðs; d0 Þ \ Sc . From (41) we know that Fs ðZk ; Þ sd0 ; j0 ðÞ þ zkj0 :
ð45Þ
ðk0 þ q0 þ k 1Þ" > q0 " ð37Þ
for all j ¼ f1; . . . ; mðsÞg. Here we have used the monotone increasing property of a min-max function in terms of its parameters. Dually, for s0 2 Sc , we have Fs0 ðZk ; Þ Fs0 ðZk ; Þ
ð44Þ
ð42Þ
But we have from (29) that Fs ðZk ; Þ ¼ Fsk0 þq0 þk ðX;Þ ðk0 þq0 þkÞðtðÞ"Þ. Together with (42), this implies
sd0 ;j0 ðÞ þ zkj0 ðk0 þ q0 þ kÞðtðÞ "Þ ðk0 þ q0 þ k 1ÞðtðÞ "Þ: ð43Þ
The second inequality above is true since (12) also holds for Fs ðZk ; Þ. Hence, by (36) we can establish that Fs ðZk ; Þ Fs ðZk ; Þ zskþ1 ¼ Fs ðZk ; Þ ¼ zskþ1 :
ð47Þ
Next let us use a CNF of Fs0 to establish zskþ1 : zskþ1 0 0
ð48Þ
Since the CNF of Fs0 in (10) ^ _ i ð ð b; Fi ðX; Þ ¼ j ðÞ þ xj ÞÞ;
ð49Þ
b¼1;...lðiÞ j2Bði;bÞ
is a minimum of lðsÞ items, for Zk , there must be a b0 and B0 ðs0 ; b0 ÞW such that Fs0 equals to one item, that is, 0 Fs0 ðZk ; Þ ¼ ð j2B0 ðs0 Þ ð bs 0 ; j ðÞ þ zkj ÞÞ. We claim that B0 ðs0 ; b0 Þ Sc . We also use proof by contradiction to show this. Suppose on the contrary, _ 0 ð bs 0 ;j ðÞ þ zkj ÞÞ ð50Þ Fs0 ðZk ; Þ ¼ ð j2B0 ðs0 ;b0 Þ
holds for a subset B0 ðs0 ; b0 Þ not included in Sc . Then there would be a j0 2 B0 ðs0 ; b0 Þ \ S. From (50) we know that 0
Fs0 ðZk ; Þ bs 0 ;j0 ðÞ þ zkj0 :
ð51Þ
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Robust Stability of Min-Max Systems
But we have from (30) that Fs0 ðZk ;Þ ¼ Fsk00 þq0 þk ðX;Þ ðkþk0 þq0 Þðmaxt2Sc t þ"Þ. Together with (51), this implies 0
bs 0 ;j0 ðÞ þ zkj0 ðk0 þ q0 þ kÞðmaxc t þ "Þ:
3). Combining 1) and 2), we have proved through induction that (31) and (32) are true for k ¼ 1; 2; . . .. As a consequence, we know that for all s 2 S and s0 2 Sc ,
t2S
ð52Þ Furthermore, j0 2 S implies from (29) that zkj0 ðk0 þ q0 þ k 1ÞðtðÞ "Þ
ð53Þ
and as a result 0
bs 0 ;j0 ðÞ ðk0 þ q0 þ kÞ½ðtðÞ "Þ ðmax t þ "Þ ðtðÞ "Þ c t2I
ð54Þ
ðq0 þ kÞ" > q0 " for k > 1, where the second inequality is due to (19). This contradicts Lemma 1. Thus, we can conclude that 0 for the vector Zk and an s0 2 Sc , if bW 0 and B0 ðs ; b0 Þ are 0 kþ1 k such that zs0 ¼ Fs0 ðZ ; Þ ¼ ð j2B0 ðs0 ;b0 Þ ð bs 0 ;j ðÞþ zkj ÞÞ, it must hold that B0 ðs0 ; b0 Þ Sc . Base on this fact, we can establish that _ 0 zskþ1 ¼ Fs0 ðZk ; Þ ¼ ð ð bs 0 ; j ðÞ þ zkj ÞÞ 0 k
Fs0 ðZ ; Þ:
j2B00 ðs0 Þ
0
ð55Þ
The second inequality is true since (10) also holds for Fs0 ðZk ; Þ. Hence, by (38) we can establish that Fs0 ðZk ; Þ Fs ðZk ; Þ ¼ zkþ1 : ¼ Fs0 ðZk ; Þ zskþ1 0 s ð56Þ
tðÞ " > max t þ " s ðFðZ; ÞÞ t2Sc s0 ðFðZ; ÞÞ is not a constant Thus we can conclude that ðFðZ; ÞÞ is not stable. vector and furthermore the system ðÞ Since 2 VðÞ, this contradicts the condition that ðÞ is stable for every 2 VðÞ. We have proved the desired result. Case b) The case in which tðÞ 0. In light of Lemma 3, we introduce a constant h ¼ 1 tðÞ and furthermore a new family of systems h ðÞ as follows h ðÞ ¼ fh ðÞj 2 g:
ð57Þ
where h ðÞ is defined by (14). Now from Lemma 3, to is not show that there is a 2 VðÞ such that ðÞ stable is equivalent to show that there is a 2 VðÞ is not stable. Observe that the choice such that h ðÞ of h ¼ 1 tðÞ guarantees that tðh ð ÞÞ ¼ maxf1 þ h; . . . ; n þ hg ¼ tðÞ þ h ¼ 1 > 0. By working on h ðÞ with the parameter space and follow the same line of arguments in Case a), we can establish the desired result. The proof is completed. &
Robust Stability Analysis for Autonomous Min-Max Systems Qianchuan Zhao and Da-Zhong Zheng Center for Intelligent and Networked Systems, Department of Automation and TNLIST Lab, Tsinghua University, Beijing 100084, China
Min-max systems are natural extensions of the well known (max,plus) linear discrete event dynamic system (DEDS) models. They are motivated by recent research on digital circuits and communication networks. In this paper, we study the robust stability of a family of min-max systems whose parameters are perturbed in intervals. In the spirit of Kharitonov, we first reduce the robust stability test to the check of stability on the vertex set of parameter space. We then further reduce the test to a subset derived from the structure of the system family. Keywords: Min-max systems, Kharitonov-like criteria.
robust
stability,
1. Introduction Discrete event dynamic systems (DEDS’s) in which the operations min, max and plus appear simultaneously are known as min-max systems or min-maxplus systems. Some of the recent advances can be found in [3,5,15,19–21] and references therein. Good survey papers include [16] and [4]. A variety of problems arising in digital circuits, computer networks and automated manufacturing plants can be described using these models (see [6,14] and references therein for more detailed discussions on the applications of min-max system models). Theoretically, such systems are natural extensions of timed discrete event systems
Correspondence to: Q. Zhao Tel: 86-10-62783612; Fax: 86-1062786911; E-mail: [email protected]
which contain only maximum timing constraints (or only minimum timing constraints) and which can be studied as so called linear models in terms of max-plus algebra (see e.g., [1,9,10]). A given min-max system is called stable if it has an eigenvalue , or a uniform asymptotic growing rate of the system state vector, namely, limk!1 XðkÞ= k ¼ ð; . . . ; Þ0 . This notion of stability is introduced in [8] for max-plus systems. It is equivalent to the request that the system has an eigenvalue such that the system has periodic trajectory after a finite number of transient phase. This type of systems only needs finite buffer capacity for operation and all events fire at a constant speed in a long run. Note that such a stability concept is different from the usual stability concept used in continuous variable dynamic systems (CVDSs) or hybrid dynamic systems (including piecewise affine systems (e.g., [11]) which contains minmax systems as a special case) literature which requires the convergence of the state to the origin as time tends to infinity. It is also different from the stability concept referring to the boundedness of XðkÞ representing queuing length or waiting time (see e.g., [18]) in the study of DEDS containing queues. Compared with the origin stability or boundedness of state, stability studied here is a weaker one, in the sense that if state XðkÞ converges to zero or is bounded, then divided by k, the limit will automatically go to ð; . . . ; Þ0 with ¼ 0. This relaxation in stability is more suitable for min-max systems because in practice the state is usually used to describe the timing of
Received 10 January 2007; Accepted 2 February 2008 Recommended by J. Lunze, A.S. Morse
105
Robust Stability of Min-Max Systems
repeated events and it is natural for some state variables to grow with at most linear speed as a function of ‘time’ (number of occurrence of the events). Stability of min-max systems plays an important role in solving engineering problems such as the clock schedule verification [7]. The policy iteration results in [7] or a direct calculation of cycle time vector based on the Duality Theorem [12] can be applied to check system stability, provided that we know the parameters exactly. However these results cannot be applied directly for the cases in which the parameters are not exactly known, for example due to uncertainties introduced in the modelling process, or perturbations experienced by plant parameters. Although there are extensive results on the study of robust stability in terms of plant parameters in CVDS (see e.g., [2,17,23,24], and references there in), and also some work in linear DEDS (see [25]), to our best knowledge, there is little work on robust stability analysis for minmax systems. Among existing results of min-max systems, the study of structural properties is most closely related. Balance condition [14,21], Strong connectivity [13], Inseparability [28] and Irreducibility [22]1 are all important examples of structural properties. Structural properties are sufficient to guarantee the stability of a system regardless of the specific values the parameters are taking. But they are not necessary when each parameter belongs to a specific interval because the loss of stability may happen only outside the interval of interest. In this paper, we study the robust stability of a family of min-max systems whose parameters are perturbed in intervals. In the spirit of Kharitonov [2,17], we first reduce the robust stability analysis to the check of stability on the vertex set of the parameter space. The structure of the family is then used to further restrict the scope of the vertex check.
2. Problem Formulation We follow the notations used in [7,14,28]. The operations a _ b and a ^ b are used to stand for maximum and minimum respectively: a _ b ¼ maxða; bÞ and a ^ b ¼ minða; bÞ. We use 0 to denote the transpose of a vector, that is, ðx1 ; . . . ; xn Þ0 is the column vector formed by transposing ðx1 ; . . . ; xn Þ. The upper case English letters X, and Y will be used in the rest of this paper to stand for vectors ðx1 ; . . . ; xn Þ0 and ðy1 ; . . . ; yn Þ0 . bðXÞ and tðXÞ will be used to represent mini fxi g and maxi fxi g respectively, and will be called the bottom and the top of X. 1
We proved that Inseparability is equivalent to Irreducibility in [27].
Definition 1: [7,14] A min-max function of type ðn; 1Þ is any function f : Rn ! R, which can be written as a term in the grammar: f :¼ x1 ; x2 ; . . . ; xn jf þ ajf ^ fjf _ f; ð1Þ where a 2 R will be understood as parameters. The notation used here is the Backus-Naur form wellknown in computer science [7]. The vertical bars separate the different ways in which terms can be recursively constructed. The simplest term is one of the n variables, xi . Given any term, a new one may be constructed by adding an a 2 R; given two terms, a new one may be constructed by applying a min or max function. Only these rules may be used to build terms. For example, fðx1 ; x2 ; x3 Þ ¼ ða þ ðx1 ^ x2 ÞÞ _ ðb þ x3 Þ is a min-max function of type ð3; 1Þ with two parameters (a and b). We will denote the set of parameters of a min-max function f as pf and denote the total number of parameters as mðfÞ. For the example, we have pf ¼ ða; bÞ and mðf Þ ¼ 2. Notice, expressions like 1 ^ x; 1 _ x should not be regarded as min-max functions according to Definition 1. Definition 2: A min-max function of type ðn; nÞ is any function F : Rn ! Rn , such that each component Fi is a min-max function of type ðn; 1Þ. The set of min-max functions of type (n, n) will be denoted by MM (n, n). Definition 3: An autonomous min-max system defined by a min-max function F 2 MMðn; nÞ is a dynamic system given by Xðk þ 1Þ ¼ FðXðkÞÞ
ð2Þ
where XðkÞ ¼ ðx1 ðkÞ; ; xn ðkÞÞ0 2 Rn is the it state vector. Note that (2) can be written in componentwise form 0 1 0 1 x1 ðk þ 1Þ F1 ðXðkÞÞ B C B C ... @ A ¼ @ ... A xn ðk þ 1Þ Fn ðXðkÞÞ 0 1 F1 ðx1 ðkÞ; . . . ; xn ðkÞÞ B C ... ¼@ A Fn ðx1 ðkÞ; . . . ; xn ðkÞÞ In order to study the robustness of a min-max system as defined in (2), we use a vector to denote all parameters appearing in the min-max functions F1 ; . . . ; Fn and use the notations FðX; Þ and ðÞ to indicate the dependency of F and on parameters respectively. Formally, is defined as ¼ ðpF1 ; . . . ; pFn Þ: P Its size is ni¼1 mðFn Þ.
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Q. Zhao and D.-Z. Zheng
M1 Disassembly line
Assembly line M2
perturbed min-max systems whose parameters belong to pre-determined intervals. Definition 4: A family of min-max systems denoted by ðÞ is defined by a set of systems ðÞ ¼ fðÞj 2 g;
Fig. 1. A repair shop.
where the parameter space has the form Y þ ½ ¼ i;j ; i;j ; Example: Consider a simple repair shop having two machines M1 and M2. Each asset has two components A1 and A2. Assets need to be first disassembled, then their components are repaired and assembled as shown in Fig. 1. It is required that once disassembled, an asset’s component A1 must be processed immediately. To improve machine utilization, we also require that the asset’s component A2 must not wait for the same machine repairing the component A1 of the same asset. For simplicity, we assume that the disassembly time can be omitted. So, an asset is disassembled only when there is an idle machine. We assume that on both machines, the repair time of component A1 is a1 and the repair time of component A2 is a2 . Denote the starting time for the repairment of component A1 of the k-th asset and the starting time for component A2 as s1 ðkÞ and s2 ðkÞ respectively, for k ¼ 1; 2; . . .. Denote the complete time for the repairment of the k-th asset’s component A1 and the complete time for component A2 as x1 ðkÞ and x2 ðkÞ respectively. So we have x1 ðkÞ ¼ s1 ðkÞ þ a1 and x2 ðkÞ ¼ s2 ðkÞ þ a2 . When we assume that there is no starvation at the output of the disassembly line and there is no blocking at the assembly line, we have s1 ðkþ1Þ ¼ x1 ðkÞ^x2 ðkÞ and s2 ðkþ1Þ ¼ x1 ðkÞ_x2 ðkÞ. As a convention, we set x1 ð0Þ ¼ 0 and x2 ð0Þ ¼ 0 as the initial condition which means that both machines are ready at time t ¼ 0. In summary, the system’s dynamics can be modelled as a min-max system defined by the following min-max function. FðXÞ ¼
ðx1 ^ x2 Þ þ a1 ðx1 _ x2 Þ þ a2
ð3Þ
In this case, n ¼ 2. The parameter vector is ¼ ða1 ; a2 Þ; where pF1 ¼ ða1 Þ, pF2 ¼ ða2 Þ. Although the components of could Pnbe located using one index taking values from 1 to i¼1 mðFn Þ, we prefer to use two indices i and j to locate them. The first index i will be used to indicate which min-max function the parameter belongs to. The second index j elaborates the location of the parameter in the function once i is fixed. So, we define i;j ¼ pFi ðjÞ. As usual, we need to introduce a family of systems of interest. In this paper, we will study the class of
ð4Þ
ð5Þ
i;j
where a and aþ are lower and upper bounds of parameter a. The range of index i is f1; . . . ; ng and the range of index j is 1; . . . ; mðFi Þ. The robustness property we will study in this paper is stability. Let us introduce stability for a min-max system first and then introduce the robust stability for a family of min-max systems. Definition 5: A given system in form of (2) is stable, if there exists a real such that lim Fki ðXÞ=k ¼ ; i ¼ 1; . . . ; n:
k!1
The number is called the eigenvalue of the system. The limit limk!1 Fk ðXÞ=k is known as cycle time vector and will be denoted as ðÞ. It is independent of the starting point X 2 Rn (as a consequence of the non-expansive nature of min-max functions, see e.g., [16])2. So, the stability condition can also be written in vector form ðÞ ¼ ð; . . . ; Þ0 : with the convention that i ðÞ ¼ limk!1 Fki ðXÞ=k. Remark 1: For every min-max system, the limit ¼ ðlimk!1 Fki ðXÞ=k; i ¼ 1; . . . ; nÞ0 limk!1 XðkÞ=k always exists due to the Duality Theorem (see [12]). So, for an unstable min-max system, its cycle time vector must be a vector whose components are nonidentical. Note that not all min-max systems with finite initial state are automatically stable. In fact, even for an autonomous (max,plus) system (a system that only contains max and plus operation without min operation) with an initial state whose entries are all finite, it is not the case that the system is always stable. For example, the (max,plus) system maxðx1 ðkÞ þ 1; x2 ðkÞÞ x1 ðk þ 1Þ ¼ x2 ðk þ 1Þ x2 ðkÞ 2
This may not be true if we allow some components of X to be equal to
þ1 or 1.
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where x1 ð1Þ ¼ 1; x2 ð1Þ ¼ 0 has the state trajectory x1 ðkÞ ¼ k; x2 ðkÞ ¼ 0 for k ¼ 1; 2; . . .. The two state variables x1 and x2 have a different rate of increase: 6 limk!1 x2 ðkÞ=k ¼ 0. Thus, acclimk!1 x1 ðkÞ=k ¼ 1 ¼ ording to our definition, the system is not stable. Definition 6: A family of min-max systems ðÞ is said to have robust stability, if ðÞ is stable for all 2 .
3. Main Results We reduce the robust stability test problem for a family of min-max systems ðÞ to a finite test on the vertex set of . The vertex set of , denoted by VðÞ is defined as Y þ f VðÞ ¼ i;j ; i;j g: i;j
Theorem 1: A family of systems ðÞ defined by FðX; Þ in form of (2) has robust stability if and only if ðÞ is stable for every 2 VðÞ. This theorem is a Kharitonov-like result in the sense that it allows the robustness test for stability of the whole family to be reduced to the test on the vertex set. Proof: The ONLY IF part is obvious since VðÞ . We should focus on the IF part. We use proof by contradiction. The details are given in the Appendix. & To illustrate the application of our theorem, consider again our example. Example (cont’d): Consider the family of min-max systems given in form of (3), if the parameters ai belong þ to interval ½a i ; ai , i ¼ 1; 2, the parameter space can be written as þ þ ¼ ½a 1 ; a1 ½a2 ; a2
and the vertex set of its parameter space is þ þ VðÞ ¼ fa 1 ; a1 g fa2 ; a2 g which in general contains 22 elements. According to Theorem 1, to judge whether the family of systems has robust stability, we need only to check stability for these parameters. For the special cases where some of the intervals collapse to a single point, the size of the vertex set may be smaller. Consider for example, the special case where a1 ¼ 1 and a2 2 ½1; 2. We only need to check 2 vertices distinguished by the value of a2 : VðÞ ¼ fð1; 1Þ; ð1; 2Þg. Direct calculation shows that ðÞ is unstable for the cases a2 ¼ 2 and stable for a2 ¼ 1 (with a1 ¼ 1). In general, the check of stability on a vertex can be done by the policy iteration results in [7] or a direct calculation of cycle time vector based on the Duality Theorem [12].
Although the number of tests is finite according to Theorem 1, it is interesting to ask whether we can reduce the number of tests further since the size of the vertex set is exponential. The following theorem partially answers this question. For a given min-max function F, introduce a monotone Boolean function3 FðX; 0Þ by setting all parameters to 0 and regard ^ as ‘AND’ and _ as ‘OR’. Here we use 0 to represent a parameter vector whose elements are uniformly zero. Let Bn be the n-dimensional Boolean space. For the example in (3), the corresponding equation FðX; 0Þ ¼ X has the form x1 ^ x 2 x1 ¼ : ð6Þ x2 x1 _ x2 Obviously, the equation FðX; 0Þ ¼ X always has two trivial solutions 0 ¼ ð0; 0; . . . ; 0Þ0 and 1 ¼ ð1; 1; . . . ; 1Þ0 , the constant Boolean vector whose elements are 0 or 1. At the same time, for our example, the Boolean equation FðX; 0Þ ¼ X has one non-trivial solution, namely X ¼ ð0; 1Þ0 . In general, we will denote the set of nontrivial solutions to the Boolean equation FðX; 0Þ ¼ X by ðFÞ, that is, ðFÞ ¼ fXjFðX; 0Þ ¼ X; X 2 Bn ; X 6¼ 0; X 6¼ 1g: So, for our example in (3), we have ðFÞ ¼ fð0; 1Þ0 g. The importance of ðFÞ lies in the fact that these solutions can be used to narrow down the scope of parameter vertices we should check. The reduced vertex set VðFÞ ðÞ of the parameter space is defined as follows. [ fj8i s:t: zi ¼ 0; i;j VðFÞ ðÞ ¼ Z2ðFÞ þ ¼ i;j and 8i s:t: zi ¼ 1; i;j ¼ i;j g:
ð7Þ
In words, VðFÞ ðÞ are all the vertices in the parameter space such that whose parameters match one of the non-trivial solutions Z of the Boolean equation FðX; 0Þ ¼ X in the following way: for i ¼ 1; . . . ; n, the parameters (i;j ¼ pFi ðjÞ, j ¼ 1; . . . ; mðFi Þ) belonging to function Fi take their upper bounds þ i;j when zi ¼ 1 and take their lower bounds þ i;j when zi ¼ 0. Note that VðFÞ ðÞ only depends on the structure of F, not on the value of the parameters. For our example, the set VðFÞ can then be determined as VðFÞ ¼ f ¼ 0 þ ða 1 ; a2 Þg because there is only one element Z ¼ ð0; 1Þ in ðFÞ and it requires the parameters in F1 to be set to lower bounds and the parameters in F2 to be set to upper bounds. 3 Recall that monotone Boolean functions are Boolean functions with no negation operation.
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Theorem 2: A family of systems ðÞ defined by FðX; Þ in the form of (2) has robust stability if and only if ðÞ is stable for every 2 VðFÞ ðÞ. Proof: The ONLY IF part is obvious since VðFÞ ðÞ . We should focus on the IF part. Suppose ðÞ is stable for all 2 VðFÞ ðÞ but there is a 2 such that ð Þ is not stable. That is, for this , ¼ ðFðX; ÞÞ is not a constant vector. Define a vertex as (17) in the proof of Theorem 1 for all i 2 f1; . . . ; ng and j 2 f1; . . . ; mðFi Þg, ( þ i;j ; if i 2 S; ð8Þ i;j ¼ i;j ; if i 2 Sc : where S ¼ fsjs ¼ tðÞg and Sc ¼ fs0 js0 < tðÞg are two subsets of f1; . . . ; ng decided by the vector . is From the proof of Theorem 1, we know that ðÞ unstable. So it is sufficient to show that satisfies 2 VðFÞ ðÞ. This will lead to a contradiction with the assumption that ðÞ is stable for all 2 VðFÞ ðÞ which completes the proof. To establish the desired result, we need to cite a fact that has been used in [28] (Lemma 4 in [28]): For ¼ ðFðX; ÞÞ, it must be true that Fð; 0Þ ¼ . Now define a Boolean vector Z such that for i ¼ 1; . . . ; n, ( zi ¼
0;
if i < tðÞ;
1;
if i ¼ tðÞ:
ð9Þ
Following similar argument of Theorem 1 in [28], we can show that FðZ; 0Þ ¼ Z, that is Z 2 ðFÞ. It is clear that zi ¼ 0 if and only if i 2 Sc and zi ¼ 1 if and only if i 2 S. Hence, according to the construction of (17), it is also true that for all i, i;j ¼ i;j if and only if zi ¼ 0 if and only if z ¼ 1. Hence satisfies and i;j ¼ þ i i;j & 2 VðFÞ ðÞ. The proof is completed. Remark: Although the size of the vertex set VðÞ is exponential in terms of the number of parameters, Theorem 2 claims that we need only to check stability on a usually smaller vertex subset VðFÞ ðÞ. To determine the set ðFÞ is actually to solve Boolean equation which is an extensively studied topic (see e.g., [26]). Remark: There may be some vertices outside VðFÞ such that the system is unstable. But according to Theorem 2, we don’t need to check them. Example (cont’d): Consider the family of min-max systems given by the min-max function (3) with a1 ¼ 1 and a2 2 ½1; 2 again. According to Theorem 2, to test the robust stability of FðX; Þ, we only need to test vertices in VðFÞ . For this specific example, VðFÞ
þ contains a single vertex. VðFÞ ¼ f ¼ ða 1 ; a2 Þ ¼ ð1; 2Þg. And the test result, as above mentioned, is that þ ðða 1 ; a2 ÞÞ is unstable. This agrees with our result for the exhaustive checking of the whole vertex set VðÞ. Under the unstable configuration (a1 ¼ 1; a2 ¼ 2), suppose that M1 repairs component A1 of the first asset, then according to our operation rule, it will continue to process component A1 of all other assets since M1 always finishes earlier than M2. Component A2 of all assets will be processed by M2. The two machines in Fig. 1 have different processing speeds. M1 works faster than M2. Asset’s components will accumulate in the input buffer of M2 and also will accumulate in output buffer of M1. To avoid buffer overflow, control must be introduced to stabilise the system. One possible way to do so is to require M1 and M2 to wait for each other, that is, only after an asset’s both components finish processing on machines, M1 and M2 start to process next asset’s components.
4. Conclusions In this paper, a Kharitonov-like result is established for the robust stability test of a family of autonomous min-max systems whose parameters are given by intervals. It is sufficient to check only a subset of the vertex set of the parameter space to guarantee the robust stability of the family. This subset depends on the structure of the family. The study of robust stability under correlated perturbations and possible applications of these robust stability analysis results to synthesis robust controllers are future research topics.
Acknowledgments This work was supported by NSFC (Grant Nos. 60074012, 60274011 and 60574067, 60721003, 60736027) and National Key Project of China, Fundamental Research Funds from Tsinghua University and Chinese Scholarship Council, Ministry of Eduction of China. The authors would like to thank both anonymous reviewers for their suggestions to improve the quality of the manuscript and thank an anonymous reviewer of a different journal for pointing out a mistake in the original proof of Theorem 1.
References 1. Baccelli F, Cohen G, Olsder GJ, Quadrat J.-P. Synchronization and Linearity. John Wiley & Sons, Chichester, 1992
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2. Barmish BR. New Tools for Robustness of Linear Systems. Macmillan, New York, 1994 3. Chen W, Tao Y. Observability and reachability of nonlinear deds and its coloring graph. Chin Sci Bull, 2001; 46 (8): 62–64 4. Cheng Y. A survey of the theory of min-max systems. In: Huang G.-B, Huang DS, Zhang X.-P (eds), ICIC 2005, Part II, LNCS 3645, Springer-Verlag, Berlin, Heidelberg, 2005, pp. 616–625 5. Cheng Y, Zheng D.-Z. A cycle time computing algorithm and its application in the structural analysis of min-max. Discrete Event Dyn Syst Theory Appl, 2004; 14(1): 5–30 6. Cochet-Terrasson J, Gaubert S, Gunawardena J. Dynamics of min-max functions. Technical Report HPL-BRIMS97-13, HP Lab, 1997 7. Cochet-Terrasson J, Gaubert S, Gunawardena J. A constructive fixed point theorem for min-max functions. Dyn Stab Syst, 1999; 14: 407–433 8. Cohen G, Moller P, Quadrat J.-P, Viot M. Linear system theory for discrete event systems. In Proceedings of the 23rd Conference on Decision and Control, Las Vegas, NV, 1984; pp. 539–544 9. Cohen G, Moller P, Quadrat J.-P, Viot M. Algebraic tools for the performance evaluation of discrete event systems. Proc IEEE, 1989; 77(1): 39–58 10. Cuninghame-Green RA. Minimax Algebra. Lecture Notes in Economics and Mathematical Systems, Vol. 166. Springer-Verlag, Berlin, 1979 11. Ferrari-Trecate G, Cuzzola FA, Mignone D, Morari M. Analysis of discrete-time piecewise affine and hybrid systems. Automatica, 38:2139–2146, 2002. 12. Gaubert S, Gunawardena J. The duality theorem for min-max functions. Comptes Rendus Acad Sci, 1998; 326: 43–48 13. Gaubert S, Gunawardena J. Existence of eigenvectors for monotone homogeneous functions. Technical Report HPLBRIMS-99-08, HP Lab, 1999 14. Gunawardena J. Min-max functions. J DEDS; 1994, 4: 377–406 15. Gunawardena J (ed). Idempotency. Cambridge University Press, Cambridge, 1998 16. Gunawardena J. From max-plus algebra to non-expansive mappings: a nonlinear theory for discrete event systems. Theor Comput Sci, 2003; 293: 141–167 17. Kharitonov VL. Asymptotic stability of an equilibrium position of a family of systems of differential equations. Differential’nye Urauneniya, 1978; 11: 2086–2088 18. Perkins JR, Kumar PR. Stable, distributed, real-time scheduling of flexible manufac-turing/assembly/diassembly systems. IEEE Trans Autom Control, 1989; 34(2): 139–148 19. De Schutter B, and van den Boom T. Model predictive control for max-min-plus systems. In: Boel R, Stremersch G (eds), Discrete Event Systems: Analysis and Control, Kluwer Academic Publishers, Boston, 2000, pp. 201–208 20. Tao Y, Chen W. Cycle time assignment of min-max systems. Int J Control, 2003, 76(18): 1790–1799 21. Tao Y.-G, Chen W, Yin Y.-X. Balance properties and stabilization of min-max systems. Int J Autom Comput, 2006; 1: 76–83 22. van der Woude J, Subiono. Conditions for the structural existence of an eigenvalue of a bipartite (min,max, þ)system. Theor Comput Sci, 2003; 293(1):13–24 23. Wang L. Robust stability of a class of polynomial families under nonlinearly correlated perturbations. Syst Control Lett, 1997; 30:25–30
24. Wang L, Huang L. Finite verification for robustness of discrete-systems under struc-tured perturbations. Chin Sci Bull, 1994; 39(11): 886–891 25. Wang L, Zheng D.-Z. Robustness of linear discrete event dynamic systems. J Control Theory Appl (in Chinese), 1990; 7(2): 27–31 26. Wang Y, McCrosky C. Solving boolean equations using rosop forms. IEEE Trans Comput, 47(2):171–177, 1998. 27. Zhao Q. A remark on inseparability of min-max systems. IEEE Trans Autom Control, 2004; 49(6): 967–970 28. Zhao Q, Zheng D.-Z, Zhu X. Structure properties of minmax systems and existence of global cycle time. IEEE Trans Autom Control, 2001; 46(1):148–151
Appendix Before giving the proof, we need two lemmas. It has been established in [14] that every min-max function f can be written in conjunctive normal form (CNF) given by a minimum of a finite set of max-plus functions. Max-plus functions contains only ‘max’ and ‘ þ ’ operations. CNFs are not unique. For a minmax function f of type ðn; 1Þ, we apply the recursively the following five rules to construct a CNF for it based on the grammar generating the min-max function. If f is of form ða þ xi Þ or simply xi for some i, where a is a parameter, output f as a CNF for itself. (ii) If f is of form f ¼ f1 ^ . . . ^ fl , where each of fi is in form of (i), output f ¼ f1 ^ . . . ^ fl as a CNF for f. (iii) If f ¼ a þ g where g ¼ g1 ^ . . . ^ gl is in form of (ii), define fi ¼ a þ gi and output f ¼ f1 ^ . . . ^ fl as a CNF for f. (iv) If f is in form of ðg1 ^ g2 Þ, where g1 and g2 are already in CNF such that g1 ¼ g11 ^ . . . ^ g1l , and 2 ^ 0 g2l0 , then, let output f ¼ gV 2 ¼ g1 ^ . . .V l 1 ð i¼1 gi Þ ^ ð lj¼1 g2j Þ as a CNF for f. (v) If f is in form of ðg1 _ g2 Þ, where g1 and g2 are already in CNF such that g1 ¼ g11 ^ . . . ^ g1l , and . ^ g2l0 , then, define fij ¼ g1i _ g2j and g2 ¼ g21 ^ . .V output f ¼ i; j¼1 ðfij Þ as a CNF for f. The range of the index i is from 1 to l and j from 1 to l0 . (i)
Since with each rule (iii)-(v), one operation can be removed, the CNF construction procedure will terminate in finite steps for any min-max function. For example, for the min-max function f ¼ ½a þ ðx1 ^ ðb þ x2 ÞÞ _ ðc þ x3 Þ, we first apply rule (i) to decide that terms x1 , ðb þ x2 Þ and ðc þ x3 Þ are in CNF. According to rule (ii), x1 ^ ðb þ x2 Þ is in CNF. Applying rule (iii), we obtain ða þ x1 Þ ^ ða þ b þ x2 Þ as a CNF of ½a þ ðx1 ^ ða þ b þ x2 ÞÞ. Finally, applying rule (v), we obtain ½ða þ x1 Þ _ ðc þ x3 Þ^
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½ða þ b þ x2 Þ _ ðc þ x3 Þ as a CNF for f. It is important to note that constants added to xi in CNF do not need to be a single parameter. For example, the constant added to ða þ b þ x2 Þ is a þ b, a sum of two parameters. In general, let us assume after applying the five rules and collecting all terms sharing the same xi , a CNF of Fi is given in the following form, Fi ðX; Þ ¼
^
ð
_
i ð b;j ðÞ þ xj ÞÞ;
Lemma 4: For every real number h, a system given in form of 0
1 0 1 F1 ðXðkÞÞ x1 ðk þ 1Þ @ A ¼ @ ... A ... xn ðk þ 1Þ Fn ðXðkÞÞ
is stable iff the system h defined in (14) is stable. 0
ð10Þ
b¼1;...lðiÞ j2Bði;bÞ
where i 2 f1; . . . ; ng, X 2 Rn , 2 . Here we use i ðÞ to denote the constant added to xj . b;j Since we are studying min-max functions with parameters taking values within bounded intervals, it is easy to imagine that the constants added to the variables x1 ; . . . ; xn are also bounded. In fact we can give an upper bound for them. Let be ¼ 1 þ max max
mðF Xi Þ
2 i2f1;...;ng
ji;j j:
ð11Þ
j¼1
We have the following lemma on the size of the constants added to the variables x1 ; . . . ; xn in the CNF given in (10). i ðÞ in (10), it holds that Lemma 1: For every b;j i j b;j ðÞj < .
Proof: Let us observe the effects of rules (iii-v) in constructing CNFs. We can see that rule (iii) causes the adding of a parameter to the constant added to xi ; rule (iv) may cause two constants belonging to the same xi combine into the minimum of both; rule (v) may cause two constants belonging to the same xi combine into the maximum of both. In any case, the i ðÞ is absolute value of the resulting constant b;j always upper bounded by the sum of the absolute i ðÞj < : & value of all parameters, as a result j b;j Dually, we can construct disjunctive normal forms (DNFs) for min-max functions. Let Fi ðX; Þ ¼
_
ð
^
ð id;j ðÞ þ xj ÞÞ
ð12Þ
d¼1;...rðiÞ j2Dði;dÞ
be a DNF of Fi . Here we use id;j ðÞ to denote the constant added to xj . We have the following lemma. Lemma 2: For every id;j ðÞ in (12), it holds that j id;j ðÞj < . Lemma 3: [14] For every min-max function f of type ðn; 1Þ and real number h, it holds that fðx1 þ h; . . . ; xn þ hÞ ¼ fðx1 ; . . . ; xn Þ þ h.
ð13Þ
B Xh ðk þ 1Þ ¼ @ 0 B ¼@
xh1 ðk þ 1Þ ...
1 C A
xhn ðk þ 1Þ F1 ðXh ðkÞÞ þ h
1
ð14Þ
C ... A h Fn ðX ðkÞÞ þ h
where Xh ðkÞ is the state vector and the initial condition is xh1 ð1Þ ¼ x1 ð1Þ þ h; . . . ; xhn ð1Þ ¼ xn ð1Þ þ h. Proof: Due to Lemma 3, we can easily show by induction that the state vector of h is related to XðkÞ by Xh ðkÞ ¼ ðx1 ðkÞ þ kh; . . . ; xn ðkÞ þ khÞ. Thus the cycle time vector of h equals to ð1 ðÞ þ h; . . . ; n ðÞ þ hÞ0 : So whenever all components of ðÞ are the same, all components of ðh Þ are the same. & Proof of the sufficient part of Theorem 1: Suppose on the contrary, ðÞ is stable for all 2 VðÞ but there is a 2 such that ð Þ is not stable. That is, for this , ðð ÞÞ ¼ ðFðX; ÞÞ ¼ is not a vector whose entries are all equal. We shall show that there is also not stable, that exists a 2 VðÞ such that ðÞ is, ðFðX; ÞÞ is not a vector whose entries are all equal. The proof is divided into two cases: the case in which the maximum value of the components of (denoted as tðÞ) is positive and the case in which tðÞ is non-positive. Case a) The case in which tðÞ > 0. Based on the vector , we define two non-empty index sets S ¼ fsjs ¼ tðÞg and Sc ¼ fs0 js0 < tðÞg. It is clear that S \ Sc ¼ ; and S [ Sc ¼ f1; . . . ; ng. It is also clear that tðÞ maxs2Ic s > 0. Let " be a positive number such that tðÞ " > 0
ð15Þ
and s =3: " ½tðÞ max c s2I
ð16Þ
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Robust Stability of Min-Max Systems
Since both tðÞ > 0 and tðÞ maxs2Ic s > 0 such " always exists. It is clear that
Now let us establish that for k ¼ 1; 2; . . ., and for all s 2 S, s0 2 Sc
s þ "Þ > " > 0: ðtðÞ "Þ ðmax c s2I
Define a vertex ¼ ði;k Þ of as follows: for all i 2 f1; . . . ; ng and j 2 f1; . . . ; mðFi Þg, ( i;j ¼
þ i;j ;
if i 2 S;
i;j ;
if i 2 S : c
ð17Þ
ð18Þ
for all s 2 S and s0 2 Sc . Our proof is constructive. Define recursively Fk ðX; Þ; k ¼ 1; 2; . . . as Fk ðX; Þ ¼ FðFk1 ðX; Þ; Þ and F0 ðX; Þ ¼ X. According to the definitions of cycle time vector and sets S and Sc , for " > 0 chosen to satisfy (15) and (16), there exists a k0 > 0 such that for all k k0 , it holds that k0 " > ðtðÞ "Þ;
ð25Þ
zks0 < ðk0 þ q0 þ k 1Þðmaxc t þ "Þ
ð26Þ
t2S
Obviously these lead to the desired result that > s0 ðFðZ; ÞÞ since s ðFðZ; ÞÞ k ¼ lim zs s ðFðZ; ÞÞ k!1 k ðk0 þ q0 þ k 1ÞðtðÞ "Þ lim k!1 k ¼ ðtðÞ "Þ;
all parameters appearing in Fs , s 2 S are That is, in , set to upper bounds of their intervals, and all parameters appearing in Fs0 , s0 2 Sc are set to lower bounds of their intervals. is not stable by showing We shall show that ðÞ that > s0 ðFðX; ÞÞ; s ðFðX; ÞÞ
zks > ðk0 þ q0 þ k 1ÞðtðÞ "Þ
ð27Þ and k ¼ lim zs0 s0 ðFðZ; ÞÞ k!1 k ðk0 þ q0 þ k 1Þðmaxt2Sc t þ "Þ lim k!1 k ¼ ðmaxc t þ "Þ: t2S
ð28Þ
ð19Þ Note that we already have for k ¼ 1; 2; . . .
and Fks ðX; Þ > kðtðÞ "Þ;
ð20Þ
zks > ðk0 þ q0 þ k 1ÞðtðÞ "Þ
ð29Þ
Fks0 ðX; Þ < kðmaxc t þ "Þ;
ð21Þ
zks0 < ðk0 þ q0 þ k 1Þðmaxc t þ "Þ
ð30Þ
t2S
t2S
for a fixed initial state X and for all s 2 S and s0 2 Sc . Let Fk0 ðX; Þ ¼ Y. Let q0 ¼ dð="Þe, where dae is the least integer no less than a (e.g., d3:2e ¼ 4). Thus q0 " :
ð22Þ
Let Z ¼ F ðX; Þ. Start from the state Z, we define two sequence of vectors Zk and Zk , k ¼ 1; 2; . . . as follows.
since Zk ¼ Fk0 þq0 þk1 ðX; Þ. To prove (25) and (26), we use an induction method and prove a stronger conclusion, that is, for all k ¼ 1; 2; . . ., and s 2 S, s0 2 Sc , zks zks
ð31Þ
zks0 zks0
ð32Þ
k0 þq0
Zkþ1 ¼ Fk ðZ; Þ; kþ1
Z
¼ F ðZ; Þ; k
ð23Þ
1). For k ¼ 1, both (31) and (32) are true because z1s ¼ z1s ¼ zs ¼ Fks 0 þq0 ðX; Þ
ð33Þ
z1s0 ¼ z1s0 ¼ zs0 ¼ Fks00 þq0 ðX; Þ
ð34Þ
ð24Þ
for k ¼ 1; 2; . . . ; with Z1 ¼ Z1 ¼ Z. We have Zkþ1 ¼ It is straight forward to FðZk ; Þ and Zkþ1 ¼ FðZk ; Þ. see Zk ¼ Fk0 þq0 þk1 ðX; Þ since Z ¼ Fk0 þq0 ðX; Þ.
according to the definition of Z.
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Q. Zhao and D.-Z. Zheng
2). Assume that (31) and (32) are true for u k. We shall prove that they are also true for u ¼ k þ 1. For s 2 S, for Zskþ1 ¼ Fs ðZk ; Þ;
ð35Þ
all parameters appearing in Fs , we observe that in , are set to upper bounds of their intervals. As a result, we have Fs ðZk ; Þ Fs ðZk ; Þ
ð36Þ
The second inequality is due to the choice of " such that tðÞ " > 0. Furthermore, due to the induction assumption, j0 2 Sc implies from (30) that zkj0 ðk0 þ q0 þ k 1Þðmaxc t þ "Þ: t2S
As a result, we have from (16) and (22) that
sd0 ; j0 ðÞ ðk0 þ q0 þ k 1Þ½ðtðÞ "Þ ðmaxc t þ "Þ t2S
since s; j ¼ þ s; j s; j
ð38Þ
Now we use the DNF of Fs given in form of (12) _ ^ ð ð sd; j ðÞ þ xj ÞÞ ð39Þ Fs ðX; Þ ¼
for k > 1. This contradicts Lemma 2. Thus, we can conclude that for the vector Zk and an s 2 S, if d0 and are such that zskþ1 ¼ Fs ðZk ; Þ ¼ DV0 ðs; d0 Þ s k ð j2D0 ðs;d0 Þ ð d0 ;j ðÞ þ zj ÞÞ, it must hold that D0 ðs; d0 Þ S. Based on this fact, we can establish that ^ ð sd0 ;j ðÞ þ zkj ÞÞ zskþ1 ¼ Fs ðZk ; Þ ¼ ð
j2D0 ðs;d0 Þ
^
ð sd0 ;j ðÞ ð j2D0 ðs;d0 Þ
þ zki ÞÞ Fs ðZk ; Þ: ð46Þ
d¼1;...rðsÞ j2DðsÞ
to establish zskþ1 zskþ1 :
ð40Þ
Since the DNF of Fs is a maximum of rðsÞ items, for every X, there must be a corresponding d0 and D0 ðs; d0 Þ such V that Fs equals to one item, that is, Fs ðX; Þ ¼ ð j2D0 ðs;d0 Þ ð sd0 ;j ðÞ þ xj ÞÞ. We claim that for the vector Zk ,Vif d0 and D0 ðs; d0 Þ are such that zskþ1 ¼ Fs ðZk ; Þ ¼ ð j2D0 ðs;d0 Þ ð sb0 ;j ðÞþ zkj ÞÞ, it must hold that D0 ðs; d0 Þ S. We use proof by contradiction to show this. Suppose on the contrary, ^ ð sd0 ;j ðÞ þ zkj ÞÞ ð41Þ Fs ðZk ; Þ ¼ ð j2D0 ðs;d0 Þ
holds for a subset D0 ðs; d0 Þ not included in S. Then there would be a j0 2 D0 ðs; d0 Þ \ Sc . From (41) we know that Fs ðZk ; Þ sd0 ; j0 ðÞ þ zkj0 :
ð45Þ
ðk0 þ q0 þ k 1Þ" > q0 " ð37Þ
for all j ¼ f1; . . . ; mðsÞg. Here we have used the monotone increasing property of a min-max function in terms of its parameters. Dually, for s0 2 Sc , we have Fs0 ðZk ; Þ Fs0 ðZk ; Þ
ð44Þ
ð42Þ
But we have from (29) that Fs ðZk ; Þ ¼ Fsk0 þq0 þk ðX;Þ ðk0 þq0 þkÞðtðÞ"Þ. Together with (42), this implies
sd0 ;j0 ðÞ þ zkj0 ðk0 þ q0 þ kÞðtðÞ "Þ ðk0 þ q0 þ k 1ÞðtðÞ "Þ: ð43Þ
The second inequality above is true since (12) also holds for Fs ðZk ; Þ. Hence, by (36) we can establish that Fs ðZk ; Þ Fs ðZk ; Þ zskþ1 ¼ Fs ðZk ; Þ ¼ zskþ1 :
ð47Þ
Next let us use a CNF of Fs0 to establish zskþ1 : zskþ1 0 0
ð48Þ
Since the CNF of Fs0 in (10) ^ _ i ð ð b; Fi ðX; Þ ¼ j ðÞ þ xj ÞÞ;
ð49Þ
b¼1;...lðiÞ j2Bði;bÞ
is a minimum of lðsÞ items, for Zk , there must be a b0 and B0 ðs0 ; b0 ÞW such that Fs0 equals to one item, that is, 0 Fs0 ðZk ; Þ ¼ ð j2B0 ðs0 Þ ð bs 0 ; j ðÞ þ zkj ÞÞ. We claim that B0 ðs0 ; b0 Þ Sc . We also use proof by contradiction to show this. Suppose on the contrary, _ 0 ð bs 0 ;j ðÞ þ zkj ÞÞ ð50Þ Fs0 ðZk ; Þ ¼ ð j2B0 ðs0 ;b0 Þ
holds for a subset B0 ðs0 ; b0 Þ not included in Sc . Then there would be a j0 2 B0 ðs0 ; b0 Þ \ S. From (50) we know that 0
Fs0 ðZk ; Þ bs 0 ;j0 ðÞ þ zkj0 :
ð51Þ
113
Robust Stability of Min-Max Systems
But we have from (30) that Fs0 ðZk ;Þ ¼ Fsk00 þq0 þk ðX;Þ ðkþk0 þq0 Þðmaxt2Sc t þ"Þ. Together with (51), this implies 0
bs 0 ;j0 ðÞ þ zkj0 ðk0 þ q0 þ kÞðmaxc t þ "Þ:
3). Combining 1) and 2), we have proved through induction that (31) and (32) are true for k ¼ 1; 2; . . .. As a consequence, we know that for all s 2 S and s0 2 Sc ,
t2S
ð52Þ Furthermore, j0 2 S implies from (29) that zkj0 ðk0 þ q0 þ k 1ÞðtðÞ "Þ
ð53Þ
and as a result 0
bs 0 ;j0 ðÞ ðk0 þ q0 þ kÞ½ðtðÞ "Þ ðmax t þ "Þ ðtðÞ "Þ c t2I
ð54Þ
ðq0 þ kÞ" > q0 " for k > 1, where the second inequality is due to (19). This contradicts Lemma 1. Thus, we can conclude that 0 for the vector Zk and an s0 2 Sc , if bW 0 and B0 ðs ; b0 Þ are 0 kþ1 k such that zs0 ¼ Fs0 ðZ ; Þ ¼ ð j2B0 ðs0 ;b0 Þ ð bs 0 ;j ðÞþ zkj ÞÞ, it must hold that B0 ðs0 ; b0 Þ Sc . Base on this fact, we can establish that _ 0 zskþ1 ¼ Fs0 ðZk ; Þ ¼ ð ð bs 0 ; j ðÞ þ zkj ÞÞ 0 k
Fs0 ðZ ; Þ:
j2B00 ðs0 Þ
0
ð55Þ
The second inequality is true since (10) also holds for Fs0 ðZk ; Þ. Hence, by (38) we can establish that Fs0 ðZk ; Þ Fs ðZk ; Þ ¼ zkþ1 : ¼ Fs0 ðZk ; Þ zskþ1 0 s ð56Þ
tðÞ " > max t þ " s ðFðZ; ÞÞ t2Sc s0 ðFðZ; ÞÞ is not a constant Thus we can conclude that ðFðZ; ÞÞ is not stable. vector and furthermore the system ðÞ Since 2 VðÞ, this contradicts the condition that ðÞ is stable for every 2 VðÞ. We have proved the desired result. Case b) The case in which tðÞ 0. In light of Lemma 3, we introduce a constant h ¼ 1 tðÞ and furthermore a new family of systems h ðÞ as follows h ðÞ ¼ fh ðÞj 2 g:
ð57Þ
where h ðÞ is defined by (14). Now from Lemma 3, to is not show that there is a 2 VðÞ such that ðÞ stable is equivalent to show that there is a 2 VðÞ is not stable. Observe that the choice such that h ðÞ of h ¼ 1 tðÞ guarantees that tðh ð ÞÞ ¼ maxf1 þ h; . . . ; n þ hg ¼ tðÞ þ h ¼ 1 > 0. By working on h ðÞ with the parameter space and follow the same line of arguments in Case a), we can establish the desired result. The proof is completed. &