Passivity and Stability of Switched Systems Under Quantization

Transcription

1 Passivity and Stability of Swithed Systems Under Quantization Feng Zhu Department of Eletrial Engineering University of Notre Dame Notre Dame, IN, Han Yu Department of Eletrial Engineering University of Notre Dame Notre Dame, IN, Panos J. Antsaklis Department of Eletrial Engineering University of Notre Dame Notre Dame, IN, Mihael J. MCourt Department of Eletrial Engineering University of Notre Dame Notre Dame, IN, ABSTRACT Passivity theory is a well-established tool for analysis and synthesis of dynamial systems. Reently, this work has been extended to swithed and hybrid systems where passivity and stability results of single systems as well as interonneted systems are derived. However, the results may no longer hold when quantization is present as is the ase with digital ontrollers or ommuniation hannels. The ontribution in this paper is to introdue a ontrol framework under whih passivity for swithed and non-swithed systems an be maintained. This framework enters on the use of an input-output oordinate transformation to reover the passivity property. In order to present these results, bakground material is provided on passive quantization and output strit passivity for swithed and non-swithed systems. The proposed framework is first presented for non-swithed systems and then generalized to swithed systems. Categories and Subjet Desriptors J. Physial Sienes and Engineering]: Engineering General Terms Theory Keywords Passivity, Swithed Systems, Quantization Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for profit or ommerial advantage and that opies bear this notie and the full itation on the first page. To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior speifi permission and/or a fee. HSCC 1, April 17 19, 01, Beijing, China. Copyright 01 ACM /1/04...$ INTRODUCTION The notion of passivity, whih originated in eletrial network theory, is a haraterization of system input/output behavior based on a generalized notion of energy. Along with Lyapunov funtion tehniques, passivity theory is widely used in analysis and ontrol of nonlinear systems 4, 10, 11, 13]. It is well known that passive systems are stable. Additionally, the parallel interonnetion and the negative feedbak interonnetion of two passive systems is still a passive system. These results provide open-loop onditions to guarantee losed-loop stability. These well known results are summarized along with some reent results in 7]. These results have been extended to swithed systems in 14, 15, 16, 17,, 9]. Although traditional passivity theory has been applied suessfully in various lassial nonlinear systems, this property is vulnerable to disretization, quantization and other fators introdued by digital ontrollers or ommuniation hannels in modern ontrol systems. In digital ontrol system design, a ontinuous-time system is first disretized into asampled-datasystem.however,itispointedoutin1,3, 3, 19, 5] that passivity is not preserved under disretization, whih means the disretized system may not be passive even if the original ontinuous-time system is passive. Exatly how muh passivity is lost under standard disretization has been quantified in 1]. The passivity degradation under the standard disretization an be haraterized in terms of passivity indies and sampling time. In 19, 3], a novel average passivity for disrete-time systems was proposed in order to preserve the passivity property losslessly under any sampling time. Besides preserving passivity in disrete-time, stability and stabilization of disrete-time passive systems were also onsidered in reent work 1, 0]. The problem of finding the maximum sampling time preserving passivity for linear disrete-time systems was onsidered in 1]. It was shown that the feedbak system is exponentially stable if the timevarying asynhronous sampling times embedded in feedbak onnetion are bounded by the maximum sampling time. Two passivity-based ontrol strategies for the problem of stabilizing sampled-data systems were presented in 0]. In addition to disretization, the effet of quantization also

2 needs to be onsidered when digital ontrollers interat with the environment by means of analog-to-digital onverters or digital-to-analog onverters that have a finite resolution. Moreover, quantization is neessary when the information between plants and ontrollers is transmitted through ommuniation networks. In fat, the problem of ontrol using quantized feedbak has been an ative researh area for a long time. Most of the work 8, 1, 4, 6, 18] onentrates on understanding and mitigating the effets of quantization for feedbak stability and stabilization. The existing results on passivity and quantization effets mainly fous on ertain speifi problems, depending on what kind of systems are onsidered. In signal proessing systems 5], passivity analysis and passifiation of LTI systems with quantization was treated as an unertainty desribed by integral quadrati onstraints. In networked ontrol systems, onditions were derived 9] under whih the losed-loop networked ontrol system is passive in the presene of sensor quantization and network indued delay. The problem of losed-loop stability for input-affine passive systems with quantized output feedbak was investigated in ]. Reent results 6, 7] used passivity to ahieve L stability in the presene of ommuniation delays and signal quantization for networked ontrol systems. To the authors best knowledge, there is no published results on either preserving passivity under quantization in general or stability onditions for swithed systems under quantization. In this paper, the main ontributions are the derivation of onditions under whih the passive struture of an output stritly passive (OSP) system an be preserved under quantization and its appliation in stability for passive swithed systems with passive quantizers. The passivity preservation relies on an input/output transformation on the quantized input and output. The result shows that one an find suh transformation so that the same passivity index of the original OSP system, with respet to the transformed input and output, will be reovered. The result is relatively general sine we only require the system to be OSP and the quantizers to be passive, whih haraterize many pratial quantizers. Although the passivity preserving ondition is initially derived for non-swithed systems, it an be extended to passive swithed systems where the input/output transformation an swith between different transformations aording to the urrent ative subsystem. Therefore, passivity of passive swithed systems under quantization an be guaranteed and the stability onditions in 15, 16] an be applied. The rest of the paper is as follows. In Setion, bakground material on disrete-time passive systems and passive swithed systems is overed. The notion of passive quantizers is introdued. The onditions on preserving passivity under quantization for OSP systems are given in Setion 3. Setion 4 extends the passivity-preserving onditions for non-swithed systems to passive swithed systems and then the stability onditions on passive swithed systems are obtained. An example is provided in Setion 5 to demonstrate the methods used in this paper. Some onlusions are provided in Setion 6.. BACKGROUND MATERIAL.1 Passivity for Disrete-Time Systems The work in this paper is based on passivity for disrete- Figure 1: A general quantizer bounded by a one time non-swithed systems with time index k Z +.Asystem has input u(k) R m,outputy(k) R m,andinternal state x(k) R n and an be modeled as x(k +1) = f(x(k),u(k)) y(k) = h(x(k),u(k)). Adisrete-timesystemispassiveifitstoresanddissipates energy supplied to the system without generating its own energy. The passivity property is typially demonstrated by finding a positive energy storage funtion and showing that the energy stored in the system at any time step is bounded by the energy supplied to the system. Definition 1. A disrete-time system (1) is passive if there exists a positive energy storage funtion V (x) (V (x) > 0, x 0)suhthatthefollowinginequalityholdsforall k k 0 V (x(k)):=v (x(k +1)) V (x(k)) u T (k)y(k) ρy T (k)y(k) () for ρ 0. When ρ>0 this system is alled output stritly passive.. Passive Quantizers Consider a quantizer q( ) with an input v and an output u, wherev R and u U. U R is a quantized set whose elements are distint quantized levels. Definition. 6] A quantizer is alled a passive quantizer if its input v and output u satisfy where u = q(v) and 0 a b<. (1) av uv bv (3) The notion of a passive quantizer 6] is based on oni systems theory 8]. A passive quantizer is a speial ase of a memoryless oni system. This an be seen in Fig. 1, where aquantizersatisfying(3)hasitsinputandoutputmapping bounded in a one haraterized by two lines with slope a and b. The quantizer is alled passive sine the ondition uv 0holdsforallinputsv. This is the general ondition for a memoryless nonlinearity to be passive 13]. The notion of passivity for quantizers an apture many quantizers

3 Figure : (a) A uniform quantizer with infinite quantization levels (b) A uniform quantizer with finite quantization levels Figure 3: (a) A logarithmi quantizer with infinite quantization levels (b) A logarithmi quantizer with finite quantization levels used in pratie, suh as the uniform mid-tread quantizer (Fig. ), the logarithmi quantizer (Fig. 3) and many nonstandard quantizers (Fig. 1). We an find the values of a and b from a quantizer s input and output mapping. For example, we an show that a = 0,b = for a uniform mid-tread quantizer with infinite/finite quantization levels; a = 0,b = 1 + δ for a logarithmi quantizer with finite quantization levels; and a =1 δ, b =1+δ for a logarithmi quantizer with infinite quantization levels, where 1 > δ > 0isaonstant quantization gain. It is worth pointing out that a quantized system in Fig. 4isnotneessarilyapassivesystemevenifthequantizers Q and Q p are passive quantizers. This leads us to resort input-output transformations introdued in Setion III to preserve passivity. Figure 4: A general system with input and output quantization.3 Passivity for Swithed Systems A nonlinear swithed system onsists of a finite set of subsystems with nonlinear dynamis. The finite number of subsystems an be enumerated, {1,,..., P}. At any point in time, a single subsystem i is ative and the dynamis are nonlinear and time-invariant. The time-varying nature of these systems omes from the swithing behavior. The swithing signal σ(k) is a funtion that maps the time to the index of the ative subsystem, σ : Z + {1,...,P}. This funtion is pieewise onstant and only hanges at swithing instants. The model with the swithing signal is given by x(k +1) = f σ(k) (x(k),u(k)) y(k) = h σ(k) (x(k),u(k)). The swithing instants an be listed in order k 1, k,et. Alternatively, the notation k ip will be used to denote the p th time that subsystem i beomes ative. For example, the first subsystem (i =1)beomesativeforthefirsttime(p =1) at time k 0 (k 0 = k 11 ). The seond subsystem i =beomes ative at time k 1 (k 1 = k 1 )andsoforth. Byusingthesetwo notations in onjuntion, it is possible to list ompletely the times that a system beomes ative as well as the times it beomes inative. Subsystem i beomes ative the p th time at time k ip and then inative at time k (ip+1). That same subsystem beomes ative again at time k i(p+1). An indiator set will be defined to signify regions where apartiularsubsystemisative.considersubsystemi that is ative from k i1 to k (i1 +1), k i to k (i +1), et. The set of times I i an be defined to indiate those time intervals where subsystem i is ative, (4) K i I i = {k ip,...,k (ip+1)}. (5) p=1 This notation will be used to draw a distintion between the ative and inative time intervals of a system. The notion of passivity for swithed systems used in this paper is based on previous work on deomposable dissipativity for swithed systems. This approah has been used in ontinuous-time 9, ] and in disrete-time 14]. The onept of deomposable dissipativity is based on the fat that systems typially store energy differently when they are ative ompared to when they are inative. The solution is to deompose the supply rate into an ative portion and an inative portion. When a subsystem is inative, it may have adifferentsupplyratedependingonwhihothersubsystem is ative. The definition given here is a speial ase of 14]. While that work presented a very general definition, the authors didn t onsider stability of interonneted systems. Traditionally, stability of feedbak interonnetions is one of the main benefits of dissipativity theory. In deomposable dissipativity, the multiple energy storage funtion approah is taken. This allows for eah subsystem i to have a unique notion of energy aptured by the storage funtion V i(x). This notion of energy is positive, i.e. for all i, V i(x) > 0forallx 0. Thenotionofsuppliedenergy for a subsystem i while it is inative may be unique for eah ative subsystem j i. Thisresultsinseveralinative energy supply rates for eah i and j. These rates may be a funtion of input, output, state, and time and will be denoted as ω j i (u, y, x, k). When eah subsystem is inative, the following inequality holds for eah ative subsystem j at an appropriate time t I j ( i) Passivity for disrete-time swithed systems is given in the following definition. Reall that a funtion α : R + R + is lass K if α(0) = 0, α is non-dereasing, and α is radially unbounded. Definition 3. Consider a disrete-time swithed system (4). This system is passive if there exists a positive storage

4 funtion V i(x), foreahsubsystemi, withthepropertythat for some K funtions α i and α i, α i( x ) V i(x) α i( x ), suh that the following onditions hold for all i. 1. During the ative time period k I i of eah subsystem i, thesystemispassive(ρ i 0) V i(x(k +1)) V i(x(k)) u T y ρ iy T y. (6). When eah subsystem i is inative, it is dissipative with respet to a ross supply rate that may be speifi to the ative subsystem j. Fork I j V i(x(k +1)) V i(x(k)) ω j i (u, y, x, k). (7) 3. The ross supply rates are absolutely summable for all swithing sequenes i and j i, k=k 0 ω j i (u, y, x, k) <L, (8) where L is an arbitrarily large finite onstant. When ρ i > 0 for all i, theswithedsystemisalledoutput stritly passive. This definition is a natural extension of passivity for nonswithed systems. Consider the ase when there exists a ommon storage funtion for the swithed system suh that equation (6) holds for all i. Inthisase,passivityforswithed systems redues to the traditional notion of passivity for non-swithed systems. 3. PRESERVING PASSIVITY UNDER QUANTIZATION 3.1 Proposed Passifiation Sheme The main problem addressed in this paper is the problem of preserving passivity with signal quantization at the system input, the system output, or both (Fig. 4). As mentioned previously, the quantizers of interest Q and Q p are passive and memoryless with Q : a u Q u Q y Q b y Q, with 0 a <b < ; Q p : a pu Q p u Qp y Qp b pyq p, with 0 a p <b p < ; (9) where u Q represents the input of the quantizer Q and y Q is the output of Q. The same holds for Q p. If the input to the quantizer is vetor, the quantization funtion ats omponent-wise on the input vetor. One an verify yq b uq and yqp b p uqp. (10) The passifiation sheme proposed in this paper is shown in Fig. 5. As mentioned, H is a disrete-time output stritly passive system suh that V (k) =V (k +1) V (k) u T (k)y (k) ρ y T (k)y (k), (11) where u,y R m,0<ρ <, V R + is the storage funtion of H. Figure 5: Proposed sheme to preserve passivity under quantization The blok M shown in Fig. 5 is an input/output oordinate transformation suh that ] ] ] um ỹ m11i = M = m m 1I m ]ỹ, (1) y m ũ m I m ũ m 1I m where m ij R, u m,y m R m and ũ, ỹ R m. An appropriate transformation will be found in order to maintain the passivity property of H C. 3. Main Results on Preserving Passivity In this setion, we apply the proposed set-up in Fig. 5 to show how passivity of the system H is preserved under quantization. Similar set-up to reover passivity of the original system over ommuniation networks under network indued delays and signal quantization have been reported in 6, 7]. The result is stated in Theorem 1. Theorem 1. Consider an OSP system H C in the proposed sheme shown in Fig. 5 with passive quantizers Q and Q p.ifatransformationm is hosen suh that m 1 =0, m 11 =b m 11m 1 = b, m 1 = b b p ρ ρ m, (13) then the subsystem H :ũ ỹ is output stritly passive suh that V (k) =V (k +1) V (k) ũ T (k)ỹ (k) ρ ỹ T (k)ỹ (k). Proof. The system H being output stritly passive implies the following V (k) =V (k +1) V (k) u T (k)y (k) ρ y T (k)y (k) = 1 ρ u(k) ρ y (k) ] T u(k) ρ y (k) ] + 1 u T (k)u (k) ρ ρ yt (k)y T (k) 1 u(k) ρ ρ y(k). (14) Sine the quantizers funtion omponent-wise on the input vetors, in view of (9), one an verify that um = m m u mi b yi = b y (15) i=1 i=1

5 where u mi is the omponent of vetor u m and y i is the omponent of vetor y.weanrewritethisas y 1 u b m. (16) Similarly, we an find u b p ym. (17) Substituting (16) and (17) into (14), gives V (k) b p ym(k) ρ ρ um(k). (18) Considering the transformation M, b { um(k) =m 11ỹ (k)+m 1ũ (k) y m(k) =m 1ỹ (k)+m ũ (k), equation (18) an be written as V (k) b p m 1ỹ (k)+m ũ (k) ρ ρ m 11ỹ (k)+m 1ũ (k) b (19) (0) Figure 6: Implementation of M in Theorem 1 Remark 1. Although Theorem 1 is derived based on disretetime OSP systems, the result remains valid for ontinuoustime OSP systems and the same transformation an be applied to preserve passivity. Remark. For the ase where only one of the quantizers is needed, one an hoose b =1when only input quantizer Q p is present or b p =1when only output quantizer Q is present. Remark 3. Sine H is an OSP system, the negative feedbak interonnetion of H with another OSP system H p, as shown in Fig. 7, is also passive and thus the stability ondition an be derived from traditional passive systems theory. The same idea is extended to swithed systems in Setion IV. thus ( b p V (k) m 1m ρ m 11m ρ b 1 )ũ T (k)ỹ (k) ( ρ m b 11 b m1) ỹ(k) ρ (1) ( ρ m b 1 b ) m ũ(k) ρ. With the parameters of M as hosen in (13), one an verify that V (k) ũ T (k)ỹ (k) ρ ỹ T (k)ỹ (k), () whih shows that H C is OSP. The implementation of the transformation M hosen in Theorem 1 is illustrated in Fig. 6. The transformation hosen is a speifi one that preserves passivity. In fat, the hoie of transformation M is not unique. One an find adifferenttransformationfrom(13),whihgivesdesigners freedom to hoose from various transformation andidates aording to different speifiations. In general, any M is allowable as long as it is invertible and satisfies the result (). Figure 7: Negative feedbak interonnetion of two OSP systems 4. STABILITY OF PASSIVE SWITCHED SYSTEMS WITH QUANTIZATION 4.1 Stability of Passive Swithed Systems Passive systems form an important lass of dynamial systems. For one, these systems are ommon in pratie. Additionally, passivity an be used to simplify analysis. Passivity is a property that implies stability and the property is preserved when systems are ombined in feedbak. Combining these two results gives open-loop onditions for losed-loop stability. Additionally, large sale systems an be shown to be stable if eah omponent is passive and the omponents are sequentially ombined in feedbak or in parallel. The following results are disrete-time extensions of the work

6 presented in 15]. They will appear in 17]. The first result onerns stability of a single passive swithed system. Theorem. Apassivedisrete-timeswithedsystemis stable for zero input (u(k) =0, k). The passivity property an be used when onsidering interonnetions of systems. The following result shows stability of the feedbak interonnetion of two passive systems. Theorem 4. Consider an output stritly passive disretetime swithed system H C (4). This system is plaed in the struture (Fig. 5) with passive quantizers defined by the onstants a, b, a p,andb p.thisontrolstruturepreservesthe output strit passivity property of system H C if the transformation M(k) is hosen aording to the following timevarying equations m 1(k) =0, m 11(k) =b (5) m 11m 1(k) = b ρ(k), m 1(k)(t) = b b p ρ (k) m (k), (6) Proof. Sine H C is OSP, for eah subsystem i there exists a V i to satisfy the passive inequality with ρ i > 0for i {1,...,P}, V i(x(k +1)) V i(x(k)) + u T (k)y (k) ρ iy (k) T y (k). (7) The quantizers satisfy the following inequalities, Figure 8: The negative feedbak interonnetion of two systems. Theorem 3. The feedbak interonnetion (Fig. 8) of two passive swithed systems G 1 and G forms a passive swithed system. As in the non-swithed ase, these results an be used to verify losed loop stability by showing that the two systems in feedbak are passive. This result an also be used from adesignperspetive. Whenontrollingapassiveswithed system, any passive ontroller is stabilizing without additional onditions. This allows for a large lass of ontrollers to be applied diretly inluding traditional PI ontrollers. 4. Passifiation of Quantized Swithed Systems The work presented in Setion 3 an be extended to swithed systems. The struture of the passifiation sheme remains the same (Fig. 5) with the system H C being modeled as a swithed system aording to the dynamis (4). Now that the system dynamis are time-varying, the transformation M must also be time-varying M(k) = ] m11(k)i m m 1(k)I m. (3) m 1(k)I m m (k)i m The matrix M(k) will be pieewise onstant,belonging to afinitesetofonstantmatries. Therewillbeatmostone onstant matrix for eah subsystem of the given swithed system. The transformation M an swith as H C swithes. In order for this to be allowable, the swithing signal of H C must be known or measurable in real time. From the perspetive of this paper, the system H C is a designed ontroller so it should be possible to measure the swithing signal. Additionally, the set of ρ i that define the OSP swithed system should be known. A funtion ρ(k) an be defined suh that ρ(k) =ρ i for ative subsystem i. (4) This funtion is pieewise onstant and hanges as the swithing signal hanges. This funtion is used to demonstrate passivity in the following theorem. u m b y and u b p y m. Applying Theorem 1, the OSP struture of eah ative subsystem is preserved at eah time step by the transformation M(k). The storage funtions V i are also preserved with the struture. Now the inative behavior an be analyzed. For eah inative subsystem i and for all ative subsystems j i, there exists a ross supply rate ω j i.foreahone,amodifiedsupply rate an be introdued suh that ω j i (ũ, ỹ,x,k)=ωj i (u,y,x,k), i, j. (8) These new ross supply rates imply and V i(x(k +1)) V i(x(k)) + ω j i (ũ, ỹ,x,k) (9) k=k 0 ω j i (ũ, ỹ,x,k) <L, (30) where L is an arbitrarily large finite onstant given by (8). Sine these hold for all i and j, theinativebehaviorisdissipative and the supply rates are still absolutely summable. All the onditions for the swithed system to be passive are satisfied. The proposed sheme maintains passivity of the swithed systems. As mentioned earlier, this hoie of transformation M(k) is not unique. The onditions listed in the theorem are suffiient to preserve passivity after the quantization effet but there is an entire lass of transformations that will also preserve passivity. This result an be used to preserve passivity of a single system. This an be used with previous results to show stability of feedbak interonnetions (Fig. 8). When this system is ombined in negative feedbak with another passive swithed system, the overall interonnetion is a passive swithed system so is stable using Theorem and 3. An example is provided in the following setion to demonstrate how this result an be used. 5. EXAMPLE The work presented in this paper is a method of maintaining passivity for disrete-time swithed systems with quantization. The following example illustrates how this method

7 an be applied to a pratial system. A linear example was hosen, however, the results are valid for nonlinear swithed systems. The swithed system H C hosen is a swithed system with two subsystems. The first subsystem of H C is modeled by the following dynamis x(k +1)= ] x(k)+ u(k) (31) ] y(k) = ] x(k)+u(k). (3) The seond subsystem of H C is x(k +1)= ] x(k)+ 1 0] u(k) (33) y(k) = ] x(k)+0.94u(k). (34) This system an be shown to be a passive swithed system using the definition given in this paper. The storage funtions to show passivity (6) are ] V 1(x) =x T (k) x(k) (35) ] V (x) =x T (k) x(k) (36) with ross supply rates ω 1 (u, y, x, k) =u T (k)y(k) (x 1 + x ) (37) ω 1(u, y, x, k) =u T (k)y(k)+ 5 x 1. (38) These rates satisfy (7-8). The system is OSP with ρ 1 =0.0 and ρ =0.95. Both input and output quantization are applied to the ontroller. The quantizers are uniform with quantization interval 0.1. It an be shown that these are passive quantizers with a =0andb =. The transformation M(k) antakeonvaluesintheset {M 1,M } where ] M 1 = (39) M = ], (40) given by (5-6). Transformation M(k) =M 1 when subsystem i =1isativeandM(k) =M when subsystem i = is ative. The swithed ontroller with quantization and transformation M(k)wassimulatedinfeedbakwithapassiveplant. The plant has the following dynamis x(k +1)= ] x(k)+ 0] u(k) (41) y(k) = ] x(k)+u(k). (4) The feedbak interonnetion of these two systems forms a passive swithed system. When simulated, both the state of the plant and the ontroller onverge to a set near the origin for arbitrary swithing. The onvergene of the plant state and output are as shown in Fig. 9 with swithing signal Fig. 10. This example demonstrates the methods introdued in this paper. The example hosen was straightforward, being a linear swithed system with two subsystems. However, x1 x y time (k) time (k) time (k) Figure 9: The first two panels show stability of the plant state x 1 and x. The third panel shows the system output y. subsystem time(k) Figure 10: The swithing signal of ontroller H that swithes between subsystems 1 and is shown. these methods apply to nonlinear swithed systems with any arbitrary finite number of subsystems. 6. CONCLUSION In this paper, we introdued a sheme to preserve the output strit passivity property of a system with passive input and output quantization by using an input-output oordinate transformation. Then we showed that the same sheme an be applied to swithed systems and thus the stability of interonneted passive swithed systems an be guaranteed from the results. The example demonstrated how these methods an be applied to a pratial quantized swithed system. ACKNOWLEDGMENTS The support of the National Siene Foundation under Grant No. CNS is gratefully aknowledged.

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