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An Optimal Compensation Framework for Linear Quadratic Gaussian Control over Lossy Networks Jen-te Yu and Li-Chen Fu, Fellow, IEEE  Abstract—In this paper we study the compensation problem of LQG control over two lossy networks under TCP-like protocols and propose a new static and latest-control based compensation framework. Compared to two popular strategies, the zero-input and hold-input compensators and the recent generalized hold-input compensator by Moayedi et al. [8], [13], our new one is more general. The contribution of the new framework is in three aspects. Firstly, it takes the problem of LQG control over lossy networks to a new level by suggesting that the original problem should be more properly posed as a two-variable instead of a one-variable optimization problem. Secondly, it links the compensator gain selection and the minimization of quadratic cost together and bridges the gap between the two by providing an optimal gain selection. Thirdly, it incorporates the above three classes of compensators as special cases. The issue as to why none of the zero-input and hold-input compensation strategies can be claimed to be better than the other is to a great extent settled. Performance comparisons of the proposed method and the predictive outage compensator by Henriksson et al. [17] and the generalized hold-input strategy by Moayedi et al. [8], [13] are made through numerical examples. Index Terms—LQG, lossy networks, dropout rate, quadratic cost, optimal, hold-input, zero-input, compensator.

I. INTRODUCTION

I

T has been witnessed that more and more control systems and their extended real-world applications today are distributed across networks located at different sites physically; networked control systems (NCS) have attracted a lot of research effort and attention in the past four decades. Important issues such as data losses, corruption, packet-reordering and transmission delays have been extensively studied in respect of their impacts, such as performance and stability, on the NCS. Linear quadratic (LQ) control and filtering (LQG) across lossy networks are two important subjects among others. Usually the loss of a transmitted signal over a lossy network can be modeled as an independent and identically distributed Bernoulli random process whose transmission loss (or Jen-te Yu is a doctoral student with the Department of Electrical Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). Li-Chen Fu is with the Department of Electrical Engineering and Department of Computer Science and Information Engineering, National Taiwan University, Taipei, Taiwan (e-mail: [email protected]). This work is supported by Ministry of Science and Technology of Republic of China under Grant MOST 103-2221-E-002-199-MY2.

conversely arrival) probability/rate is known, from statistical data for example. Sinopoli et al. [1] investigated Kalman filtering with lossy intermittent observations. Under TCP-like and UDP-like protocols, Schenato et al. [2] extended that work to optimal controls. Similar problems were studied by Imer et al. in [3]. As is well-known from the literature, two compensation strategies are quite popular and commonly used: the zero-input and the hold-input. The former strategy means that zero value is adopted whereas the latter applies directly the latest-control with no modification whenever a transmitted signal gets lost. Following the above classification, the works [2] and [3] belong to the zero-input class and extension using previous control was discussed briefly in [3] also. Bae et al. [4] considered a signal loss compensation problem for a rehabilitation system using a modified LQG controller plus a disturbance observer, in which the zero-input compensator was employed. Yu et al. [7] took a switched system approach to tackle the stabilization problem of networked controls, applying hold-input policy. Using state feedback controllers, Zhang and Yu [12] considered the problem of exponential stabilization of networked systems subject to guaranteed cost and bounded packet losses. Again, for compensation purpose the previously received signals were used directly without any modification there. In a LQG setting, Kogel and Findeisen [14] proposed an approach named extended input schemes to handle the compensation problem aforementioned where they used heuristic methods and bilinear matrix inequalities to determine the suboptimal controllers. From a different perspective, Gommans et al. [15] presented a linear matrix inequality based framework to improve robustness of stability of the above two popular compensation strategies with respect to data dropouts. Both worst-case bounds and stochastic information based models were considered for the data loss and the local dropout history was also utilized. The compensator proposed there was a dynamic one and acted as a model-based closed-loop observer and an open-loop predictor interchangeably, requiring no acknowledgement of successful data transmissions over the networks. Although using zero-input or hold-input is straightforward and intuitive, Schenato [11] has reported that none of the above two compensators can be claimed superior to the other. The recent account [21] came to the same conclusion. Detailed

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compensator synthesis procedures of these two strategies can be found in [11], [21], including an in-depth performance analysis on the scalar case [11], [21]. See Gommans et al. [15] for some comparisons and discussions. There exist other performance criteria. For example, the comparison between the above two popular compensators made by Guo et al. [16] was from H∞ control, and the conclusion was similar. Antunes et al. [22] approached the problem from a protocol’s perspective. In this paper we revisit the compensation problem of discrete-time LQG control across two lossy networks under TCP-like protocols. Signal dropout occurs at two places: one is from controller to actuator and the other is from sensor to estimator/Kalman filter. See Fig. 1 for the overall system configuration. Compensators based on static/dynamic, linear/nonlinear combination of several past controls at different previous time instants can be constructed in principle, but they are more complex. On the other hand, static and latest-control based compensators are structurally much simpler (possibly the simplest) and appealing from implementation/cost standpoint. We therefore limit our focus on the latter in this paper. It turns out however, as will be shown under our new framework, that the hold-input should not be used directly without modification. Likewise, the zero-input compensation may not be a good policy either. An improvement on the hold-input strategy was put forward by Moayedi et al. [8] and [13]. They suggested that hold-input compensator should be multiplied by an optimized scalar, and hence named it “generalized hold-input”. Another class of compensators, the predictive type, was proposed by Henriksson et al. [17], which uses technique of transfer function matching to predict the lost signals. The contribution of this paper is in three folds. Firstly, it takes the compensation problem of LQG control over two lossy networks to a new level by suggesting that the original problem should be more properly posed as a two-variable instead of a one-variable optimization problem. Secondly, it links the compensator gain selection and the minimization of the quadratic cost functional and provides an optimal gain selection. Thirdly, it proposes a more general framework that treats the zero-input, the hold-input, and the recent one by Moayedi et al. [8], [13] as special cases. The rest of the paper is organized as follows. The system model is presented in Section II, where the basic underlying assumptions are given. The two popular compensation strategies are also discussed in this section, namely, the zero-input and the hold-input compensators. In Section III, a more general framework to tackle the compensation problem is proposed. Section IV presents the optimal design, including optimal gain equations for both the controller and the compensator. We will illustrate that the two popular compensators and the one by Moayedi et al. [8], [13] are just special cases of ours in Section V. Some comments on the latter will be given also. Three numerical examples are given in Section VI to validate the proposed approach and compare the performances of different compensators. Finally, a conclusion is provided in Section VII. It is a well-known fact [11] that the Riccati equation fails to

have a solution if the data dropout rate goes above a critical value. It is therefore assumed throughout this paper that the data loss rate falls below the critical level. We also note that if the second-order derivative, i.e. the Hessian matrix is positive definite at the stationary point, then the corresponding cost value at that point is a minimum, according to optimization theory [19]-[20]. We adopt the following notations in this paper: x stands for the state of the dynamical system, u is the control input, L stands for state feedback gain, N refers to the compensator gain, Pr represents the probability, and E stands for expected value. J represents a quadratic cost value; matrices X and U represent the weighting matrices for state and control respectively in the associated cost functional, Tr stands for the trace of a matrix, ν represents the average dropout rate of the signal from controller to actuator, and γ denotes the average dropout rate of the signal from the sensor to the Kalman filter. Matrices Q and R represent the covariance of the process and sensor noise respectively. Matrix P denotes the error covariance of state estimation. Subscripts k and m when associated with signals in the context of a dynamical system are time instants. A gain with subscript “z.i.”, “h.i.”, “M”, or “opt” stands for that of zero-input, hold-input, Moayedi et al. [8], [13], and the optimal one respectively. A variable in the Kalman filter with a superscript “–” represents that before measurement update. Superscript “+” stands for the Moore-Penrose inverse. Hessian matrix is denoted by Θ. II. SYSTEM MODEL AND THE TWO POPULAR COMPENSATORS It is assumed throughout the paper that the signal dropping probability/rate is known. That is, the following are given where Pr(·) and E[·] stand for probability and expectation respectively.  k  0,1, Pr( k  0)   , [ k ] 1 

 k  0,1, Pr( k  0)   , [  k ] 1  

(1)

Consider the following discrete linear dynamical system and its associated Kalman filter (2) xk 1  Axk  Buk  k , k (0, Q)

yk  C xk  k , k  k 1

xˆ

 k 1

P

(3)

(0, R)

 Axˆk  Buk

(4)

 APk A  Q T

(5)

xˆk  xˆ   k Pk C R ( yk  Cxˆ )  k

1

T

 k

(6)

Pk  P  (1   ) P C (CP C  R) CP  k

 k

T

 k

T

1

 k

(7) where xk is the state, uk is the control, yk is the output,  k and  k , are process noise and measurement noise respectively; and Pk is the error covariance with error defined as ek  xk  xˆk . The output and the control signals will be sent to the Kalman filter and the actuator respectively through two lossy networks under TCP-like protocols. See Fig. 1 for the overall configuration wherein the binary (Bernoulli) random variables  k and  k are used to depict if the transmitted signals are received. For example, when  k  1 , it means the signal indeed arrives at its



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Yk+1. Combining (16) with (11)-(15) we get

destination, otherwise  k  0 . The two popular strategies take the following forms zero-input: uk   k ukc ,

J kf1   ( zkT1Yk 1 zk 1 )  g k 1

hold-input: uk   k ukc  (1  k ) uk 1

(8)

The superscript “c” used here is to depict that the signal is specifically generated by the controller as contrasted to the one out of the actuator that enters into the plant. uk

Actuator

k  0

 k 1

ukc

3


Plant

Sensor

Acknowledgement

Controller

xˆ k

yk

 k 1

k0

Kalman Filter

Fig. 1. Configuration of the LQG control over two lossy networks.

III. A MORE GENERAL COMPENSATION FRAMEWORK Consider the following control where N is a matrix gain (9) uk   k ukc  (1  k ) Nuk 1 To begin, let us define an augmented state [11] x  zk   k  . uk 1  We may write the dynamics of the augmented system as   x  x  B zk 1  H ( k ) zk  k   ukc   k  , z0   0    0  I  u1   0  0

(10)

 g k 1  Tr (Y11 k 1Q)   zkT H 0T Yk 1 H 0 zk B  (1  ) zkT H1T Yk 1 H1 zk  2(1  ) zkT H1T Yk 1   ukc I  T

B B  (1  )ukcT   Yk 1   ukc I  I  f T J k  ( zk Yk zk )  g k

(17)

  H 0T Yk 1 H 0  (1  ) H1T Yk 1 H1        min  zT 0 X   zk  ukc , N  k     T  0  N UN           2(1  ) z T H T Y  B  u c  k 1 k  1 k     I    T     B B  (1  )ukcT  U    Yk 1    ukc  I  I             g k 1  Tr (Y11 k 1Q )  (18) IV. THE OPTIMAL COMPENSATOR

(11)

We limit our focus on the infinite horizon case in this paper and consider static gains only. The expected total cost is

 A (1  k ) BN   A BN  A 0  H ( k )    , H 0   0 N  , H1   0 0  . (12) 0 (1   ) N     k  

   J  J 0      xkT X xk  ukT U uk   . (19)  k 0  With mild assumptions (the signal loss rate falling below the critical level, the pair (A, X 1/2 ) being detectable, and the feedback gain L being stabilizing) it is expected that the total cost would be finite and Y would exist [1], [2], [11]. Note that saddle points may exist in general. To ensure that the obtained solution indeed yields a minimum cost, we will introduce an additional condition on the second-order derivative, i.e. on the Hessian matrix. According to optimization theory, we know that the cost value at the stationary point is a minimum if the Hessian matrix at that point is positive definite [18]-[20]. Under these conditions we now present the main results of this paper. Theorem 1: The optimal static controller gain L and compensator gain N for the infinite horizon case are given by

To tackle the problem, we employ the dynamic programming technique [2], [11]. Define the cost-to-go as follows where subscript k and superscript f stand for the starting and final time instants respectively  f  J kf      xmT X m xm  umT U m um   . (13)  mk  The weighting matrices in the quadratic cost functional are chosen as X m = X and U m = U, except U f = 0. The original problem should be more properly posed as a two-variable instead of a one-variable optimization problem, and we therefore consider the following

J kf  min   J kf1  xkT X k xk  ukT U k uk  uk , N

(14)

xkT X k xk ukT U k uk 0 X  2 cT c  zkT   zk   k uk Uuk 2 T 0 (1   ) N U N k   cT  2 k (1  k )uk 0 U N  zk .

(15)

Assumption 1: There exist a symmetric positive semi-definite matrix Yk and a scalar g k such that the cost-to-go can be expressed as (16) J kf  ( zkT Yk zk )  gk . See [2] and [11]. In the sequel, Yk will be partitioned whenever necessary; e.g. Y11 k+1 stands for the (1,1) block of partitioned

1

T T  B B  B  A   L U   Y     Y    I I I       0    U 1 ( BT Y11  Y12T ) A

(20)

N  L Ac L , Ac  A  BL . (21) Proof: We derive the governing equations for the finite horizon case first. Extension to infinite horizon follows next. Based on them, optimality conditions are imposed. Also derived is the second-order derivative, i.e. the Hessian matrix that is required to be positive definite, which ensures that gains obtained indeed are optimal [18]-[20]. The control is in a state feedback form using the estimated state provided by the Kalman filter



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(22) ukc  Lxˆk where L stands for a stabilizing state feedback gain. After substituting this equation into (18), together with the fact [2] (23) (ek xˆkT )  0, ek  xk  xˆk , one can find the governing equations for Yk and g k as follows 0  X T Yk      H 0 Yk 1 H 0 T 0  N UN  

 T  B T  H1 Yk 1 H1  H1 Yk 1    L 0  I    T   T B   +(1  )   L 0   Yk 1 H1  I    T    B B T     L 0   U    Y     L 0   I  k 1  I             g k  g k 1  Tr (Y11 k 1 Q)

(24)

   B  B  (25)  (1  )Tr  LT  U    Yk 1    LPk  .   I  I     One should solve (24)-(25) backward in time using the following final conditions [2] X 0  Yf   (26)  , g f  0. 0 0 The expected initial state is typically assumed to be distributed uniformly over a sphere satisfying the following condition   x0 x0T   I . (27) For the infinite horizon case, equation (24) becomes  X  AT Y11 A AT (Y11 B  Y12 ) N  Y   T T  T N TU N  N ( B Y11  Y12 ) A 

 Y11 0 ,Y   T 0  Y12

T

(29)

T

Y12   AT (Y11B  Y12 )N

(30)

Ac  A  BL



T

 2.

(38)

 2 J  2  L  T 2   J   L N 

2 J   L  N  T  2  J  N 2 

(39)

2 J 2 J  2(1  ) I  U ,  2 (1  )( LLT )  U 2 2 L N



(40)

2 J  2 (1  ) ( NL  LAc )T U   I  L  N





(41)

N  NT . (42) According to optimization theory, if the Hessian matrix (39) at the stationary point is positive definite, then the corresponding cost is minimum [18]-[20]. This ends our proof. Remarks: We refer the reader to [18] regarding the issue as to how one could algorithmically avoid the saddle points. Note that when the signal dropout probabilities/rates of the two lossy networks under consideration go to zero, the above results recover that of the standard LQG control naturally [10].

(31)

V. THREE EXISTING COMPENSATORS AS SPECIAL CASES

(32)

One can see from (8) that the zero-input compensator ( N = 0) and the hold-input compensator ( N = I ) are just two special cases under our general formulation. The new framework thoroughly clarifies why neither of them can be claimed to be better than the other, as both are predetermined and in fact seem arbitrary, failing to take optimality of the total quadratic cost into consideration. The compensator gain (denoted as  below) in the generalized hold-input strategy recently proposed by Moayedi et al [8], [13] is a scalar, which can be expressed as uk   k ukc  (1  k ) uk 1 (43)

Y22   N T UN . (33) Let the optimality condition for the controller gain L hold J Tr (Y11 ) 0  . (34) L L Based on (34), together with equations (31)-(33), we may obtain the optimal state feedback gain (20). Given (20), one may rewrite equations (31)-(32) as follows Y11  X  A T Y11 A  (1  )[ AcT Y11 Ac  LT Y12T Ac  AcT Y12 L  LT (U  Y22 )L ],

+

Given (31), (33), (35), (36) and (38), the Hessian matrix Θ can be obtained as follows where Γ is a permutation matrix

T

T

B  B U  U    Y  . I  I  According to (28), the following three identities hold Y11  X  AT Y11 A  (1  )M

T –1

  L(UN )   ( Ac LT )  U

Y12  , (28) Y22 

 B  A  A  B M  LTUL  LT   Y      Y   L I  0 0 I

T

Given (33), (35)-(37) and the fact L ( L L ) =L , we get the compensator gain (21). Note that the conditions (34) and (37) are of first order and existence of saddle points is possible. We therefore require one more condition (of the second order) that the Hessian matrix at the stationary point be positive definite [19]-[20]. Mathematically, the two matrix gains L and N can be treated as two vectors in the context of optimization. To elaborate this more, we employ the following two identities, where ψ1, ψ2 and ψ3 are compatible matrices, () stands for matrix-vectorization, and  denotes the Kronecker product

 1 2 3   3T  1   2 , Tr ( 1 2 )   1T

T

 X  AT Y11 A  M  (1  )  0 

(36) Y12   LT UN . Similarly, the following optimality condition for N must hold  J Tr (Y11 ) 0  . (37) N N

(35)

We may recover this compensator simply by letting



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(44) N  I. Equations (33) and (36) for this case can be rewritten as (45) Y12   LT U , Y22   2 U. Following the optimality condition (37) together with (44)-(45) we may get the optimal value for  as Tr  LT UL  Ac  AcT   (46)   . 2Tr  LT UL  Similarly, the Hessian matrix for this special case can be written as  2 J 2 J    2 L    L T  (47) T  2 2    J  J    L    2 

2 J 2 J  2(1  ) I  U ,  2 (1  )Tr ( LT UL) 2  2 L

(48)

2 J  2 (1  )  2 I  Ac  AcT   U  L. (49) L  Some comments on this compensator are in order. Firstly, they required that the value of  fall into the range from 0 to 1 [8], [13]. Obviously, there is no justification for this range constraint from optimality standpoint. If the optimal  happens to be positive, there is no reason why its value should be less than 1. Secondly, the optimal  may turn out to be negative. The latter can be easily seen from the case of a scalar plant where the optimal  , according to (46), is equal to A c . It is expected that our method will outperform theirs, simply because a scalar gain can hardly compete with a matrix gain. The latter will be illustrated through the second numerical example given below. VI. NUMERICAL EXAMPLES We provide three numerical examples to make performance comparisons. The first example considers a general case. The second example is a continuation of the first one except that it focuses on the compensators and the theoretical total quadratic costs are computed. The third example is given to contrast performance difference be-tween the proposed method and the one by Henriksson et al. [17]. A. General Case–Example 1 In the first example, a general case is considered. The data and gains are given as follows. To differentiate between different compensators, we use subscripts “opt”, “M”, “z.i.”, and “h.i.” to denote them, which are short for optimal, the one by Moayedi et al. [8], [13], the zero-input, and the hold-input respectively.  0.61 0.53 1.30   0.12 0.55  1 0  A   1.15 0.03 0.96  , B   0.86 0.08  , C T  0 1  ,  0.78 0.24 0.02   1.16 0.60  0 0

  0.25,   0.15, Q  0.05I3 , R  0.05I 2 , P0  0.05I3  0.6280 0.0035 0.5897 Lopt   ,  0.2979 0.3699 0.9690

 0.6298 LM    0.3392  0.6063 Lz.i.    0.4009

0.0415 0.7426 , 0.5296 1.4630 0.0578 0.7758 , 0.5425 1.4471

 0.3660 0.1788 0.9636 Lh.i.   ,  0.8094 0.5915 1.2255  0.0267 0.5782 Nopt    ,   0.1196.  0.1191 0.2441 As the random noises are permanently exciting the system, we only show partial trajectories to illustrate schematically how the four compensators perform differently. See Fig. 2 for the results. The optimal scalar  is negative in this example. B. Focusing on Compensators–Example 2 Since the main difference between various compensation strategies lies in adopting different compensators, we continue the previous example here but focus on the compensators. Here, we consider the infinite horizon case with   0 and assume no process and measurement noises for ease of exposition. The theoretical cost values are computed for comparison purpose. J opt  11.5382, J M  13.5930, J z.i.  13.6579, J h.i.  24.6938. Note that it is impossible to find any positive scalar gain  that will yield a lower cost value than JM given above. C. Comparison to Henriksson et al. [17]–Example 3 The third example is to illustrate the performance difference between the proposed method and the one by Henriksson et al. [17]. An explicit model following problem is considered, where the reference state is chosen to be oscillatory and slowly convergent. Static controllers L r and L p are used with the superscripts “r” and “p” denoting the reference model and the plant respectively. See Fig. 3 for overall system configuration, where G r (z) and G p (z) represent respective transfer functions. xr   Ar 0  0 xk 1  Axk  Buk , xk   kp  , A   , B   p, p B  0 A   xk 

      J      ekT W ek  ukT U uk        xkT X xk  ukT U uk   ,  k 0   k 0  W  W   ek   I  I  xk , W  0, U  0, X   .  W W   The gains L r and L p are computed using the design technique presented in [10]. In the simulation, consecutive signal dropout is purposely added to the system. Given below are the two compensators with d – in (51) representing the latest known dk in their method [17] uk   k ukc  (1  k ) Nuk 1 , p

(50)

p

L G ( z) d . (51) 1  Lp G p ( z ) The data and the gains are given as follows. Ar  0.95, Br  0, A p  1.25, B p  1.95, x0r  1, x0p  0, uk   k ukc  (1  k )

W  1, U  0.05, Lr  0.4644, Lp  0.6329, N  0.624. From Fig. 4, one can see the big performance difference



TAC TN-14-0675

during the time period where consecutive signal dropout occurs. One possible explanation is that the latest known signal d – stored in the buffer is used directly without any modification. 40 40 35

ACKNOWLEDGEMENT The authors would like to thank the reviewers and the associate editor for their valuable comments and suggestions which greatly help improve the quality of the paper. REFERENCES

Hold-input

[1]

30 30 Zero-input

25

Moayedi et al. [8], [13]

J 020 20 20 15

[2]

This paper

[3]

10 10 5

00 0

5 5

10 10

15 15

k

20 20

Fig. 2. Performance comparison between four compensators–Example 1.

G r ( z)

xkr

dk

Lr

+

uk

G p ( z)

xkp

[4] [5] [6] [7]

 k 1

k  0

[8]

Lp

ukc

Fig. 3. Configuration of the explicit model following problem–Example3.

[9]

14

[10]

Henriksson et al. [17]

12 10

[11]

8

xkr , p

6

[12]

Reference state

4

This paper

2

[13]

0 -2

[14]

c

Consecutive loss of uk

-4 0

0.2

6


0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

k Fig. 4. Performance comparison of Henriksson et al. [17] and this paper.

VII. CONCLUSIONS In this paper we revisited the static and latest-control based compensation problem for LQG control over two lossy networks. A new framework was proposed which suggested that the original problem should be more properly posed as a two-variable instead of one-variable optimization problem. We showed that its performance went beyond that of the two popular compensation strategies: the zero-input and the hold-input, the generalized hold-input proposed by Moayedi et al. [8], [13], and the predictive one by Henriksson et al [17]. The salience of our new framework lies in the fact that it links the compensator’s gain selection to minimization of the total cost in the most general manner in its class. Three numerical examples were provided to make performance comparisons. They showed that the proposed new compensator outperformed the other four. The promising result highly supports the feasibility of the proposed framework.

[15] [16]

[17]

[18] [19] [20] [21] [22]

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