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Quantized Output Feedback Stabilization by Luenberger Observers
Proceedings of the 20th World Congress The International Federation of Congress Automatic Control Proceedings of the 20th World The International Federation of Automatic Control Toulouse, France, July 2017 The International Federation of Congress Automatic Control Proceedings of the 20th9-14, World Toulouse, France, July 9-14, 2017 Available online at www.sciencedirect.com The International of Automatic Control Toulouse, France,Federation July 9-14, 2017 Toulouse, France, July 9-14, 2017
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IFAC PapersOnLine 50-1 (2017) 2577–2582 Quantized Output Feedback Stabilization Quantized Output Feedback Stabilization Quantized Output Feedback Stabilization by Luenberger Observers Quantized Output Feedback Stabilization by Luenberger Observers by Luenberger Observers by Luenberger Observers ∗ ∗∗ ∗∗ Masashi Wakaiki ∗ , Tadanao Zanma ∗∗ , and Kang-Zhi Liu ∗∗
Masashi Wakaiki ∗∗ , Tadanao Zanma ∗∗ ∗∗ , and Kang-Zhi Liu ∗∗ Masashi Wakaiki , Tadanao Zanma , and Kang-Zhi Liu ∗∗ ∗ ∗ Masashi Wakaiki , Tadanao Zanma ∗∗ , and Kang-Zhi Liu ∗∗ ∗ Graduate school of System Informatics, Kobe University, Kobe Graduate school of System Kobe ∗ ∗ Graduate school of (email: System Informatics, Informatics, Kobe University, University, Kobe Kobe 657-0013, Japan [email protected]). ∗∗∗ 657-0013, Japan (email: [email protected]). Graduate school of System Informatics, Kobe University, Kobe Department of Electrical and Electronic Engineering, Chiba 657-0013, Japan (email: [email protected]). ∗∗ Department of Electrical and Electronic Engineering, Chiba ∗∗ 657-0013, ∗∗ Japan (email: [email protected]). University, Chiba 263-8522, Japan (email: [email protected], Department of Electrical and Electronic Engineering, Chiba University, Chiba 263-8522, (email: [email protected], ∗∗ Department of Electrical Japan and Electronic Engineering, Chiba [email protected]). University, Chiba 263-8522, Japan (email: [email protected], [email protected]). University, [email protected]). 263-8522, Japan (email: [email protected], [email protected]). Abstract: We study a stabilization problem for systems with quantized output feedback. The Abstract: We study a stabilization problem for for systems systems with with quantized quantized output output feedback. feedback. The The Abstract: Wefrom studya aLuenberger stabilization problem state estimate observer is used for control inputs and quantization centers. state estimate from a Luenberger observer is used for control inputs and quantization centers. Abstract: Wefrom study aLuenberger stabilization problem systems with quantized output feedback. The First consider the case when the is and provide data-rate conditions for state estimate observer is for used for control and quantization centers. First we we consider thea case when only only the output output is quantized quantized andinputs provide data-rate conditions for state estimate from acase Luenberger observer is used for where control inputs and quantization centers. stabilization. We next generalize the results to the case both of the plant input and output First we consider the when only the output is quantized and provide data-rate conditions for stabilization. We next generalize the the results to the case whereand both of thedata-rate plant input and output First we consider the case when only output is quantized provide conditions for are quantized and where controllers the quantized estimate of the plant output encoders stabilization. generalize the send results the case where both the plant input to and output are quantized quantizedWe andnext where controllers send theto quantized estimate of of the plant output to encoders stabilization. We next generalize the results tothe thenumerical case where both of the plant input to and output are and where controllers send the quantized estimate of the plant output encoders as quantization centers. Finally, we present comparison of the derived data-rate as quantization centers. Finally, we send present the numerical comparison of theoutput derived data-rate are quantized and where controllers theand quantized estimate of the plant to data-rate encoders conditions with those in the earlier studies a time response an inverted pendulum. as quantization centers. Finally, we present the numerical comparison of the derived conditions with those in Finally, the earlier studies and time response of an inverted pendulum. as quantization we studies presentand the aanumerical comparison of the derived data-rate conditions with centers. those in the earlier time response of an inverted pendulum. © 2017, IFAC (International Federation of Automatic Hosting by Elsevier Ltd. All rights reserved. conditions with those in the earlier studies andoutput aControl) time feedback, response of an inverted pendulum. Keywords: Quantization, model-based control, deadbeat control, networked Keywords: Quantization, model-based control, output feedback, deadbeat control, networked control systems, systems, linear systems. systems. Keywords: Quantization, model-based control, output feedback, deadbeat control, networked control linear Keywords: Quantization, model-based control, output feedback, deadbeat control, networked control systems, linear systems. control1.systems, linear systems. INTRODUCTION et al. al. (2014); (2014); Liberzon Liberzon (2003a); (2003a); Niu Niu et et al. al. (2009); (2009); Xia Xia et et al. al. 1. INTRODUCTION et 1. INTRODUCTION (2010), its state estimate was exploited only to generate et al. (2014); Liberzon (2003a); Niu et al. (2009); Xia et al. (2010), its state estimate was exploited only to Xia generate 1. INTRODUCTION (2014); Liberzon (2003a); Niu et al. (2009); et al. theal.plant plant input, and quantization center wastothe the origin. (2010), itsinput, state estimate was exploited only generate Control loops loops in in aa practical practical network network contain contain channels channels over over et the and aa quantization center was origin. Control (2010), its state estimate was exploited only to generate To reduce quantization errors, output estimates play an the plant input, and a quantization center was the origin. Control loops in a practical network contain channels over which only only aa finite finite number number of of bits bits can can be be transmitted. transmitted. Due Due To reduce quantization errors, output estimates play an which the plant input, and a quantization center was the origin. To reduce quantization errors,centers. output estimates play an important role as quantization However, relatively Control loops intransmission a practical network contain channels over to such limited capacity, we should quantize which only a finite number of bits can be transmitted. Due important role as quantization centers. However, relatively to suchonly limited transmission capacity, we should quantize quantization outputcontrol estimates play an littlereduce work has has been done errors, on quantized quantized based on the the important role as quantization centers. However, relatively which a sending finite number bits can be Due To data before them of out through network. Howto such limited transmission capacity, weaatransmitted. should quantize little work been done on control based on data before sending them out through network. Howimportant role as quantization centers. However, relatively quantization center constructed by a Luenberger observer. little work has been done on quantized control based on the to such limited transmission capacity, quantize ever, large quantization errors lead to towethe the deterioration data before sending themerrors out through a should network. How- quantization center constructed by a Luenberger observer. ever, large quantization lead deterioration work hascenter been done on quantized control based on the quantization constructed by a Luenberger observer. data before sending them outway through a network. How- little ever, large quantization errors lead to the deterioration of control performance. One to reduce quantization The notable exception is an output encoding method by of control performance. One way to reduce quantization quantization center constructed by a Luenberger observer. The notable exception is an output encoding method by ever, large quantization errors lead to theexploit deterioration errors under data-rate constraints is to output of control performance. One way to reduce quantization Sharon and Liberzon (2008, 2012), where output estimates The notable exception is an output encoding method by errors under data-rate constraints isreduce to exploit output Sharon and Liberzon (2008, 2012), where output estimates of control performance. One way In tois quantization estimates as quantization centers. this paper, weoutput adopt The errors under data-rate constraints topaper, exploit notable exception is an output encoding method by Sharon and Liberzon (2008, 2012), where output estimates are used as quantization centers and where the class of estimates as quantization centers. In this we adopt are used usedand as Liberzon quantization centers and where theestimates class of of errors under data-rateasconstraints isthis topaper, exploit output Luenberger observers output estimators due to their estimates as quantization centers. In we adopt Sharon (2008, 2012), where output are as quantization centers and where the class observers includes a Luenberger observer whose estimate Luenberger observers as output estimators due to their observers includes a Luenberger observer whose estimate estimates as observers quantization centers. In an thisencoding paper, Luenberger as output estimators duewe toadopt their are simple structure structure and aim aim to design design strategy used as quantization and where the estimate classand of is periodically periodically initialized bycenters a pseudo-inverse pseudo-inverse observer, observers includes a Luenberger observer whose simple and to an encoding strategy is initialized by a observer, and Luenberger observers as output estimators due strategy to their observers for stabilization. simple structure and aim to design an encoding includes a Luenberger observer whose estimate Sharon and Liberzon (2008, 2012) provided a sufficient is periodically initialized by a pseudo-inverse observer, and for stabilization. Sharon and Liberzon Liberzon (2008, 2012) provided provided aa sufficient sufficient simple structure and aim to design an encoding strategy is for stabilization. periodically initialized bywith a pseudo-inverse and Sharon and (2008, 2012) condition for stabilization stabilization unbounded observer, disturbances. A fundamental limitation of data rate for stabilization condition for with unbounded disturbances. for stabilization. A fundamental limitation of data rate for stabilization Sharon and Liberzon (2008, 2012) provided a sufficient condition for stabilization with unbounded disturbances. Compared with these studies, the advantages of the prowasfundamental first obtained obtained by Wong Wong and Brockett Brockett (1999), and and Compared with these studies, the advantages of the proA limitation of data rate for stabilization was first by and (1999), condition for stabilization withthe unbounded disturbances. posed encoding encoding method are twofold. First, pseudoCompared with these studies, advantages of proA fundamental limitation of data rate for were stabilization inspired by this result, data-rate limitations studied was first obtained by Wong and Brockett (1999), and posed method are twofold. First, aa the pseudoinspired byobtained this result, data-rate limitations were studied Compared with these studies, the advantages of the proinverse observer is not required. Second, the proposed posed encoding method are twofold. First, a pseudowas first by Wong and Brockett (1999), and for linear linearby time-invariant systems in in Tatikondawere andstudied Mitter inverse observer is not required. Second, the proposed inspired this result, data-rate limitations for time-invariant systems Tatikonda and Mitter encoding method are twofold. First, a proposed pseudoinverse observer is not required. Second, theboth method can be easily extended to the case of input inspired by this result,systems data-rate limitations were for linear time-invariant systems in Tatikonda andstudied Mitter posed (2004), for stochastic in Nair and Evans (2004), method observer can be easily extended to the case of both input (2004), fortime-invariant stochastic systems in Nair and Evans (2004), is not required. Second, theboth proposed and output output quantization. method can quantization. be easily extended to the case of input for systems Tatikonda and (2004), Mitter andlinear for for uncertain systems in Okano Okano and Ishii (2014). Al- inverse (2004), stochastic systems in in Nair and Evans and and for uncertain systems in and Ishii (2014). Almethod can quantization. be easily extended to the case of both input output (2004), for stochastic systems in and Nairzooming-out and Evans (2004), though the so-called zooming-in encodand for the uncertain systems in Okano Ishii (2014). Al- and In this this paper, we present present an an output output encoding encoding method method for for though so-called zooming-in and and zooming-out encodand output quantization. In paper, we and for uncertain systems in Okano and Ishii (2014). Although the so-called zooming-in and zooming-out encoding method method developed developed in in Brockett Brockett and and Liberzon Liberzon (2000); (2000); In the stabilization of sampled-data systems with discretethis paper, we present an output encoding method for ing thethis stabilization of sampled-data systems with discretethough the so-called zooming-in zooming-out encoding method developed in Brockett and Liberzon (2000); Liberzon (2003b) provides only and sufficient conditions for In paper, weobservers. present an output encoding method for timestabilization Luenberger The proposed encoding method the of sampled-data systems with discreteLiberzon (2003b) provides only sufficient conditions for time Luenberger observers. The proposed encoding method ing method developed in Brockett and Liberzon (2000); stabilization, this encoding procedure is simple and hence Liberzon (2003b) provides only sufficient conditions for the stabilization of sampled-data systems with discreteis based on the zooming-in technique and employs estitime Luenberger observers. The proposed encoding method stabilization, this encoding procedure is simple and hence is based based on on the the observers. zooming-inThe technique and employs estiLiberzon (2003b) provides only sufficient conditions for time was extended extended to various various systems, e.g.,isnonlinear nonlinear systems stabilization, this encoding procedure simple and hence proposed encoding method is zooming-in technique and employs estimatesLuenberger generated from a Luenberger Luenberger observer for both both stawas to systems, e.g., systems mates generated from a observer for stastabilization, this encoding procedure is simple and hence was extended to various systems, e.g., nonlinear systems in Liberzon Liberzon (2006); (2006); Liberzon Liberzon and and Hespanha Hespanha (2005), (2005), syssys- mates is based on the zooming-in technique and employs estibilization and quantization. First we consider only output generated from a Luenberger observer for both stain bilization and quantization. First weobserver considerfor only output was extended to various systems, e.g., nonlinear systems in Liberzon (2006); Liberzon and in Hespanha (2005), systems with external external disturbances Liberzon and Neˇ i´cc mates generated from a Luenberger both staquantization and assume that encoders also contain contain estibilization andand quantization. First we consider only output tems with disturbances in Liberzon and Neˇ ssi´ quantization assume that encoders also estiin Liberzon (2006); Liberzon and Hespanha (2005), sys(2007); Sharon and (2008, 2012), and recently tems with external disturbances in Liberzon and Neˇ s i´ c bilization and quantization. First we consider only output mators. Simple sufficient conditions for stabilization are quantization and assume that encoders also contain esti(2007); Sharon and Liberzon (2008, 2012), and recently mators. Simple Simple sufficient conditions for also stabilization are tems with external disturbances in (2014); Liberzon and Neˇ si´c quantization switched/hybrid systems in Liberzon Liberzon Wakaiki and (2007); Sharon and Liberzon (2008, 2012), and recently andboth assume that encoders contain estimators. sufficient conditions for stabilization are obtained in the case of general Luenberger observers switched/hybrid systems in (2014); Wakaiki and obtainedSimple in the both case of general Luenberger observers (2007); Sharon and Liberzon (2008, (2014); 2012), recently switched/hybrid systems in Liberzon Wakaiki and Yamamoto (2016); Yang and Liberzon (2015).and Readers are mators. sufficient conditions for stabilization are and of of deadbeat deadbeat observers. obtained in the both case of general Luenberger observers Yamamoto (2016); Yang and Liberzon (2015). Readers are and observers. switched/hybrid systems in Liberzon (2014); Wakaiki and referred to the survey papers by Ishii and Tsumura (2012); Yamamoto (2016); Yang and Liberzon (2015). Readers are obtained in the both case of general Luenberger observers and of deadbeat observers. referred to the survey papers by Ishii and Tsumura (2012); Second we generalize generalize the results results of of general general Luenberger Luenberger Yamamoto (2016); Yang and Liberzon (2015). Readers are and Nair et et to al.the (2007), and the by books by Liberzon (2003c); referred survey papers Ishiiby and Tsumura (2012); of deadbeat observers. Second we the Nair al. (2007), and the books Liberzon (2003c); observersweto togeneralize the situation situation where both of the the plant plant input Second the where resultsboth of general Luenberger referred to the survey papers by Ishii and Tsumura (2012); Matveev and Savkin (2009) on this topic for further inforNair et al. (2007), and the books by Liberzon (2003c); observers the of input Matveev and(2007), Savkinand (2009) onbooks this topic topic for further further infor- Second we generalize the results of general Luenberger and output are quantized. Moreover, in the second case, observers to the situation where both of the plant input Nair et al. the by Liberzon (2003c); Matveev and Savkin (2009) on this for information. and output output are quantized. Moreover, in second case, mation. to are thequantized. situation where both the plant input and Moreover, inof the the second case, encoders do not estimate the plant state by themselves, Matveev mation. and Savkin (2009) on this topic for further infor- observers encoders do not estimate the plant state by themselves, Although a Luenberger observer has been widely used for and output are quantized. Moreover, in the second case, encoders do not estimate the plant state by themselves, but controllers send the quantized estimate of the plant mation. Although aa Luenberger Luenberger observer has widely used but controllers send the quantized estimate ofthemselves, the plant Although observer has been beene.g., widely used for for encoders quantized output output feedback feedback stabilization, in Ferrante Ferrante do notsend estimate the plantestimate state by of but controllers the quantized the plant quantized stabilization, e.g., in Although aoutput Luenberger observer has beene.g., widely used for but controllers send the quantized estimate of the plant quantized feedback stabilization, in Ferrante quantized output feedback stabilization, e.g., in Ferrante2632 Copyright © 2017 IFAC Copyright © 2017, 2017 IFAC 2632Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 2632 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 2632Control. 10.1016/j.ifacol.2017.08.101
Proceedings of the 20th IFAC World Congress 2578 Masashi Wakaiki et al. / IFAC PapersOnLine 50-1 (2017) 2577–2582 Toulouse, France, July 9-14, 2017
output to the encoders. In contrast with the quantization of the plant output, we quantize the plant input and the output estimate by using the origin as the quantization center, which reduces computational resources in the components of the plant side. We see that if the closed-loop system without quantization is stable, then there exists an encoding method such that the closed-loop system in the presence of three types of quantization errors is also stable. This paper is organized as follows. In the next section, we study the case when only the plant output is quantized and obtain two data-rate conditions for general Luenberger observers and deadbeat observers. In Section 3, the proposed encoding method is extended to the case when both of the plant input and output are quantized. Section 4 is devoted to the numerical comparison of the obtained data-rate conditions with those of the earlier studies by Liberzon (2003b); Sharon and Liberzon (2008, 2012) and the time response of an inverted pendulum. We provide concluding remarks in Section 5. Due to space constraints, all proofs have been omitted and can be found in Wakaiki et al. (2017). Notation and Definitions: The symbol Z+ denotes the set of nonnegative integers. Let λmin (P ) and r(P ) denote the smallest eigenvalue and the spectral radius of P ∈ Rn×n , respectively. We denote by In by the identity matrix of size n. For a vector v = [v1 · · · vn ]� ∈ Rn , we denote its maximum norm by |v| = max{|v1 |, . . . , |vn |} and the corresponding induced norm of a matrix M ∈ Rm×n by �M � = sup{|M v| : v ∈ Rn , |v| = 1}. Plant:
We consider a continuous-time linear system � x(t) ˙ = Ax(t) + Bu(t) ΣP : (1) y(t) = Cx(t), where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input, and y(t) ∈ Rp is the output. This plant is connected with a controller through a time-driven encoder and zero-order hold (ZOH) with period h > 0. Define xk := x(kh) and yk := y(kh) for every k ∈ Z+ , and also set � h Bd := eAs Bds. (2) Ad := eAh , 0
Throughout this paper, we place the following assumptions: Assumption 1. (Initial state bound). A constant Est > 0 satisfying |x(0)| ≤ Est is known. Assumption 2. (Stabilizability and detectability). The discretized system (Ad , Bd , C) is stabilizable and detectable. Remark 3. We can obtain an initial state bound Est by the “zooming-out” procedure in Liberzon (2003b). 2. OUTPUT QUANTIZATION In this section, we consider the scenario where only the output yk is quantized and where the encoder has computational resources to estimate the plant state. 2.1 Controller Let qk ∈ Rp be the quantized value of the sampled output yk and K ∈ Rn×m , L ∈ Rn×p be a feedback gain and an
ZOH
u(t)
Plant
y(t)
Encoder Network
uk
Decoder Controller
qk
Fig. 1. Closed-loop system with output quantization. observer gain, respectively. For the plant ΣP in (1), we use a discrete-time Luenberger observer for feedback control and output quantization: ⎧ ˆk+1 = Ad x ˆk + Bd uk + L(qk − yˆk ) ⎨x = C x ˆ y ˆ ΣC : (3) k k ⎩ uk = −K x ˆk
where x ˆk ∈ Rn is the state estimate and yˆk ∈ Rp is the output estimate. We set the initial estimate xˆ0 to be x ˆ0 = 0. Each of the encoder and the controller contains the above Luenberger observer, and those observers are assumed to be synchronized. Through the zero-order hold, the control input u(t) is generated as (kh ≤ t < (k + 1)h). u(t) = uk Fig. 1 shows the closed-loop system with quantized output.
2.2 Output Encoding Suppose that we obtain an error bound Ek such that |yk − yˆk | ≤ Ek . The next subsection is devoted to the computation of a bound sequence {Ek } for stabilization.
For each k ∈ Z+ , we divide the hypercube (4) {y ∈ Rp : |y − yˆk | ≤ Ek } p into N equal boxes and assign a number in {1, . . . , N p } to each divided box by a certain one-to-one mapping. Since x ˆ0 = 0, we see from Assumption 1 that the error ˆk satisfies |e0 | = |x(0)| ≤ Est . Thus we can set ek := xk − x E0 := �C�Est .
The encoder sends to the controller the number q¯k of the divided box containing yk , and then the controller generates qk equal to the center of the box with number q¯k . If yk lies on the boundary on several boxes, then we can choose any one of them. This encoding strategy leads to Ek (5) =: μk . |yk − qk | ≤ N 2.3 Computation of Bound Sequence {Ek } Here we obtain bound sequences {Ek } and data-rate conditions for stabilization. We first consider general Luenberger observers and next focus on deadbeat observers.
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Use of General Luenberger Observers: The proposed encoding strategy with the following bound sequence {Ek } achieves the exponential convergence of the state under a certain data-rate condition. Theorem 4. Let Assumptions 1 and 2 hold. Define the ¯ by matrices R and R ¯ := Ad − Bd K. R := Ad − LC, R (6) Let the observer gain L and the feedback gain K satisfy ¯ < 1, and r(R) < 1, r(R) � � � � �CR� � ≤ M0 ρ� , �CR� L� ≤ M ρ� . (7)
for some M0 , M > 0 and ρ < 1. If we pick N ≥ 2 so that M < N, (8) 1−ρ then the proposed encoding method with a bound sequence {Ek } defined by ⎧ M ⎪ ⎨M0 Est ρ + E0 k = 0 N � � (9) Ek+1 := M ⎪ ⎩ ρ+ Ek k ≥ 1. N achieves the exponential convergence of the state x and the estimate x ˆ.
Use of Deadbeat Observers: In the rest of this section, we focus on deadbeat observers. If the pair (C, Ad ) is observable, then there exists a matrix L ∈ Rn×p such that Rη = (Ad − LC)η = 0, (10) where η is the observability index of (C, Ad ). Construction methods of such an observer gain L have been developed for deadbeat control; see, e.g., Chapter. 5 of O’Reilly (1983). Using the property (10), we obtain an alternative error bound sequence {Ek } for stabilization. Theorem 5. Let Assumptions 1 and 2 hold. Assume that ¯ as in (C, Ad ) is observable. Define the matrices R and R (6), and let the observer gain L and the feedback gain K ¯ < 1, where η is the observability satisfy Rη = 0 and r(R) index of (C, Ad ). Set a constant α� to be � � �CR� L� α� := (11) N for � = 0, . . . , η − 1. If we pick N ≥ 2 so that a matrix F defined by ⎡ ⎤ α0 α1 . . . αη−1 0 ⎥ ⎢1 ⎥ F := ⎢ (12) .. ⎣ ⎦ . 0 1 0 satisfies r(F ) < 1, (13) then the proposed encoding method with a bound sequence {Ek } defined by ⎧ � k � �CR� L� � � � ⎪ ⎪ k+1 � � Est + ⎪ Ek−� 0 ≤ k ≤ η−2 ⎪ ⎨ CR N �=0 Ek+1 := η−1 � � � �CR� L� ⎪ ⎪ ⎪ ⎪ Ek−� k ≥ η−1 ⎩ N �=0 (14) achieves the exponential convergence of the state x and the estimate x ˆ. Remark 6. As (9), (14) also has the form of linear timeinvariant recursion.
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Remark 7. Sharon and Liberzon (2008, 2012) proposed the quantizer based on a psuedo-inverse observer. That quantizer achieves closed-loop stability if N ≥ 2 satisfies � � �CAd C† � < N, (15) where
⎡
C CA d ⎢ C := ⎢ ⎣ ...
CAη−1 d
⎤
⎥ −η+1 ⎥A , ⎦ d
� �−1 � C† := C� C C ,
and the total data size is N p . In the state feedback case of Liberzon (2003b), the counterpart of (15) is �Ad � < N,
(16)
n
and the total data size is N .
3. INPUT AND OUTPUT QUANTIZATION In this section, we quantize both of the plant input and output. Moreover, we assume in the previous section that the encoder has computational resources to estimate the plant state, while we here study the scenario where the controller sends the quantized output estimate to the encoder. Hence the encoder does not need to compute or store the estimate. 3.1 Controller Let K ∈ Rn×m , L ∈ Rn×p be a feedback gain and an observer gain, respectively. Let qk ∈ Rp denote the quantized value of the sampled output yk . We denote by Q1 and Q2 qunatization functions of the output estimate yˆ and the control input u, respectively. For the plant ΣP in (1), we construct the following observer-based controller: ⎧ ˆk + Bd uk + L(qk − Q1 (ˆ yk )) ˆk+1 = Ad x ⎨x yˆk = C x ˆk (17) Σ�C : ⎩ uk = −K x ˆk where Ad and Bd are defined as in (2). We set the initial estimate x ˆ0 to be x ˆ0 = 0. Compared with the controller ΣC in (3), the controller Σ�C uses the quantized output yk ) instead of the original output estimate estimate Q1 (ˆ yˆk . The control input u(t) is produced as u(t) = Q2 (uk )
(kh ≤ t < (k + 1)h).
Note that the controller can compute x ˆk and hence yˆk , uk at time k − 1. Fig. 2 illustrates the closed-loop system we consider in this section. Remark 8. Instead of (17), we can use different controllers such as ⎧ ˆk + Bd Q2 (uk ) + L(qk − Q1 (ˆ yk )) ˆk+1 = Ad x ⎨x yˆk = C x ˆk Σ��C : ⎩ uk = −K x ˆk .
The major reason to use (17) is that if x ˆ0 = 0, then we have k−1 � x ˆk = (Ad − Bd K)k−� (q� − Q1 (y� )) (18) �=0
and hence yˆk and uk can be described by q� −Q1 (y� ), which makes encoding methods simple.
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Q2 (uk )
ZOH
Decoder
u(t)
Q1 (ˆ yk )
center, whereas the quantization centers for the output estimate yˆk and the input uk are the origin, which allows the plant side to have less computational resources. Remark 9. Although the encoder does not have to estimate output measurements, the bounds Ek , E1,k , and E2,k should be computed in the plant side. However, these bounds can be calculated by simple difference equations (21) as shown in the next subsection. If components in the plant side do not have computational resources enough to implement those difference equations, the controller can send sufficiently accurate values of the bounds even in the presence of quantization, because the dimension of each bound is one.
y(t)
Plant
Encoder
Network Decoder
Encoder yˆk uk
Controller
qk
Fig. 2. Closed-loop system with input and output quantization.
3.4 Computation of Bound Sequence {(Ek , E1,k , E2,k )}
3.2 Output Encoding
The following theorem is an extension of Theorem 4. Theorem 10. Let Assumptions 1 and 2 hold, and define R ¯ as in (6). Let the observer gain L and the feedback and R ¯ < 1 and gain K satisfy r(R) < 1, r(R) ¯ � L� ≤ M1 ρ¯� , �K R ¯ � L� ≤ M2 ρ¯� �CR� � ≤ M0 ρ� , �C R
Suppose that we obtain an error bound Ek such that yk )| ≤ Ek . A bound sequence {Ek } satisfying |yk − Q1 (ˆ this condition is obtained in Section 3.4. Instead of (4), the encoder computes quantized measurements by dividing the hypercube yk )| ≤ Ek } (19) {y ∈ Rp : |y − Q1 (ˆ p into N equal boxes. The difference between (4) and (19) is the quantization center. In (4), the encoder has the state estimate x ˆk , and hence the quantization center can be yˆk . On the other hand, the encoder here employs the quantized yk ) reported by the controller as the output estimate Q1 (ˆ quantization center. The rest of the output encoding is the same as in Subsection 3.2. 3.3 Estimate and Input Encoding The controller sends to the plant the quantization data of the control input uk and the output estimate yˆk . Suppose that we have bounds E1,k and E2,k such that |ˆ yk | ≤ E1,k and |uk | ≤ E2,k . Such a bound sequence {(E1,k , E2,k )} is obtained in Section 3.4. The bounds and levels of the quantization of yˆ, u are given by (E1,k , N1 ) and (E2,k , N2 ). Namely, the controller computes the quantized output estimate and the quantized input by dividing the hypercubes y | ≤ E1,k } {ˆ y ∈ Rp : |ˆ {u ∈ Rm : |u| ≤ E2,k }
into N1p and N2m equal boxes and assigns a number in {1, . . . , N1p } and {1, . . . , N2m } to each divided box by a certain one-to-one mapping, respectively. The decoder in the plant side generates Q1 (ˆ yk ) and Q2 (uk ) equal to the center of the boxes with number reported by the controller. yk ) and Q2 (uk ) satisfy Thus Q1 (ˆ E1,k |ˆ yk − Q1 (ˆ yk )| ≤ N1 E2,k |uk − Q2 (uk )| ≤ . N2 Since x ˆ0 = 0, we can set the initial values E1,0 := 0 E2,0 := 0. The encoding strategy (19) of the output yk uses the yk ) as the quantization quantized output estimate Q1 (ˆ
�CR� Bd � ≤ M3 ρ� , �CR� L� ≤ M4 ρ� hold for some M0 , M1 , M2 , M3 , M4 > 0 and ρ¯ ≤ ρ < 1. Define constants α0 and α1 by � � M1 M4 M 2 M3 M := (N − 1) · + N1 N2 N1 M4 + (N − 1)M1 β0 := ρ + N N1 M β1 := N α0 := ρ + β0 α1 := β1 − ρβ0 . If we pick N, N1 , N2 ≥ 2 so that � � α0 α 1 F := (20) 1 0 satisfies r(F ) < 1, then the proposed encoding with a bound sequence {(Ek , E1,k , E2,k )} defined � � ⎧ (N − 1)M1 E0 ⎪ ⎪ M E ρ + M + 4 ⎨ 0 st N1 N Ek+1 := β E + β E ⎪ 0 k 1 k−1 ⎪ ⎩ α0 Ek + α1 Ek−1 (N − 1)M1 Ek E1,k+1 := ρ¯E1,k + N (N − 1)M2 Ek E2,k+1 := ρ¯E2,k + N
method by k=0 k=1 k≥2 k≥0 k ≥ 0.
(21) achieves the exponential convergence of the state x and the estimate x ˆ. Remark 11. Although we here generate Q1 (ˆ yk ) and Q2 (uk ) separately, one can quantize yˆk and uk simultaneously. This simultaneous quantization reduces the computational cost of the controller, but the data-rate condition for stabilization becomes conservative. Therefore, we do not proceed along this line. Remark 12. There always exist quantization levels N , N1 , and N2 such that the matrix F in (20) satisfies r(F ) < 1. In fact, as N , N1 , and N2 increase to infinity, we have
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4.1 Comparison of Data-Rate Conditions First we consider the quantization of only the plant output and compare the data-rate conditions of three types of observers: the steady-state Kalman filter (7) with process noise covariance 10−3 In and measurement noise covariance 10−1 Ip , the deadbeat observer (13), and the pseudo-inverse observer (15). In addition to the output feedback case, we also investigate the state encoding case (16). In Table 1, we show the comparison of the minimum quantization level for exponential convergence, which are N p in the output feedback case and N n in the state feedback case. Note that steady-state Kalman filter and the deadbeat observers are represented by a linear time-invariant state equation but pseudo-inverse observers are not. The first example is an inverted pendulum whose dynamics is given by (1) with ⎡ ⎤ ⎡ ⎤ 0 1 0 0 0 ⎢0 −20.06 53.26 −1.096⎥ ⎢35.28⎥ A := ⎣ , B := ⎣ (22) 0 0 0 1 ⎦ 0 ⎦ 0 −20.01 98.41 −2.025 35.18 C := [1 0 1 0] . The state [x1 x2 x3 x4 ] =: x are the arm angle, the arm angular velocity, the pendulum angle, and the pendulum angular velocity. The input u is the motor voltage. Additionally, we borrow a 2-mass motor drive with one output and three states from Ji and Sul (1995), a pneumatic cylinder with one output and three states from Kimura et al. (1996), and a batch reactor with two output and four states from Rosenbrock (1974). In Table 1, we see that the Kalman filter requires less datarate than the deadbeat observer and the pseudo-inverse observer for the pneumatic cylinder. This is because the cylinder have their unstable poles only on the imaginary axis. Hence the observer gain of the Kalman filter is small, which decreases M in (7). Although deadbeat observers and pseudo-inverse observers have the same property: finite-time state reconstruction in the idealized situation without quantization, the data-rate condition (15) by pseudo-inverse observers is better than that (13) by deadbeat observers. This is because pseudo-inverse observers employ output measurements directly for state reconstruction, whereas deadbeat observers summarize output information by their states. Moreover, compared with the state feedback case (16), the output feedback case requires small data sizes in most numerical examples because the state dimension n and the output dimension p satisfy n > p.
From Theorem 10, the state x exponentially converges to the origin under the proposed encoding strategy with (N, N1 , N2 ) = (151, 301, 1101) for which F in (20) satisfies r(F ) = 0.9906. Supposing that we obtain an initial state bound E0 = 0.15, we compute a time response for the initial state x(0) = [0 0 0.1 0]� . The plot of the arm angle x1 and the pendulum angle x3 is in Figs. 3 and 4. Fig. 5 illustrates the motor voltage u. From Figs. 3 and 4, we observe that the arm and pendulum angles decrease to zero in the presence of three types of quantization errors. Since the initial estimate x ˆ0 = 0, the quantization errors of the output estimate and the input are small at first. In fact, the output estimate bound {E1,k } and the input bound {E2,k } take the maximum value at about time t = 0.6, and the effect of the quantization errors appears from time t = 0.5 in Figs. 3, 4, and 5. 5. CONCLUSION We studied quantized output feedback stabilization by Luenberger observers. Data-rate conditions for general Luenberger observers are characterized by the spectral radius of the system matrix of the error dynamics. On the 0.1
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Arm angle x1 [rad]
4. NUMERICAL EXAMPLES
sampling period be h = 0.03 sec. We set the feedback gain K to be the quadratic regulator whose weighting matrices of the state and the input are diag(100, 0, 300, 0) and 1, respectively. The observer gain L is the Kalman filter whose covariances of the process noise and measurement noise are 10−3 Ip and 10−1 Ip , respectively.
0
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Fig. 3. Arm angle x1 with/without quantization. 0.15 0.15
4.2 Time response of Inverted Pendulum We consider again the inverted pendulum (22) in the previous subsection and compute its time response under the quantization of the plant input and output. The controller is assumed to send to the encoder the quantized value of the output estimate for quantization centers. Let the
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α0 → ρ and α1 → ρ2 , and hence the eigenvalues of F tend to ρ < 1.
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Fig. 4. Pendulum angle x3 with/without quantization.
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Table 1. Minimum quantization levels for exponential convergence. Kalman filter (7)
Deadbeat observer (13)
Pseudo-inverse observer (15)
State encoding (16)
62 = 36 51 = 5 21 = 2 102 = 100
42 = 16 71 = 7 71 = 7 42 = 16
42 = 16 31 = 3 31 = 3 42 = 16
44 = 256 33 = 27 23 = 8 34 = 81
Inverted pendulum (h = 0.03) 2-mass motor drive (h = 10−3 ) Pneumatic cylinder (h = 10−3 ) Batch reactor (h = 0.1)
other hand, a data-rate condition for deadbeat observers is determined by the behavior of the error dynamics for η steps, where η is the observability index of the plant. The proposed encoding method was also extended to the case where both of the plant input and output are quantized and where the encoder does not have an estimator for generating quantization centers. Future work involves addressing more general systems such as nonlinear systems and switched systems. REFERENCES Brockett, R.W. and Liberzon, D. (2000). Quantized feedback stabilization of linear systems. IEEE Trans. Automat. Control, 45, 1279–1289. Ferrante, F., Gouaisbaut, F., and Tarbouriech, S. (2014). Observer-based control for linear systems with quantized output. In Proc. ECC’14. Ishii, H. and Tsumura, K. (2012). Data rate limitations in feedback control over network. IEICE Trans. Fundamentals, E95-A, 680–690. Ji, J.K. and Sul, S.K. (1995). Kalman filter and LQ based speed controller for torsional vibration suppression in a 2-mass motor drive system. IEEE Trans. Ind. Electron., 42, 564–571. Kimura, T., Fujioka, H., Tokai, K., and Takamori, T. (1996). Sampled-data H∞ control for a Pneumatic cylinder system. In Proc. IEEE 35th CDC. Liberzon, D. (2003a). Hybrid feedback stabilization of systems with quantized signals. Automatica, 39, 1543– 1554. Liberzon, D. (2003b). On stabilization of linear systems with limited information. IEEE Trans. Automat. Control, 48, 304–307. Liberzon, D. (2003c). Switching in Systems and Control. Boston: Birkh¨ auser. Liberzon, D. (2006). Quantization, time delays, and nonlinear stabilization. IEEE Trans. Automat. Control, 51, 1190–1195.
Motor voltage u [V]
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Fig. 5. Motor voltage u with/without quantization.
Liberzon, D. (2014). Finite data-rate feedback stabilization of switched and hybrid linear systems. Automatica, 50, 409–420. Liberzon, D. and Hespanha, J.P. (2005). Stabilization of nonlinear systems with limited information feedback. IEEE Trans. Automat. Control, 50, 910–915. Liberzon, D. and Neˇsi´c, D. (2007). Input-to-state stabilization of linear systems with quantized state measurement. IEEE Trans. Automat. Control, 52, 767–781. Matveev, A.S. and Savkin, A.V. (2009). Estimation and Control over Communication Networks. Birkh¨ auser, Boston. Nair, G.N. and Evans, R.J. (2004). Stabilizability of stochastic linear systems with finite feedback data rates. SIAM J. Control Optim., 43, 413–436. Nair, G.N., Fagnani, F., Zampieri, S., and Evans, R.J. (2007). Feedback control under data rate constraints: An overview. Proc. IEEE, 95, 108–137. Niu, Y., Jia, T., Wang, X., and Yang, F. (2009). Outputfeedback control design for NCSs subject to quantization and dropout. Inf. Sci., 179, 3804–3813. Okano, K. and Ishii, H. (2014). Stabilization of uncertain systems with finite data rates and Markovian packet losses. IEEE Trans. Control Network Systems, 1, 298– 307. O’Reilly, J. (1983). Observer for Linear Systems. New York: Academic. Rosenbrock, H.H. (1974). Computer-Aided Control System Design. New York: Academic Press. Sharon, Y. and Liberzon, D. (2008). Input-to-state stabilization with quantized output feedback. In Proc. HSCC’08. Sharon, Y. and Liberzon, D. (2012). Input to state stabilizing controller for systems with coarse quantization. IEEE Trans. Automat. Control, 57, 830–844. Tatikonda, S. and Mitter, S. (2004). Control under communication constraints. IEEE Trans. Automat. Control, 49, 1056–1068. Wakaiki, M. and Yamamoto, Y. (2016). Stabilization of switched linear systems with quantized output and switching delays. To appear in IEEE Trans. Automat. Control. Wakaiki, M., Zanma, T., and Liu, K.Z. (2017). Quantized output feedback stabilization by Luenberger observers. arXiv:1703.06567. Wong, W.S. and Brockett, R.W. (1999). Systems with finite communication bandwidth constraints II: Stabilization with limited information feedback. IEEE Trans. Automat. Control, 44, 1049–1053. Xia, Y., Yan, J., Shang, J., Fu, M., and Liu, B. (2010). Stabilization of quantized systems based on Kalman filter. Control Eng. Pract., 20, 954–962. Yang, G. and Liberzon, D. (2015). Stabilizing a switched linear system with disturbance by sampled-data quantized feedback. In Proc. ACC’15.
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IFAC PapersOnLine 50-1 (2017) 2577–2582 Quantized Output Feedback Stabilization Quantized Output Feedback Stabilization Quantized Output Feedback Stabilization by Luenberger Observers Quantized Output Feedback Stabilization by Luenberger Observers by Luenberger Observers by Luenberger Observers ∗ ∗∗ ∗∗ Masashi Wakaiki ∗ , Tadanao Zanma ∗∗ , and Kang-Zhi Liu ∗∗
Masashi Wakaiki ∗∗ , Tadanao Zanma ∗∗ ∗∗ , and Kang-Zhi Liu ∗∗ Masashi Wakaiki , Tadanao Zanma , and Kang-Zhi Liu ∗∗ ∗ ∗ Masashi Wakaiki , Tadanao Zanma ∗∗ , and Kang-Zhi Liu ∗∗ ∗ Graduate school of System Informatics, Kobe University, Kobe Graduate school of System Kobe ∗ ∗ Graduate school of (email: System Informatics, Informatics, Kobe University, University, Kobe Kobe 657-0013, Japan [email protected]). ∗∗∗ 657-0013, Japan (email: [email protected]). Graduate school of System Informatics, Kobe University, Kobe Department of Electrical and Electronic Engineering, Chiba 657-0013, Japan (email: [email protected]). ∗∗ Department of Electrical and Electronic Engineering, Chiba ∗∗ 657-0013, ∗∗ Japan (email: [email protected]). University, Chiba 263-8522, Japan (email: [email protected], Department of Electrical and Electronic Engineering, Chiba University, Chiba 263-8522, (email: [email protected], ∗∗ Department of Electrical Japan and Electronic Engineering, Chiba [email protected]). University, Chiba 263-8522, Japan (email: [email protected], [email protected]). University, [email protected]). 263-8522, Japan (email: [email protected], [email protected]). Abstract: We study a stabilization problem for systems with quantized output feedback. The Abstract: We study a stabilization problem for for systems systems with with quantized quantized output output feedback. feedback. The The Abstract: Wefrom studya aLuenberger stabilization problem state estimate observer is used for control inputs and quantization centers. state estimate from a Luenberger observer is used for control inputs and quantization centers. Abstract: Wefrom study aLuenberger stabilization problem systems with quantized output feedback. The First consider the case when the is and provide data-rate conditions for state estimate observer is for used for control and quantization centers. First we we consider thea case when only only the output output is quantized quantized andinputs provide data-rate conditions for state estimate from acase Luenberger observer is used for where control inputs and quantization centers. stabilization. We next generalize the results to the case both of the plant input and output First we consider the when only the output is quantized and provide data-rate conditions for stabilization. We next generalize the the results to the case whereand both of thedata-rate plant input and output First we consider the case when only output is quantized provide conditions for are quantized and where controllers the quantized estimate of the plant output encoders stabilization. generalize the send results the case where both the plant input to and output are quantized quantizedWe andnext where controllers send theto quantized estimate of of the plant output to encoders stabilization. We next generalize the results tothe thenumerical case where both of the plant input to and output are and where controllers send the quantized estimate of the plant output encoders as quantization centers. Finally, we present comparison of the derived data-rate as quantization centers. Finally, we send present the numerical comparison of theoutput derived data-rate are quantized and where controllers theand quantized estimate of the plant to data-rate encoders conditions with those in the earlier studies a time response an inverted pendulum. as quantization centers. Finally, we present the numerical comparison of the derived conditions with those in Finally, the earlier studies and time response of an inverted pendulum. as quantization we studies presentand the aanumerical comparison of the derived data-rate conditions with centers. those in the earlier time response of an inverted pendulum. © 2017, IFAC (International Federation of Automatic Hosting by Elsevier Ltd. All rights reserved. conditions with those in the earlier studies andoutput aControl) time feedback, response of an inverted pendulum. Keywords: Quantization, model-based control, deadbeat control, networked Keywords: Quantization, model-based control, output feedback, deadbeat control, networked control systems, systems, linear systems. systems. Keywords: Quantization, model-based control, output feedback, deadbeat control, networked control linear Keywords: Quantization, model-based control, output feedback, deadbeat control, networked control systems, linear systems. control1.systems, linear systems. INTRODUCTION et al. al. (2014); (2014); Liberzon Liberzon (2003a); (2003a); Niu Niu et et al. al. (2009); (2009); Xia Xia et et al. al. 1. INTRODUCTION et 1. INTRODUCTION (2010), its state estimate was exploited only to generate et al. (2014); Liberzon (2003a); Niu et al. (2009); Xia et al. (2010), its state estimate was exploited only to Xia generate 1. INTRODUCTION (2014); Liberzon (2003a); Niu et al. (2009); et al. theal.plant plant input, and quantization center wastothe the origin. (2010), itsinput, state estimate was exploited only generate Control loops loops in in aa practical practical network network contain contain channels channels over over et the and aa quantization center was origin. Control (2010), its state estimate was exploited only to generate To reduce quantization errors, output estimates play an the plant input, and a quantization center was the origin. Control loops in a practical network contain channels over which only only aa finite finite number number of of bits bits can can be be transmitted. transmitted. Due Due To reduce quantization errors, output estimates play an which the plant input, and a quantization center was the origin. To reduce quantization errors,centers. output estimates play an important role as quantization However, relatively Control loops intransmission a practical network contain channels over to such limited capacity, we should quantize which only a finite number of bits can be transmitted. Due important role as quantization centers. However, relatively to suchonly limited transmission capacity, we should quantize quantization outputcontrol estimates play an littlereduce work has has been done errors, on quantized quantized based on the the important role as quantization centers. However, relatively which a sending finite number bits can be Due To data before them of out through network. Howto such limited transmission capacity, weaatransmitted. should quantize little work been done on control based on data before sending them out through network. Howimportant role as quantization centers. However, relatively quantization center constructed by a Luenberger observer. little work has been done on quantized control based on the to such limited transmission capacity, quantize ever, large quantization errors lead to towethe the deterioration data before sending themerrors out through a should network. How- quantization center constructed by a Luenberger observer. ever, large quantization lead deterioration work hascenter been done on quantized control based on the quantization constructed by a Luenberger observer. data before sending them outway through a network. How- little ever, large quantization errors lead to the deterioration of control performance. One to reduce quantization The notable exception is an output encoding method by of control performance. One way to reduce quantization quantization center constructed by a Luenberger observer. The notable exception is an output encoding method by ever, large quantization errors lead to theexploit deterioration errors under data-rate constraints is to output of control performance. One way to reduce quantization Sharon and Liberzon (2008, 2012), where output estimates The notable exception is an output encoding method by errors under data-rate constraints isreduce to exploit output Sharon and Liberzon (2008, 2012), where output estimates of control performance. One way In tois quantization estimates as quantization centers. this paper, weoutput adopt The errors under data-rate constraints topaper, exploit notable exception is an output encoding method by Sharon and Liberzon (2008, 2012), where output estimates are used as quantization centers and where the class of estimates as quantization centers. In this we adopt are used usedand as Liberzon quantization centers and where theestimates class of of errors under data-rateasconstraints isthis topaper, exploit output Luenberger observers output estimators due to their estimates as quantization centers. In we adopt Sharon (2008, 2012), where output are as quantization centers and where the class observers includes a Luenberger observer whose estimate Luenberger observers as output estimators due to their observers includes a Luenberger observer whose estimate estimates as observers quantization centers. In an thisencoding paper, Luenberger as output estimators duewe toadopt their are simple structure structure and aim aim to design design strategy used as quantization and where the estimate classand of is periodically periodically initialized bycenters a pseudo-inverse pseudo-inverse observer, observers includes a Luenberger observer whose simple and to an encoding strategy is initialized by a observer, and Luenberger observers as output estimators due strategy to their observers for stabilization. simple structure and aim to design an encoding includes a Luenberger observer whose estimate Sharon and Liberzon (2008, 2012) provided a sufficient is periodically initialized by a pseudo-inverse observer, and for stabilization. Sharon and Liberzon Liberzon (2008, 2012) provided provided aa sufficient sufficient simple structure and aim to design an encoding strategy is for stabilization. periodically initialized bywith a pseudo-inverse and Sharon and (2008, 2012) condition for stabilization stabilization unbounded observer, disturbances. A fundamental limitation of data rate for stabilization condition for with unbounded disturbances. for stabilization. A fundamental limitation of data rate for stabilization Sharon and Liberzon (2008, 2012) provided a sufficient condition for stabilization with unbounded disturbances. Compared with these studies, the advantages of the prowasfundamental first obtained obtained by Wong Wong and Brockett Brockett (1999), and and Compared with these studies, the advantages of the proA limitation of data rate for stabilization was first by and (1999), condition for stabilization withthe unbounded disturbances. posed encoding encoding method are twofold. First, pseudoCompared with these studies, advantages of proA fundamental limitation of data rate for were stabilization inspired by this result, data-rate limitations studied was first obtained by Wong and Brockett (1999), and posed method are twofold. First, aa the pseudoinspired byobtained this result, data-rate limitations were studied Compared with these studies, the advantages of the proinverse observer is not required. Second, the proposed posed encoding method are twofold. First, a pseudowas first by Wong and Brockett (1999), and for linear linearby time-invariant systems in in Tatikondawere andstudied Mitter inverse observer is not required. Second, the proposed inspired this result, data-rate limitations for time-invariant systems Tatikonda and Mitter encoding method are twofold. First, a proposed pseudoinverse observer is not required. Second, theboth method can be easily extended to the case of input inspired by this result,systems data-rate limitations were for linear time-invariant systems in Tatikonda andstudied Mitter posed (2004), for stochastic in Nair and Evans (2004), method observer can be easily extended to the case of both input (2004), fortime-invariant stochastic systems in Nair and Evans (2004), is not required. Second, theboth proposed and output output quantization. method can quantization. be easily extended to the case of input for systems Tatikonda and (2004), Mitter andlinear for for uncertain systems in Okano Okano and Ishii (2014). Al- inverse (2004), stochastic systems in in Nair and Evans and and for uncertain systems in and Ishii (2014). Almethod can quantization. be easily extended to the case of both input output (2004), for stochastic systems in and Nairzooming-out and Evans (2004), though the so-called zooming-in encodand for the uncertain systems in Okano Ishii (2014). Al- and In this this paper, we present present an an output output encoding encoding method method for for though so-called zooming-in and and zooming-out encodand output quantization. In paper, we and for uncertain systems in Okano and Ishii (2014). Although the so-called zooming-in and zooming-out encoding method method developed developed in in Brockett Brockett and and Liberzon Liberzon (2000); (2000); In the stabilization of sampled-data systems with discretethis paper, we present an output encoding method for ing thethis stabilization of sampled-data systems with discretethough the so-called zooming-in zooming-out encoding method developed in Brockett and Liberzon (2000); Liberzon (2003b) provides only and sufficient conditions for In paper, weobservers. present an output encoding method for timestabilization Luenberger The proposed encoding method the of sampled-data systems with discreteLiberzon (2003b) provides only sufficient conditions for time Luenberger observers. The proposed encoding method ing method developed in Brockett and Liberzon (2000); stabilization, this encoding procedure is simple and hence Liberzon (2003b) provides only sufficient conditions for the stabilization of sampled-data systems with discreteis based on the zooming-in technique and employs estitime Luenberger observers. The proposed encoding method stabilization, this encoding procedure is simple and hence is based based on on the the observers. zooming-inThe technique and employs estiLiberzon (2003b) provides only sufficient conditions for time was extended extended to various various systems, e.g.,isnonlinear nonlinear systems stabilization, this encoding procedure simple and hence proposed encoding method is zooming-in technique and employs estimatesLuenberger generated from a Luenberger Luenberger observer for both both stawas to systems, e.g., systems mates generated from a observer for stastabilization, this encoding procedure is simple and hence was extended to various systems, e.g., nonlinear systems in Liberzon Liberzon (2006); (2006); Liberzon Liberzon and and Hespanha Hespanha (2005), (2005), syssys- mates is based on the zooming-in technique and employs estibilization and quantization. First we consider only output generated from a Luenberger observer for both stain bilization and quantization. First weobserver considerfor only output was extended to various systems, e.g., nonlinear systems in Liberzon (2006); Liberzon and in Hespanha (2005), systems with external external disturbances Liberzon and Neˇ i´cc mates generated from a Luenberger both staquantization and assume that encoders also contain contain estibilization andand quantization. First we consider only output tems with disturbances in Liberzon and Neˇ ssi´ quantization assume that encoders also estiin Liberzon (2006); Liberzon and Hespanha (2005), sys(2007); Sharon and (2008, 2012), and recently tems with external disturbances in Liberzon and Neˇ s i´ c bilization and quantization. First we consider only output mators. Simple sufficient conditions for stabilization are quantization and assume that encoders also contain esti(2007); Sharon and Liberzon (2008, 2012), and recently mators. Simple Simple sufficient conditions for also stabilization are tems with external disturbances in (2014); Liberzon and Neˇ si´c quantization switched/hybrid systems in Liberzon Liberzon Wakaiki and (2007); Sharon and Liberzon (2008, 2012), and recently andboth assume that encoders contain estimators. sufficient conditions for stabilization are obtained in the case of general Luenberger observers switched/hybrid systems in (2014); Wakaiki and obtainedSimple in the both case of general Luenberger observers (2007); Sharon and Liberzon (2008, (2014); 2012), recently switched/hybrid systems in Liberzon Wakaiki and Yamamoto (2016); Yang and Liberzon (2015).and Readers are mators. sufficient conditions for stabilization are and of of deadbeat deadbeat observers. obtained in the both case of general Luenberger observers Yamamoto (2016); Yang and Liberzon (2015). Readers are and observers. switched/hybrid systems in Liberzon (2014); Wakaiki and referred to the survey papers by Ishii and Tsumura (2012); Yamamoto (2016); Yang and Liberzon (2015). Readers are obtained in the both case of general Luenberger observers and of deadbeat observers. referred to the survey papers by Ishii and Tsumura (2012); Second we generalize generalize the results results of of general general Luenberger Luenberger Yamamoto (2016); Yang and Liberzon (2015). Readers are and Nair et et to al.the (2007), and the by books by Liberzon (2003c); referred survey papers Ishiiby and Tsumura (2012); of deadbeat observers. Second we the Nair al. (2007), and the books Liberzon (2003c); observersweto togeneralize the situation situation where both of the the plant plant input Second the where resultsboth of general Luenberger referred to the survey papers by Ishii and Tsumura (2012); Matveev and Savkin (2009) on this topic for further inforNair et al. (2007), and the books by Liberzon (2003c); observers the of input Matveev and(2007), Savkinand (2009) onbooks this topic topic for further further infor- Second we generalize the results of general Luenberger and output are quantized. Moreover, in the second case, observers to the situation where both of the plant input Nair et al. the by Liberzon (2003c); Matveev and Savkin (2009) on this for information. and output output are quantized. Moreover, in second case, mation. to are thequantized. situation where both the plant input and Moreover, inof the the second case, encoders do not estimate the plant state by themselves, Matveev mation. and Savkin (2009) on this topic for further infor- observers encoders do not estimate the plant state by themselves, Although a Luenberger observer has been widely used for and output are quantized. Moreover, in the second case, encoders do not estimate the plant state by themselves, but controllers send the quantized estimate of the plant mation. Although aa Luenberger Luenberger observer has widely used but controllers send the quantized estimate ofthemselves, the plant Although observer has been beene.g., widely used for for encoders quantized output output feedback feedback stabilization, in Ferrante Ferrante do notsend estimate the plantestimate state by of but controllers the quantized the plant quantized stabilization, e.g., in Although aoutput Luenberger observer has beene.g., widely used for but controllers send the quantized estimate of the plant quantized feedback stabilization, in Ferrante quantized output feedback stabilization, e.g., in Ferrante2632 Copyright © 2017 IFAC Copyright © 2017, 2017 IFAC 2632Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2017 IFAC 2632 Peer review under responsibility of International Federation of Automatic Copyright © 2017 IFAC 2632Control. 10.1016/j.ifacol.2017.08.101
Proceedings of the 20th IFAC World Congress 2578 Masashi Wakaiki et al. / IFAC PapersOnLine 50-1 (2017) 2577–2582 Toulouse, France, July 9-14, 2017
output to the encoders. In contrast with the quantization of the plant output, we quantize the plant input and the output estimate by using the origin as the quantization center, which reduces computational resources in the components of the plant side. We see that if the closed-loop system without quantization is stable, then there exists an encoding method such that the closed-loop system in the presence of three types of quantization errors is also stable. This paper is organized as follows. In the next section, we study the case when only the plant output is quantized and obtain two data-rate conditions for general Luenberger observers and deadbeat observers. In Section 3, the proposed encoding method is extended to the case when both of the plant input and output are quantized. Section 4 is devoted to the numerical comparison of the obtained data-rate conditions with those of the earlier studies by Liberzon (2003b); Sharon and Liberzon (2008, 2012) and the time response of an inverted pendulum. We provide concluding remarks in Section 5. Due to space constraints, all proofs have been omitted and can be found in Wakaiki et al. (2017). Notation and Definitions: The symbol Z+ denotes the set of nonnegative integers. Let λmin (P ) and r(P ) denote the smallest eigenvalue and the spectral radius of P ∈ Rn×n , respectively. We denote by In by the identity matrix of size n. For a vector v = [v1 · · · vn ]� ∈ Rn , we denote its maximum norm by |v| = max{|v1 |, . . . , |vn |} and the corresponding induced norm of a matrix M ∈ Rm×n by �M � = sup{|M v| : v ∈ Rn , |v| = 1}. Plant:
We consider a continuous-time linear system � x(t) ˙ = Ax(t) + Bu(t) ΣP : (1) y(t) = Cx(t), where x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input, and y(t) ∈ Rp is the output. This plant is connected with a controller through a time-driven encoder and zero-order hold (ZOH) with period h > 0. Define xk := x(kh) and yk := y(kh) for every k ∈ Z+ , and also set � h Bd := eAs Bds. (2) Ad := eAh , 0
Throughout this paper, we place the following assumptions: Assumption 1. (Initial state bound). A constant Est > 0 satisfying |x(0)| ≤ Est is known. Assumption 2. (Stabilizability and detectability). The discretized system (Ad , Bd , C) is stabilizable and detectable. Remark 3. We can obtain an initial state bound Est by the “zooming-out” procedure in Liberzon (2003b). 2. OUTPUT QUANTIZATION In this section, we consider the scenario where only the output yk is quantized and where the encoder has computational resources to estimate the plant state. 2.1 Controller Let qk ∈ Rp be the quantized value of the sampled output yk and K ∈ Rn×m , L ∈ Rn×p be a feedback gain and an
ZOH
u(t)
Plant
y(t)
Encoder Network
uk
Decoder Controller
qk
Fig. 1. Closed-loop system with output quantization. observer gain, respectively. For the plant ΣP in (1), we use a discrete-time Luenberger observer for feedback control and output quantization: ⎧ ˆk+1 = Ad x ˆk + Bd uk + L(qk − yˆk ) ⎨x = C x ˆ y ˆ ΣC : (3) k k ⎩ uk = −K x ˆk
where x ˆk ∈ Rn is the state estimate and yˆk ∈ Rp is the output estimate. We set the initial estimate xˆ0 to be x ˆ0 = 0. Each of the encoder and the controller contains the above Luenberger observer, and those observers are assumed to be synchronized. Through the zero-order hold, the control input u(t) is generated as (kh ≤ t < (k + 1)h). u(t) = uk Fig. 1 shows the closed-loop system with quantized output.
2.2 Output Encoding Suppose that we obtain an error bound Ek such that |yk − yˆk | ≤ Ek . The next subsection is devoted to the computation of a bound sequence {Ek } for stabilization.
For each k ∈ Z+ , we divide the hypercube (4) {y ∈ Rp : |y − yˆk | ≤ Ek } p into N equal boxes and assign a number in {1, . . . , N p } to each divided box by a certain one-to-one mapping. Since x ˆ0 = 0, we see from Assumption 1 that the error ˆk satisfies |e0 | = |x(0)| ≤ Est . Thus we can set ek := xk − x E0 := �C�Est .
The encoder sends to the controller the number q¯k of the divided box containing yk , and then the controller generates qk equal to the center of the box with number q¯k . If yk lies on the boundary on several boxes, then we can choose any one of them. This encoding strategy leads to Ek (5) =: μk . |yk − qk | ≤ N 2.3 Computation of Bound Sequence {Ek } Here we obtain bound sequences {Ek } and data-rate conditions for stabilization. We first consider general Luenberger observers and next focus on deadbeat observers.
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Use of General Luenberger Observers: The proposed encoding strategy with the following bound sequence {Ek } achieves the exponential convergence of the state under a certain data-rate condition. Theorem 4. Let Assumptions 1 and 2 hold. Define the ¯ by matrices R and R ¯ := Ad − Bd K. R := Ad − LC, R (6) Let the observer gain L and the feedback gain K satisfy ¯ < 1, and r(R) < 1, r(R) � � � � �CR� � ≤ M0 ρ� , �CR� L� ≤ M ρ� . (7)
for some M0 , M > 0 and ρ < 1. If we pick N ≥ 2 so that M < N, (8) 1−ρ then the proposed encoding method with a bound sequence {Ek } defined by ⎧ M ⎪ ⎨M0 Est ρ + E0 k = 0 N � � (9) Ek+1 := M ⎪ ⎩ ρ+ Ek k ≥ 1. N achieves the exponential convergence of the state x and the estimate x ˆ.
Use of Deadbeat Observers: In the rest of this section, we focus on deadbeat observers. If the pair (C, Ad ) is observable, then there exists a matrix L ∈ Rn×p such that Rη = (Ad − LC)η = 0, (10) where η is the observability index of (C, Ad ). Construction methods of such an observer gain L have been developed for deadbeat control; see, e.g., Chapter. 5 of O’Reilly (1983). Using the property (10), we obtain an alternative error bound sequence {Ek } for stabilization. Theorem 5. Let Assumptions 1 and 2 hold. Assume that ¯ as in (C, Ad ) is observable. Define the matrices R and R (6), and let the observer gain L and the feedback gain K ¯ < 1, where η is the observability satisfy Rη = 0 and r(R) index of (C, Ad ). Set a constant α� to be � � �CR� L� α� := (11) N for � = 0, . . . , η − 1. If we pick N ≥ 2 so that a matrix F defined by ⎡ ⎤ α0 α1 . . . αη−1 0 ⎥ ⎢1 ⎥ F := ⎢ (12) .. ⎣ ⎦ . 0 1 0 satisfies r(F ) < 1, (13) then the proposed encoding method with a bound sequence {Ek } defined by ⎧ � k � �CR� L� � � � ⎪ ⎪ k+1 � � Est + ⎪ Ek−� 0 ≤ k ≤ η−2 ⎪ ⎨ CR N �=0 Ek+1 := η−1 � � � �CR� L� ⎪ ⎪ ⎪ ⎪ Ek−� k ≥ η−1 ⎩ N �=0 (14) achieves the exponential convergence of the state x and the estimate x ˆ. Remark 6. As (9), (14) also has the form of linear timeinvariant recursion.
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Remark 7. Sharon and Liberzon (2008, 2012) proposed the quantizer based on a psuedo-inverse observer. That quantizer achieves closed-loop stability if N ≥ 2 satisfies � � �CAd C† � < N, (15) where
⎡
C CA d ⎢ C := ⎢ ⎣ ...
CAη−1 d
⎤
⎥ −η+1 ⎥A , ⎦ d
� �−1 � C† := C� C C ,
and the total data size is N p . In the state feedback case of Liberzon (2003b), the counterpart of (15) is �Ad � < N,
(16)
n
and the total data size is N .
3. INPUT AND OUTPUT QUANTIZATION In this section, we quantize both of the plant input and output. Moreover, we assume in the previous section that the encoder has computational resources to estimate the plant state, while we here study the scenario where the controller sends the quantized output estimate to the encoder. Hence the encoder does not need to compute or store the estimate. 3.1 Controller Let K ∈ Rn×m , L ∈ Rn×p be a feedback gain and an observer gain, respectively. Let qk ∈ Rp denote the quantized value of the sampled output yk . We denote by Q1 and Q2 qunatization functions of the output estimate yˆ and the control input u, respectively. For the plant ΣP in (1), we construct the following observer-based controller: ⎧ ˆk + Bd uk + L(qk − Q1 (ˆ yk )) ˆk+1 = Ad x ⎨x yˆk = C x ˆk (17) Σ�C : ⎩ uk = −K x ˆk where Ad and Bd are defined as in (2). We set the initial estimate x ˆ0 to be x ˆ0 = 0. Compared with the controller ΣC in (3), the controller Σ�C uses the quantized output yk ) instead of the original output estimate estimate Q1 (ˆ yˆk . The control input u(t) is produced as u(t) = Q2 (uk )
(kh ≤ t < (k + 1)h).
Note that the controller can compute x ˆk and hence yˆk , uk at time k − 1. Fig. 2 illustrates the closed-loop system we consider in this section. Remark 8. Instead of (17), we can use different controllers such as ⎧ ˆk + Bd Q2 (uk ) + L(qk − Q1 (ˆ yk )) ˆk+1 = Ad x ⎨x yˆk = C x ˆk Σ��C : ⎩ uk = −K x ˆk .
The major reason to use (17) is that if x ˆ0 = 0, then we have k−1 � x ˆk = (Ad − Bd K)k−� (q� − Q1 (y� )) (18) �=0
and hence yˆk and uk can be described by q� −Q1 (y� ), which makes encoding methods simple.
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Q2 (uk )
ZOH
Decoder
u(t)
Q1 (ˆ yk )
center, whereas the quantization centers for the output estimate yˆk and the input uk are the origin, which allows the plant side to have less computational resources. Remark 9. Although the encoder does not have to estimate output measurements, the bounds Ek , E1,k , and E2,k should be computed in the plant side. However, these bounds can be calculated by simple difference equations (21) as shown in the next subsection. If components in the plant side do not have computational resources enough to implement those difference equations, the controller can send sufficiently accurate values of the bounds even in the presence of quantization, because the dimension of each bound is one.
y(t)
Plant
Encoder
Network Decoder
Encoder yˆk uk
Controller
qk
Fig. 2. Closed-loop system with input and output quantization.
3.4 Computation of Bound Sequence {(Ek , E1,k , E2,k )}
3.2 Output Encoding
The following theorem is an extension of Theorem 4. Theorem 10. Let Assumptions 1 and 2 hold, and define R ¯ as in (6). Let the observer gain L and the feedback and R ¯ < 1 and gain K satisfy r(R) < 1, r(R) ¯ � L� ≤ M1 ρ¯� , �K R ¯ � L� ≤ M2 ρ¯� �CR� � ≤ M0 ρ� , �C R
Suppose that we obtain an error bound Ek such that yk )| ≤ Ek . A bound sequence {Ek } satisfying |yk − Q1 (ˆ this condition is obtained in Section 3.4. Instead of (4), the encoder computes quantized measurements by dividing the hypercube yk )| ≤ Ek } (19) {y ∈ Rp : |y − Q1 (ˆ p into N equal boxes. The difference between (4) and (19) is the quantization center. In (4), the encoder has the state estimate x ˆk , and hence the quantization center can be yˆk . On the other hand, the encoder here employs the quantized yk ) reported by the controller as the output estimate Q1 (ˆ quantization center. The rest of the output encoding is the same as in Subsection 3.2. 3.3 Estimate and Input Encoding The controller sends to the plant the quantization data of the control input uk and the output estimate yˆk . Suppose that we have bounds E1,k and E2,k such that |ˆ yk | ≤ E1,k and |uk | ≤ E2,k . Such a bound sequence {(E1,k , E2,k )} is obtained in Section 3.4. The bounds and levels of the quantization of yˆ, u are given by (E1,k , N1 ) and (E2,k , N2 ). Namely, the controller computes the quantized output estimate and the quantized input by dividing the hypercubes y | ≤ E1,k } {ˆ y ∈ Rp : |ˆ {u ∈ Rm : |u| ≤ E2,k }
into N1p and N2m equal boxes and assigns a number in {1, . . . , N1p } and {1, . . . , N2m } to each divided box by a certain one-to-one mapping, respectively. The decoder in the plant side generates Q1 (ˆ yk ) and Q2 (uk ) equal to the center of the boxes with number reported by the controller. yk ) and Q2 (uk ) satisfy Thus Q1 (ˆ E1,k |ˆ yk − Q1 (ˆ yk )| ≤ N1 E2,k |uk − Q2 (uk )| ≤ . N2 Since x ˆ0 = 0, we can set the initial values E1,0 := 0 E2,0 := 0. The encoding strategy (19) of the output yk uses the yk ) as the quantization quantized output estimate Q1 (ˆ
�CR� Bd � ≤ M3 ρ� , �CR� L� ≤ M4 ρ� hold for some M0 , M1 , M2 , M3 , M4 > 0 and ρ¯ ≤ ρ < 1. Define constants α0 and α1 by � � M1 M4 M 2 M3 M := (N − 1) · + N1 N2 N1 M4 + (N − 1)M1 β0 := ρ + N N1 M β1 := N α0 := ρ + β0 α1 := β1 − ρβ0 . If we pick N, N1 , N2 ≥ 2 so that � � α0 α 1 F := (20) 1 0 satisfies r(F ) < 1, then the proposed encoding with a bound sequence {(Ek , E1,k , E2,k )} defined � � ⎧ (N − 1)M1 E0 ⎪ ⎪ M E ρ + M + 4 ⎨ 0 st N1 N Ek+1 := β E + β E ⎪ 0 k 1 k−1 ⎪ ⎩ α0 Ek + α1 Ek−1 (N − 1)M1 Ek E1,k+1 := ρ¯E1,k + N (N − 1)M2 Ek E2,k+1 := ρ¯E2,k + N
method by k=0 k=1 k≥2 k≥0 k ≥ 0.
(21) achieves the exponential convergence of the state x and the estimate x ˆ. Remark 11. Although we here generate Q1 (ˆ yk ) and Q2 (uk ) separately, one can quantize yˆk and uk simultaneously. This simultaneous quantization reduces the computational cost of the controller, but the data-rate condition for stabilization becomes conservative. Therefore, we do not proceed along this line. Remark 12. There always exist quantization levels N , N1 , and N2 such that the matrix F in (20) satisfies r(F ) < 1. In fact, as N , N1 , and N2 increase to infinity, we have
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4.1 Comparison of Data-Rate Conditions First we consider the quantization of only the plant output and compare the data-rate conditions of three types of observers: the steady-state Kalman filter (7) with process noise covariance 10−3 In and measurement noise covariance 10−1 Ip , the deadbeat observer (13), and the pseudo-inverse observer (15). In addition to the output feedback case, we also investigate the state encoding case (16). In Table 1, we show the comparison of the minimum quantization level for exponential convergence, which are N p in the output feedback case and N n in the state feedback case. Note that steady-state Kalman filter and the deadbeat observers are represented by a linear time-invariant state equation but pseudo-inverse observers are not. The first example is an inverted pendulum whose dynamics is given by (1) with ⎡ ⎤ ⎡ ⎤ 0 1 0 0 0 ⎢0 −20.06 53.26 −1.096⎥ ⎢35.28⎥ A := ⎣ , B := ⎣ (22) 0 0 0 1 ⎦ 0 ⎦ 0 −20.01 98.41 −2.025 35.18 C := [1 0 1 0] . The state [x1 x2 x3 x4 ] =: x are the arm angle, the arm angular velocity, the pendulum angle, and the pendulum angular velocity. The input u is the motor voltage. Additionally, we borrow a 2-mass motor drive with one output and three states from Ji and Sul (1995), a pneumatic cylinder with one output and three states from Kimura et al. (1996), and a batch reactor with two output and four states from Rosenbrock (1974). In Table 1, we see that the Kalman filter requires less datarate than the deadbeat observer and the pseudo-inverse observer for the pneumatic cylinder. This is because the cylinder have their unstable poles only on the imaginary axis. Hence the observer gain of the Kalman filter is small, which decreases M in (7). Although deadbeat observers and pseudo-inverse observers have the same property: finite-time state reconstruction in the idealized situation without quantization, the data-rate condition (15) by pseudo-inverse observers is better than that (13) by deadbeat observers. This is because pseudo-inverse observers employ output measurements directly for state reconstruction, whereas deadbeat observers summarize output information by their states. Moreover, compared with the state feedback case (16), the output feedback case requires small data sizes in most numerical examples because the state dimension n and the output dimension p satisfy n > p.
From Theorem 10, the state x exponentially converges to the origin under the proposed encoding strategy with (N, N1 , N2 ) = (151, 301, 1101) for which F in (20) satisfies r(F ) = 0.9906. Supposing that we obtain an initial state bound E0 = 0.15, we compute a time response for the initial state x(0) = [0 0 0.1 0]� . The plot of the arm angle x1 and the pendulum angle x3 is in Figs. 3 and 4. Fig. 5 illustrates the motor voltage u. From Figs. 3 and 4, we observe that the arm and pendulum angles decrease to zero in the presence of three types of quantization errors. Since the initial estimate x ˆ0 = 0, the quantization errors of the output estimate and the input are small at first. In fact, the output estimate bound {E1,k } and the input bound {E2,k } take the maximum value at about time t = 0.6, and the effect of the quantization errors appears from time t = 0.5 in Figs. 3, 4, and 5. 5. CONCLUSION We studied quantized output feedback stabilization by Luenberger observers. Data-rate conditions for general Luenberger observers are characterized by the spectral radius of the system matrix of the error dynamics. On the 0.1
0.1
Arm angle x1 [rad]
4. NUMERICAL EXAMPLES
sampling period be h = 0.03 sec. We set the feedback gain K to be the quadratic regulator whose weighting matrices of the state and the input are diag(100, 0, 300, 0) and 1, respectively. The observer gain L is the Kalman filter whose covariances of the process noise and measurement noise are 10−3 Ip and 10−1 Ip , respectively.
0
0
-0.05
-0.1
-0.1
-0.15
-0.2
With Quantization
-0.2
Without Quantization
-0.3 0 -0.3
0
0.5
1
1
1.5
Time
2
2
2.5
3
3
Fig. 3. Arm angle x1 with/without quantization. 0.15 0.15
4.2 Time response of Inverted Pendulum We consider again the inverted pendulum (22) in the previous subsection and compute its time response under the quantization of the plant input and output. The controller is assumed to send to the encoder the quantized value of the output estimate for quantization centers. Let the
0.05
-0.25
Pendulum angle x3 [rad]
α0 → ρ and α1 → ρ2 , and hence the eigenvalues of F tend to ρ < 1.
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0.1
0.05
0
0
-0.05
With Quantization
-0.1
Without Quantization
-0.15 0 -0.15
0
0.5
1
1
1.5
Time
2
2
2.5
3
3
Fig. 4. Pendulum angle x3 with/without quantization.
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Table 1. Minimum quantization levels for exponential convergence. Kalman filter (7)
Deadbeat observer (13)
Pseudo-inverse observer (15)
State encoding (16)
62 = 36 51 = 5 21 = 2 102 = 100
42 = 16 71 = 7 71 = 7 42 = 16
42 = 16 31 = 3 31 = 3 42 = 16
44 = 256 33 = 27 23 = 8 34 = 81
Inverted pendulum (h = 0.03) 2-mass motor drive (h = 10−3 ) Pneumatic cylinder (h = 10−3 ) Batch reactor (h = 0.1)
other hand, a data-rate condition for deadbeat observers is determined by the behavior of the error dynamics for η steps, where η is the observability index of the plant. The proposed encoding method was also extended to the case where both of the plant input and output are quantized and where the encoder does not have an estimator for generating quantization centers. Future work involves addressing more general systems such as nonlinear systems and switched systems. REFERENCES Brockett, R.W. and Liberzon, D. (2000). Quantized feedback stabilization of linear systems. IEEE Trans. Automat. Control, 45, 1279–1289. Ferrante, F., Gouaisbaut, F., and Tarbouriech, S. (2014). Observer-based control for linear systems with quantized output. In Proc. ECC’14. Ishii, H. and Tsumura, K. (2012). Data rate limitations in feedback control over network. IEICE Trans. Fundamentals, E95-A, 680–690. Ji, J.K. and Sul, S.K. (1995). Kalman filter and LQ based speed controller for torsional vibration suppression in a 2-mass motor drive system. IEEE Trans. Ind. Electron., 42, 564–571. Kimura, T., Fujioka, H., Tokai, K., and Takamori, T. (1996). Sampled-data H∞ control for a Pneumatic cylinder system. In Proc. IEEE 35th CDC. Liberzon, D. (2003a). Hybrid feedback stabilization of systems with quantized signals. Automatica, 39, 1543– 1554. Liberzon, D. (2003b). On stabilization of linear systems with limited information. IEEE Trans. Automat. Control, 48, 304–307. Liberzon, D. (2003c). Switching in Systems and Control. Boston: Birkh¨ auser. Liberzon, D. (2006). Quantization, time delays, and nonlinear stabilization. IEEE Trans. Automat. Control, 51, 1190–1195.
Motor voltage u [V]
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Fig. 5. Motor voltage u with/without quantization.
Liberzon, D. (2014). Finite data-rate feedback stabilization of switched and hybrid linear systems. Automatica, 50, 409–420. Liberzon, D. and Hespanha, J.P. (2005). Stabilization of nonlinear systems with limited information feedback. IEEE Trans. Automat. Control, 50, 910–915. Liberzon, D. and Neˇsi´c, D. (2007). Input-to-state stabilization of linear systems with quantized state measurement. IEEE Trans. Automat. Control, 52, 767–781. Matveev, A.S. and Savkin, A.V. (2009). Estimation and Control over Communication Networks. Birkh¨ auser, Boston. Nair, G.N. and Evans, R.J. (2004). Stabilizability of stochastic linear systems with finite feedback data rates. SIAM J. Control Optim., 43, 413–436. Nair, G.N., Fagnani, F., Zampieri, S., and Evans, R.J. (2007). Feedback control under data rate constraints: An overview. Proc. IEEE, 95, 108–137. Niu, Y., Jia, T., Wang, X., and Yang, F. (2009). Outputfeedback control design for NCSs subject to quantization and dropout. Inf. Sci., 179, 3804–3813. Okano, K. and Ishii, H. (2014). Stabilization of uncertain systems with finite data rates and Markovian packet losses. IEEE Trans. Control Network Systems, 1, 298– 307. O’Reilly, J. (1983). Observer for Linear Systems. New York: Academic. Rosenbrock, H.H. (1974). Computer-Aided Control System Design. New York: Academic Press. Sharon, Y. and Liberzon, D. (2008). Input-to-state stabilization with quantized output feedback. In Proc. HSCC’08. Sharon, Y. and Liberzon, D. (2012). Input to state stabilizing controller for systems with coarse quantization. IEEE Trans. Automat. Control, 57, 830–844. Tatikonda, S. and Mitter, S. (2004). Control under communication constraints. IEEE Trans. Automat. Control, 49, 1056–1068. Wakaiki, M. and Yamamoto, Y. (2016). Stabilization of switched linear systems with quantized output and switching delays. To appear in IEEE Trans. Automat. Control. Wakaiki, M., Zanma, T., and Liu, K.Z. (2017). Quantized output feedback stabilization by Luenberger observers. arXiv:1703.06567. Wong, W.S. and Brockett, R.W. (1999). Systems with finite communication bandwidth constraints II: Stabilization with limited information feedback. IEEE Trans. Automat. Control, 44, 1049–1053. Xia, Y., Yan, J., Shang, J., Fu, M., and Liu, B. (2010). Stabilization of quantized systems based on Kalman filter. Control Eng. Pract., 20, 954–962. Yang, G. and Liberzon, D. (2015). Stabilizing a switched linear system with disturbance by sampled-data quantized feedback. In Proc. ACC’15.
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