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Stabilisation of discrete‐time systems with finite‐level uniform and logarithmic quantisers
2018; Institution of Engineering and Technology; Volume: 12; Issue: 8 Linguagem: Inglês
10.1049/iet-cta.2017.1092
ISSN1751-8652
AutoresGustavo Cruz Campos, J.M. Gomes da Silva, Sophie Tarbouriech, Carlos Eduardo Pereira,
Tópico(s)Cybersecurity and Information Systems
ResumoIET Control Theory & ApplicationsVolume 12, Issue 8 p. 1125-1132 Research ArticleFree Access Stabilisation of discrete-time systems with finite-level uniform and logarithmic quantisers Gustavo Cruz Campos, Gustavo Cruz Campos DELAE, Universidade Federal do Rio Grande do Sul, Porto Alegre, BrazilSearch for more papers by this authorJoao Manoel Gomes da Silva Jr., Corresponding Author Joao Manoel Gomes da Silva Jr. jmgomes@ufrgs.br orcid.org/0000-0002-6275-2895 DELAE, Universidade Federal do Rio Grande do Sul, Porto Alegre, BrazilSearch for more papers by this authorSophie Tarbouriech, Sophie Tarbouriech LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, FranceSearch for more papers by this authorCarlos Eduardo Pereira, Carlos Eduardo Pereira DELAE, Universidade Federal do Rio Grande do Sul, Porto Alegre, BrazilSearch for more papers by this author Gustavo Cruz Campos, Gustavo Cruz Campos DELAE, Universidade Federal do Rio Grande do Sul, Porto Alegre, BrazilSearch for more papers by this authorJoao Manoel Gomes da Silva Jr., Corresponding Author Joao Manoel Gomes da Silva Jr. jmgomes@ufrgs.br orcid.org/0000-0002-6275-2895 DELAE, Universidade Federal do Rio Grande do Sul, Porto Alegre, BrazilSearch for more papers by this authorSophie Tarbouriech, Sophie Tarbouriech LAAS-CNRS, Université de Toulouse, CNRS, Toulouse, FranceSearch for more papers by this authorCarlos Eduardo Pereira, Carlos Eduardo Pereira DELAE, Universidade Federal do Rio Grande do Sul, Porto Alegre, BrazilSearch for more papers by this author First published: 13 March 2018 https://doi.org/10.1049/iet-cta.2017.1092Citations: 6AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract This study deals with the stabilisation of discrete-time linear systems subject to static finite-level quantisation on the control inputs. Two kinds of quantisers are considered: uniform and logarithmic. The modelling of the finite-level quantisation is obtained by the application of deadzone and saturation maps to an infinite-level quantiser. From this model, conditions for the synthesis of state feedback control laws guaranteeing the convergence of the trajectories to an attractor around the origin provided that the initial state belongs to a certain admissible set are proposed. These conditions can thus be incorporated in linear matrix inequality-based optimisation schemes to compute the stabilising gain while minimising the size of the attractor. 1 Introduction Quantisation of signals is inherent to a digital implementation of control systems and can be considered as a hard non-linearity in the loop. Its effects may lead to undesirable phenomena such as limit cycles, multiple equilibria or chaotic behaviour, even if the controller is supposed to be a stabilising one [1, 2]. Considering the increasing implementation of Networked Control Systems (NCS), researches on quantisation regained the attention of the control community. In particular, since in NCS the control loop elements exchange information through communication channels with limited bandwidth, quantisation plays an important role in reducing the data traffic. However, the use of coarse quantisation may considerably degrade the behaviour of the controlled system. With this motivation, the study of the effects of quantisation on control systems has attracted the attention of several researchers over the last years; see, e.g. [3–6]. It is also important to note that many of these studies have been proposed for continuous-time systems. In the context of discrete-time systems, one can cite some works dealing with the controller or observer design in the presence of uniform or logarithmic quantisers: see, e.g. [7, 8]. In particular, in [9], it has been shown that, for a quadratically stabilisable single-input system, a logarithmic quantiser is the optimal solution in terms of coarse quantisation density. However, it is also shown that the quantiser must have an infinite number of quantisation levels, which is not feasible in practice. Considering also infinite-level logarithmic quantisers, in [10] the authors have introduced the sector bound approach for quantised feedback systems giving simple formulae to the stabilisation problem considering state and output feedback controllers. The problem of finite-level quantisation can be addressed by using dynamic quantisers (see e.g. [11–13]). However, the implementation of such quantisers is in general much more complex than static ones and may not be possible in standard digital platforms. When static finite-level quantisers are considered, an important issue regards the fact that the asymptotic convergence of the trajectories to the origin cannot be ensured for open-loop unstable systems. In this case, due to an implicit input deadzone, the trajectories will converge to either periodic or chaotic orbits around the origin. The idea, in this case, is to guarantee that the trajectories converge asymptotically to a set called attractor. Considering logarithmic quantisers, this problem is addressed in [9] for single-input systems. In that reference, for a given optimal control law, the authors focus on providing bounds on the quantiser parameters to guarantee the convergence of the trajectories to a positively invariant set. On the other hand, quantiser overflow can be seen as an input saturation. It is well known that the global attractivity of the trajectories in the presence of bounded controls cannot be achieved if the open-loop system is exponentially unstable [14]. Hence, in this case, it is important to characterise sets of admissible initial conditions, which are included in the basin of attraction of the attractor. Considering single-input systems and a finite-level logarithmic quantiser, these problems are addressed in [15, 16], where a sector bound approach is applied to derive linear matrix inequality (LMI)-based conditions for estimating a set of initial conditions and an attractor, such that all the state trajectories starting in the first set will converge to the second in a finite time. In those works, the control law is supposed to be given, i.e. only the stability analysis problem is addressed. Moreover, the set of admissible states is constrained to be included in the region where the saturation of the quantiser is not active. Finite-level uniform quantisers are considered for instance in [2, 17]. In these works, based on sector-based relations, conditions to design stabilising state feedback control laws aiming at the minimisation of the attractor set are derived. In [2] the set of admissible initial states, for which the convergence to the attractor is ensured, is also characterised. It should, however, be pointed out that these papers deal only with continuous-time systems. This work addresses the synthesis of stabilising control laws for multi-input discrete-time linear systems subject to finite-level static uniform and logarithmic quantisers. It can be seen as a comprehensive version of the conference paper [18], where only the stability analysis has been addressed. Furthermore, we extend the approach to deal with different quantisation parameters in each input channel. Based on the modelling of the finite-quantisation through deadzone and saturation maps, a unified framework to deal with both uniform and logarithmic quantisations is proposed. Although the approach also applies to stable systems, we particularly focus on the case in which the plant is exponentially unstable. In this paper, considering explicitly the non-linear effects induced by the quantisation, the deadzone as well as the saturation, conditions to compute a state feedback gain to ensure that the trajectories of the non-linear closed-loop system converge asymptotically to an attractor around the origin, provided that the initial condition belongs to an admissible set, are proposed. Differently from references [15, 16], we consider the effective non-linear effect of the saturation and a formal proof regarding the positive invariance of the attractor is provided. The proposed conditions are cast in LMI-based optimisation problems to compute the controller gain in order to minimise the size of the attractor while guaranteeing the set of admissible initial states includes some pre-specified region in the state space. A numerical example to illustrate the methodology and to compare the solutions with uniform and logarithmic quantisers is provided. The paper is organised as follows. In Section 2, the generic framework and the problem statement are presented. Section 3 addresses the case of uniform quantisation, while Section 4 deals with the logarithmic case. Optimisation and computational issues are discussed in Section 5. A numerical example and some concluding remarks end the paper. Notation : Throughout the paper, I denotes the identity matrix and denotes the null matrix (equivalently the null vector) of appropriate dimensions. For a matrix , , , denote its transpose, its i th row and its trace, respectively. The matrix is the block-diagonal matrix having as diagonal blocks. In symmetric matrices, stands for symmetric blocks. For a vector , , , denote its i th component, its transpose and the component-wise absolute value operator, respectively. is the component-wise sign function, with , and the component-wise floor operator. For two sets and , denotes the set deprived of . 2 Problem statement Consider the following discrete-time system: (1)where , , are, respectively, the state, the input of the system and the initial state. A, B are real matrices of suitable dimensions and is a component-wise infinite level quantiser. In order to model the finite-level quantisation, we consider a component-wise saturation function, and a component-wise deadzone function , i.e. for it follows that: (2) (3)In this case, represents the maximum/minimum quantisation levels for the i th input channel, while denote the limits for which the i th input signal is coded as zero. In order to stabilise system (1), we consider a state feedback control law (4)By defining now the functions , and as follows: (5) (6) (7)the closed-loop dynamics is given by the non-linear system (8)where for simplicity we denote , and . From (2)–(4), two sets of interest in the state space can be defined (9) (10)The set corresponds to the region where none of the inputs is saturated in the maximal or minimal quantisation level, while corresponds to the region where the actual control input injected into the system is zero. If , the closed-loop dynamics reads (11)i.e. only the non-linear effect of the infinite-level quantiser is present. On the other hand, note that if , the system is in open loop. Hence, if the matrix A is not Schur–Cohn, the asymptotic stabilisation of the origin for the closed-loop system cannot be achieved even when the gain K is supposed to be a stabilising one (i.e. when is Schur–Cohn). Actually, in this case, either a limit cycle or a chaotic behaviour will be generated around the origin. Hence the convergence to the origin should be replaced by the convergence to an attractor set around the origin, as small as possible (i.e. referred as practical stability in [9]). Moreover, if A is not Schur–Cohn, under input saturation only local (regional) stability can be achieved [14, 19]. In this work, we focus on the case in which matrix A is not Schur–Cohn, i.e. the open-loop system is exponentially unstable. Then, the problem we aim to solve can be stated as follows. Problem 1.Given system (1), with saturation and deadzone limits given, respectively, by vectors and , and the control law (4), determine a gain K, a set and a compact set containing the origin, such that and are positively invariant sets with respect to the closed-loop system (1)–(4) (or equivalently system (8)). For every initial condition , the trajectories of the closed-loop system (1)–(4) are bounded and converge in finite time to (which is therefore an attractor of the trajectories). is as smaller as possible. The sets in Problem 1 are depicted in Fig. 1. Note that the set is implicitly included in the region of attraction of the attractor set . As aforementioned, since A is assumed not to be Schur–Cohn, the global convergence to the attractor is not possible to achieve with bounded controls. On the other hand, the set is the region where the trajectories are ultimately bounded and, if minimised can be seen as an estimate of the periodic or chaotic orbit around the origin. Fig. 1Open in figure viewerPowerPoint Sets and In the sequel, we address Problem 1 considering uniform and logarithmic finite-level quantisations. 3 Uniform quantisation The function corresponding to a uniform quantisation is defined as follows: (12)where is a positive real-scalar representing the quantisation step of the i th control input. In this case, , where is the number of positive levels between and zero. Moreover, as it can be seen in Fig. 2, the deadzone is implicitly included in the definition of q. Hence, the equation describing the closed-loop system (8), considering a finite-level uniform quantisation can be simplified to the following one: (13)with , . Fig. 2Open in figure viewerPowerPoint Uniform quantisation Before, proposing a result to address Problem 1, we recall some auxiliary conditions regarding the sector bounded functions and . Lemma 1.(see [17] and [19] for details) For as defined in (5), with defined by (12), and considering , the relations (14) (15)are verified for any diagonal positive definite matrices , .Actually, (14) corresponds to a generalisation of the condition in [17] to cope with different quantisation parameters in each input. Lemma 2.(see [17] and [19] for details) For as defined in (6) and a vector , the relation (16)is verified for any diagonal positive matrix , provided that with (17) Note that considering , from (5), condition (16) reads (18)By using Lemmas 1 and 2, the following proposition to solve Problem 1 can be stated. Proposition 1.If there exists a symmetric positive definite matrix , diagonal positive matrices , , , matrices Y, , positive scalars , and a scalar , , satisfying the following conditions: (19) (20) (21)then, for , it follows that the trajectories of the system (13) converge in finite time to the set , with (22) (23) Moreover, and are positively invariant sets with respect to (13). Proof.Considering the quadratic Lyapunov function , with , we want to prove that (24)with being a positive scalar, .In this case, it follows that is a positively invariant set and , the corresponding trajectory converges in finite time to . Note that Now applying S -procedure along with Lemmas 1 and 2, considering and , with G being a free matrix variable, if (25)it follows that (24) is satisfied, provided that , with (26)By using (13), it is possible to re-write (25) as follows: (27)with defined as follows: (28)It is straightforward to verify that (27) will be satisfied for some if we guarantee (20) and . Applying Schur complement, is equivalent to (29)Then by pre- and post-multiplying (29) by and with the change of variables , , , , we obtain relation (19).Furthermore, one has to prove that the set is positively invariant. This is proven if we guarantee that whenever (i.e. when ). It is possible to do that by verifying the following inequality, for some positive scalar : (30)On the other hand, inequality (30) is true if we are able to verify that (31)and (32)provided that .Now by choosing , it is straightforward to verify that (32) is implied by (19). Furthermore, since and , it follows that: (33)Hence (20) implies (31). Then, if (19) and (20) are satisfied, (30) is implicitly verified and thus is positively invariant with respect to system (13).Finally, the satisfaction of relation (21) implies that the ellipsoid is included in the polyhedral set defined in (26), which ensures the validity of condition (18), with , , which concludes the proof. □ 4 Logarithmic quantisation In the case of the logarithmic quantisation, each component of the decentralised vector is defined as (34) (35)The parameter is said to be the quantisation density (instead of quantisation step as in the uniform quantisation). The function corresponding to an infinite-level quantiser is depicted in Fig. 3. Fig. 3Open in figure viewerPowerPoint Logarithmic quantisation Recall now that, to obtain a finite-level quantiser, saturation and deadzone effects are applied over the infinite-level quantiser . In this case, is considered as the least positive quantisation level. Actually, is related to the deadzone limit through the equation . Moreover, the saturation limit of each quantised input is given by , where corresponds to the number of quantisation levels between and zero. To solve Problem 1, we recall a sector condition verified by the nonlinearity (it corresponds to a multi-input version of the one in [15]) Lemma 3.For with defined in (34) and (35), and , the relation (36)is verified for any diagonal positive definite matrix . For the non-linearity , one can use a similar condition to that one stated in Lemma 1, as follows: Lemma 4.For every defined in (7), the relation (37)is verified for any diagonal positive definite matrix . By using Lemmas 2–4, the following proposition to solve Problem 1 can be stated. Proposition 2.If there exists a symmetric positive definite matrix , diagonal positive matrices , , , matrices Y, , positive scalars , and a scalar , , satisfying the following conditions: (38) (39) (40)then, for , it follows that the trajectories of the system (8) converge in finite time to the set , with and as defined in (22) and (23). Moreover, and are positively invariant sets with respect to (8). Proof.Applying S -procedure along with Lemmas 2–4, if (41)with being a positive scalar, then it follows that , is satisfied, provided that .By using (8) and considering , it is possible to re-write (41) as follows: (42)with defined as follows: (43)It is straightforward to verify that (42) will be satisfied for some if we guarantee (39) and . By applying Schur complement twice, then by pre- and post-multiplying by and with the change of variables , , , , , we obtain relation (38).To prove that the set is positively invariant, it suffices to verify that, for some positive scalar , (44)Then, by choosing , following the same reasoning is done in the proof of Proposition 1, it follows that (38) and (39) imply (44).Finally, the satisfaction of relation (40) implies that the ellipsoid is included in the polyhedral set defined in (26), which concludes the proof. □ 5 Computational issues In this section, we discuss how to compute the stabilising control law by using the results stated in Propositions 1 and 2. Recalling Problem 1, the goal is to compute K in order to obtain an attractor set as small as possible. To achieve this objective, one needs to find a suitable measure for the set and minimise it subject to conditions of Propositions 1 and 2. In particular, it is possible to implicitly minimise the volume of by minimising , or equivalently, . On the other hand, it is also important to guarantee that the set has a reasonable size or that it covers a given set of admissible initial conditions . With this aim, we can consider as a polytope in the state space described by the convex hull of its vertices, i.e. . In this case, an extra condition to ensure that is given as follows: (45)Hence, in order to compute K to minimise the attractor while ensuring a guaranteed region of attractivity, i.e. regional stability, the following optimisation problems can be considered. Uniform quantisation (46) Logarithmic quantisation (47) Notice that conditions and are implicitly ensured in (46) by (19). On the other hand, conditions , , , are implicitly ensured in (47) by (38). It is important to note that conditions (19) and (21), in the uniform quantisation case, and conditions (38) and (40), in the logarithmic quantisation case, are non-linear in the decision variables (i.e. they are not LMIs), which prevents from solving directly a convex optimisation problem. Nevertheless, it is possible to overcome this problem by considering the variables , and as tuning parameters. In the uniform quantisation case, must be fixed as well [20]. Moreover, to guarantee feasibility, it is necessary that the following conditions are respected: (48)Thus, the optimal value of (46) or (47) can be obtained by solving LMI-based optimisation problems on a grid with respect to and (in the uniform case). This can be efficiently addressed by using, for instance, the Nelder–Mead simplex method (implemented in MATLAB by the fminsearch function). As aforementioned, in problems (46) and (47) we ensure that is sufficiently large by imposing that . If we seek a larger , we can enlarge the shape set . This can be done, for instance, by using a scaling factor for in order to ensure , which can be translated in the LMI constraint (49)where . In this case, while is minimised, the scalar is implicitly adjusted to fulfil (49). However, by increasing a side effect is that the optimal will also be probably increased since both sets are defined from the same matrix P. There is, in fact, a trade-off to manage. Moreover, note that for too large the conditions may be not feasible. This comes from the fact that, since the open-loop system is exponentially unstable (by assumption), the global stabilisation is indeed not possible. On the other hand, we can formulate a converse problem in which the main objective is the maximisation of , while keeping sufficiently small. For this, we can consider for instance a polyhedral shape set represented as (50)where . Thus, in order to ensure that is sufficiently small we can force , which is ensured by the following set of constraints: (51)In this case, in order to maximise the following optimisation problems can be considered: Uniform quantisation (52) Logarithmic quantisation (53) Note that the constraint ensures that , i.e. the minimisation of implies the maximisation of the . 6 Numerical example Consider the following discrete-time system derived from the continuous-time system treated in [21], with a sampling period : (54)Our objective is to compare a uniform and a logarithmic quantiser when designing a static state feedback controller that minimises the size of the attractor. With this aim, we consider that both quantisers use the same number of bits (i.e. presents the same number of quantisation levels). Considering that the most significant bit is reserved for the sign, the relation between the number of bits and the number of positive levels of the quantiser N, for each input channel, is given by (55)The levels of saturation in the control channels are also considered as being the same for both quantisers and are given by vector . Fixing all these parameters, in the uniform case, the quantisation step in each channel can be calculated by , , with being the number of quantisation levels in control input i. In the logarithmic case, the quantisation density is a degree of freedom, which will affect the smallest positive quantised value through the following relation: (56)Assuming for both channels (which yields ), the two quantisation cases are analysed in the sequel. For this, we apply the optimisation problems (46) or (47) with , with . Considering the uniform quantiser, for and given above, we obtain . A grid search allows to find the best feasible values for the tuning parameters, respectively, , , and . The solution to the optimisation problem (46) in this case yields and which has its trace equal to 86.88. In the logarithmic quantiser case, also for and given above, we choose . Recalling that , we obtain . Comparing with the deadzone limits of the uniform quantiser case, one can expect the attractor size to be considerably smaller in the logarithmic case. A grid search allows to find the best feasible values for the tuning parameters, respectively, , and . The solution to the optimisation problem (47) in this case yields and which has traced its equal to . Figs. 4, 5–6 show, respectively, the evolution of the state for an initial condition of in the uniform and logarithmic cases, and the ellipsoids obtained in both cases. Fig. 4Open in figure viewerPowerPoint States evolution in the unif. quantisation case Fig. 5Open in figure viewerPowerPoint States evolution in the log. quantisation case Fig. 6Open in figure viewerPowerPoint Ellipsoid in the uniform (outer one - in red) and logarithmic (inner one - in blue) quantisation cases with the same number of bits Notice that in this case both analysis and simulation show the superiority of logarithmic quantisers in terms of attractor size, considering the same number of quantisation levels and the same saturation limits. Nevertheless, as mentioned previously in this section, the quantisation density is a degree of freedom of the logarithmic quantiser and a good choice of this parameter must be made. One would expect that the closer to one is, a smaller would be obtained since quantisation would become denser and the quantisation non-linearity (see Fig. 3) would lie in a closer sector (i.e. would be close to zero). However, considering and the number of quantisation levels fixed, relation (56) shows that there is a trade-off between and (which is proportional to , another parameter that influences the size of ). In this example, the best choice for was found numerically. Table 1 shows the results of the optimisation problem (47) that would have been obtained with different values of , . Notice that, when , becomes larger and, as expected, becomes larger too (i.e. is smaller). On the other hand, when , becomes smaller but, since the quality of quantisation is worse (actually the sector where the lies is more opened, i.e. more uncertainty is considered), grows. Table 1 also shows that for , we obtain which is similar in size to the one obtained in the uniform case. Finally, for the deadzone limit is approximately the same as that one of the uniform quantisation, but is several times larger than the one obtained in the uniform case. This indicates that, depending on the quantisation density chosen for the logarithmic quantiser, if we fix the saturation and the deadzone limits as well as the number of quantisation levels, the uniform quantiser may provide better results. Table 1. Results obtained for different values of – logarithmic quantisation 0.4706 0.36 2569.9 0.5385 0.3 5830.8 0.7391 0.15 8773.6 0.7857 0.12 2958.0 0.8182 0.10 289.0 0.8349 0.09 83.7 0.8935 0.056 1.45 In order to illustrate the trade-off regarding the size of and , as discussed in Section 5, we replace now (45) by (49) in optimisation problems (46) and (47). Tables 2 and 3 show the results obtained for different values of the scaling factor for the uniform and logarithmic cases, respectively. Note that as increases, i.e. the larger is the shape set that must be included by , the smaller is the value of , which indicates that increases. On the other hand, it can be observed that decreases, meaning that also increases as we increase . Moreover, as expected, for too large ( in the uniform case and in the logarithmic one) no feasible solution (N.F.) can be found since the global stability is not achievable because the open-loop system is exponentially unstable. Table 2. Influence of over the sizes of and – uniform quantisation 0.5 0.0299 1369.70 3.3106 0.8 0.0685 231.77 0.6230 1.0 0.1602 86.88 0.2259 1.2 21.1349 2.34 0.0744 1.3 N.F. N.F. N.F. Table 3. Influence of over the sizes of and – logarithmic quantisation 0.5 5.5860 0.8 1.0100 1.0 0.0017 0.3754 1.17 2.9110 36.72 0.0705 1.18 N.F. N.F. N.F. 7 Conclusion This paper has tackled the stabilisation of (unstable) multi-input discrete-time linear systems involving input static finite quantisation. A unified framework to handle uniform and logarithmic quantisers has been proposed. The approach is based on the modelling of the finite quantisation through the appropriate application of deadzone and saturation operators. Based on the properties of the three types of non-linearities (i.e. infinite quantisation, deadzone and saturation), conditions to design a static state feedback controller that guarantees the convergence to an attractor set , provided the initial state belongs to a set , are developed. These conditions are incorporated in optimisation problems aiming at determining the stabilising feedback gain while minimising the size of the attractor set and guaranteeing that the set covers a pre-defined region in the state space. Although this work considered only input quantisation, the approach can be extended to cope with state quantisation and also be used to optimally tune the quantisers parameters when the gain is given. 8 Acknowledgments The work was supported by CAPES (STIC-AmSud project 88881.143275/2017-01) and CNPq (grants PQ-305979/2015-9 and Univ-422992/2016-0), Brazil. 9 References 1Delchamps D.: 'Stabilizing a linear system with quantized state feedback', IEEE Trans. Autom. 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