Joint 48th IEEE Conference on Decision and Control and 28th Chinese Control Conference Shanghai, P.R. China, December 16-18, 2009
WeBIn6.13
Optimal Tracking Performance of Discrete-Time Systems Over an Additive White Noise Channel Yiqian Li, Ertem Tuncel, Jie Chen and Weizhou Su Abstract— This paper studies the optimal tracking performance of multiple-input multiple-output (MIMO), finite dimensional, linear time-invariant discrete-time systems with a powerconstrained additive white noise (AWN) channel in the feedback path. We adopt the tracking error power as a measure of the performance and examine the best achievable performance by all two-parameter stabilizing controllers. In the due process, a scaling scheme is introduced as a means of integrating controller and channel design, and is optimized to better the tracking performance. In contrast to the standard setting where tracking of a step reference signal is conducted with no communication constraint, in which the tracking error can be made as zero for minimum phase plants, it is shown explicitly herein that the tracking performance will be additionally constrained by the plant unstable poles, as a consequence of noisy, power-constrained channels in the feedback loop.
I. I NTRODUCTION The study of feedback control over communication networks has drawn considerable attention in recent years (see, e.g., [1]). A simple yet somewhat typical scenario of such networked control systems is depicted in Fig. 1. It is generally believed, and intuitively known, that the performance of networked control systems can be constrained by the communication links in the feedback loop. Various information transmission constraints, such as data-rate limit, quantization precision, time delays and data packet drop-out, are all likely to have a negative effect. There has been especially notable recent work focused on the stabilization of networked feedback systems, based on different models of communication channels. For noiseless data rate limited channels, the minimal data rate for stabilization of linear systems has been derived [2]–[4], which reveals that the plant unstable eigenvalues in logarithmic magnitudes constitute a fundamental bound. Stabilization over eraser channels was also studied in, e.g., [5]. From an alternative approach, an additive white Gaussian noise (AWGN) channel model was considered in [6] and a stream of ensuing works. Similarly, for AWGN channels with a prescribed signal-tonoise ratio (SNR) constraint, the minimum SNR threshold required to stabilize a given plant has been obtained, which generally depends on the plant’s non-minimum phase zeros and unstable poles. It is worth noting that the use of an This research was supported in part by the NSF/USA under the grant number ECCS-0801874, and in part by the Natural Science Foundation of China under the grant numbers NSFC-60628301, NSFC-60834003. Yiqian Li, Ertem Tuncel and Jie Chen are with Department of Electrical Engineering, University of California, Riverside, CA 92521, USA
{yili,ertem,jchen}@ee.ucr.edu
Weizhou Su is with College of Automation Science and Engineering, South China University of Technology, Guangzhou, 510640, China
[email protected]
978-1-4244-3872-3/09/$25.00 ©2009 IEEE
r
Controller
Plant
Noisy Channel Fig. 1.
Feedback tracking over a noisy channel
AWGN channel not only preserves a system’s linearity, but also can readily translate the SNR constraint into one on the channel capacity, thus rendering the control analysis and design problems more amendable to conventional tools drawing upon linear systems theory. In spite of significant progress on stabilization problems, the study on performance issues of networked feedback systems is only beginning to emerge. In this vein, the disturbance attenuation performance was investigated in [7], which amounts to minimizing the variance of the plant output in response to a Gaussian disturbance over an AWGN channel. It was found therein that scaling may be used effectively to compensate the channel. In [8], the same authors investigated a similar problem using more general control strategies that may be nonlinear and time-varying. In this paper, we explore the fundamental limitation on the tracking performance imposed by the plant as well as by the communication channel in the aforementioned networked feedback setting. We examine the optimal reference tracking performance of MIMO, discrete-time systems with power-constrained AWN type channels in the feedback path. Inspired by [7], we also investigate a channel compensation strategy consisting of a pre- and post-processing scheme via constant scaling, which is seen as a simple way of integrating control and communication design. We show explicitly how this simple or even naive scheme may improve tracking performance, and how the scheme should be optimally designed. The idea lies in how to exploit the channel to the maximum extent allowable under the power constraint, and at the same time reduce the noise effect. We derive explicit expressions for the best achievable tracking performance in these settings and show that the plant unstable poles and the channel power constraint play crucial roles in the system’s stabilization and performance. The results demonstrate a sharp contrast to the standard feedback systems with no communication channels, for which it is known that zero tracking error can be achieved for minimum phase plants [9].
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II. N OTATION AND P ROBLEM F ORMULATION A. Notation We first describe the notation used throughout this paper. The expectation operator is denoted by E. z¯ denotes the conjugate of a complex number z. The transpose and conjugate transpose of a matrix A are denoted by AT and AH . Vectors are denoted by boldface letters. We shall assume that all the vectors and matrices have compatible dimensions. The open unit disc, the closed unit disc, the exterior of the closed unit disc and the unit circle are denoted by D , {z : |z| < 1}, ¯ , {z : |z| ≤ 1}, D ¯ c , {z : |z| > 1} and T, respectively. D In addition, k · k denotes the Euclidean vector norm and k · kF the Frobenius matrix norm. We shall consider the Hilbert space ½ L2 , G : G(z) measurable in T, ¾ Z π ° ° 1 2 jω °2 ° kGk2 , G(e ) F dω < ∞ 2π −π in which the inner product is defined as Z π © ª 1 hF, Gi , trace F (ejω )GH (ejω ) dω. 2π −π
[K1 K2 ]
P (z)
1/λ
+
yλ (k)
y(k)
λ
n(k) Fig. 2.
Feedback tracking over an AWN channel with scaling factors
The reference input is a real vector-valued random process r(k) = (r1 (k), . . . , rm (k))T which is zero-mean and WSS 2 and the power of ri (k) is denoted by σri . We assume that r(k) has a rational, positive-real power spectrum matrix Gr (z), then there exists a matrix function ψr (z) which is ¯ c , and it satisfies the relation [11] rational and analytic in D Gr (z) = ψr (z)ψrT (z −1 ). With the aid of this factorization, the power of r(k) can be expressed as σr2 , E{krk2 } = kψr (z)k22 .
¯ c is The set of those functions in L2 which are analytic in D denoted by H2 and the set of those functions in L2 which are analytic in D and equal to 0 at infinity is denoted by H2⊥ . It is an important fact that the subspaces H2 and H2⊥ form an orthogonal pair of L2 . In other words, for any F ∈ H2 and G ∈ H2⊥ , we have hF, Gi = 0. Finally, let RH∞ denote the set of all stable, proper, rational transfer function matrices.
Define the error signal by e(k) , r(k) − yλ (k). We want to jointly design the controller and the scaling factors that minimize the tracking error subject to the channel input power constraint:
B. Problem Formulation
For a rational transfer function matrix P , let its right and left coprime factorizations be given by
The discrete-time tracking system under consideration is depicted in Fig. 2, where P represents the given plant model, and [K1 , K2 ] is a general two-parameter controller, and their transfer function matrices are P (z) and K1 (z), K2 (z), respectively. We shall use the same symbol for the system and its transfer function and omit the frequency variable z whenever convenient. The signals r, n, y, yλ are the reference input, the channel noise, the system output and the channel input, respectively. The plant output is transmitted to the two-parameter controller via the AWN channel. Since the two-parameter controller represents the most general linear feedback structure, the optimal tracking error achieved by such a controller is the best possible performance ever achievable. Moreover, with two degrees of freedom, it offers us advantages countering the channel noise while minimizing the tracking error. Throughout this paper, we assume that all the signals are WSS processes and the system has reached its steady state. For an MIMO system, we assume that the AWN channel in Fig. 2 is parallel and has total input power constraint E{kyλ k2 } ≤ P which is to be distributed among each individual channel [10]. The noise n(k) = (n1 (k), . . . , nm (k))T is a vector of uncorrelated zero-mean white noise processes ni (k), with power spectral density Φi , 1 ≤ i ≤ m.
inf
λ>0,K∈K
E{kek2 }, subject to E{kyλ k2 } ≤ Γ.
(1)
C. Preliminaries
˜ ˜ −1 N P = N M −1 = M
(2)
˜, M ˜ ∈ RH∞ and satisfy the double Bezout where N, M, N identity ¸· ¸ · ˜ X −Y˜ M Y =I (3) ˜ M ˜ N X −N ˜ Y˜ ∈ RH∞ . The set of all stabilizing twofor some X, Y, X, parameter compensators can be characterized by the Youla parameterization [12] using the factorization (2) £ ¤ ˜ − RN ˜ )−1 × K = {K : K = K1 K2 = (X £ ¤ ˜ , Q, R ∈ RH∞ }. (4) Q Y˜ − RM We shall impose the following assumptions. Assumption 1: P is right-invertible. Assumption 1 makes sure that the number of input to the plant is at least as many as that of the output, which is necessary for the tracking problem to be well-posed [9]. Assumption 2: P does not have non-minimum phase zeros. Thus we will only focus on the performance limitations imposed by the channel and the plant unstable poles.
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¯ c , i = 1, . . . , Np . Assume that P has unstable poles pi ∈ D ˜ Then it is possible to factorize M (z) such that ˜ (z) = M ˜ m (z)B(z) ˜ M
[K1 K2 ]
(5)
˜ m (z) is minimum phase, and B(z) ˜ where M is all-pass and collects all the unstable poles. Define p p Θ , diag( Φ1 , . . . , Φm ) which collects the noise levels. We shall also need a similar factorization ˜Θ . ˜Θ = M ˜ Θm B M (6) ˜ Θm is minimum phase and admits right inverse The matrix M ¯ ˜Θ (z) is all-pass and can be constructed analytic in Dc . And B as ˜Θ , B ˜ΘN B ˜Θ(N −1) · · · B ˜Θ1 , B (7) p p · z−pi ¸ · H¸ £ ¤ ˜Θi , ωi Wi 1−p¯i z 0 ωi H . B (8) 0 I Wi ³ ´T (1) (m) The unitary vector ωi , ωi , . . . , ωi can be computed sequentially from the pole direction vectors of P , and Wi forms a unitary matrix together with ωi [13].
+ n(k) Fig. 3.
2
2
E{kyk } = where
2 kN Qψr k2
2 kT Θk2
+
(10)
,
(11)
˜N ˜r − R)M ˜. T , I − N (X
The problem is inf E{kek2 }, subject to E{kyk2 } ≤ Γ.
In this section, we study the optimal reference tracking performance over the AWN channel. We assume that the plant is sending the signal “as is” to the channel without any scaling, filtering or coding, as is shown in Fig. 3. The following theorem characterizes the optimal tracking performance in the present setting. Theorem 1: Assume that the plant P (z) has m outputs ¯ c , i = 1, . . . , Np . Define and has simple unstable poles pi ∈ D ° ³ ´°2 ° ˜ −1 − B ˜ −1 (∞) ° η , °Θ B ° . Θ Θ
Let He∗ denote the optimal value. To solve the problem, we first form the Lagrangian H(K, ²) , (1 − ²)E{kek2 } + ²(E{kyk2 } − Γ)
Then the system is stabilizable if and only if Γ > η and the best tracking performance under channel input power constraint is given by ³ ´2 p 2 √ σ − Γ − η + η if η < Γ < η+σr2 , r He∗ = (9) η if Γ ≥ η+σr2 . Proof: To begin with, observe that the transfer functions from n and r to e and y are formed as (with a slight abuse of notation) ³ ´ −1 −1 e = I − (I − P K2 ) P K1 r − (I − P K2 ) P K2 n, −1
1 inf H(K, ²). 1 − ² K∈K 0≤²≤1
He∗ = sup
(13)
From (10) and (11), (12) can be expressed as °·√ ¸ °2 ° 1 − ²(I − N Q) ° √ inf H(K, ²) = inf ° ψr ° ° ° ²N Q K²K Q²RH∞ 2 +
inf
R∈RH∞
2
kT Θk2 − ²Γ.
It is clear that the optimization over all stabilizing controllers boils down to two independent problems. Define °·√ ¸ °2 ° 1 − ²(I − N Q) ° √ H1 , ° ψr ° ° ° ²N Q 2 and H1∗ , inf Q∈RH∞ H1 . Similarly,
P K2 n.
With the aid of the Youla parameterization (4), we can further write the above transfer functions as ³ ´ ˜N ˜r − R)M ˜ n, e = (I − N Q) r + I − N (X ³ ´ ˜N ˜r − R)M ˜ n, y = N Qr + −I + N (X
(12)
where 0 ≤ ² ≤ 1 and calculate the infimum of H(K, ²) over all stabilizing controllers. The quantity gives a lower bound 1 on the best tracking performance, i.e. 1−² inf K∈K H(K, ²) ≤ ∗ He . Because of the convexity of the H2 optimization problem, the equality can be achieved by some ² and
2
P K1 r + (I − P K2 )
2
E{kek2 } = k(I − N Q) ψr k2 + kT Θk2 ,
K∈K
A. Optimal Tracking without Scaling
−1
Feedback tracking over an AWN channel
˜r is the right inverse of N ˜ . Since the reference signal where N we consider is a WSS random process, the power of e and y can be related to the L2 norm of the corresponding transfer function and the power spectral density matrices of n and r. Specifically,
III. M AIN R ESULTS
y = (I − P K2 )
y(k)
P (z)
2
H2 , kT Θk2 . and H2∗ , inf R∈RH∞ H2 . A direct calculation reveals that °µ·√ ¸ ¶ °2 ¸ · √ ° ° 1 − ²I − √1 − ²I ° N Q ψr ° H1 = ° + ° . ²I 0 2
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WeBIn6.13 It is useful to perform an inner-outer factorization [14] such that · √ ¸ − √1 − ²I N = ∆ i ∆o ²I Since the plant is minimum-phase, N is also minimumphase. Therefore it is easy to see that ∆o = N and · √ ¸ − √1 − ²I ∆i = . ²I Introduce
¸ ∆Ti (z −1 ) , Ψ(s) , I − ∆i (z)∆Ti (z −1 ) ·
then we have ΨH (ejω )Ψ(ejω ) = I. It follows that ° µ·√ ¸ ¶ °2 ¸ · √ ° ° 1 − ²I − √1 − ²I H1∗ = inf ° Ψ + N Q ψr ° ° ° 0 ²I Q²RH∞ 2 ° ·√ ¸ °2 ° ° ¡ ¢ 1 − ²I ˆ 1∗ + ° I − ∆i ∆H =H ψr ° (14) i ° ° 0 2 = ²(1 − ²)σr2
(15)
since ˆ 1∗ , H
°µ ·√ ¸ ¶ °2 ° ° 1 − ²I ° ∆H + N Q ψr ° i ° ° =0 0 Q∈RH∞ 2 inf
by properly choosing Q ∈ RH∞ such that N Q → (1 − ²)I.
(16)
˜Θ is a unitary operator in On the other hand, noting that B L2 , we have °³ ´ °2 ° ˜N ˜r − R)M ˜ Θ° H2 = ° I − N (X ° 2 ° °2 ° ° ˜ ˜ ˜ ˜ = °Θ − N (X Nr − R)MΘm BΘ ° . 2
It follows that H2∗
rk
inf H(K, ²) = ²(1 − ²)σr2 + η − ²Γ.
(21)
kΘ
ωk
r
B. Optimal Tracking with Scaling Factor
K∈K
By setting ² = 1, we obtain the stabilizability condition. Define α , ²/(1 − ²) as the Lagrange multiplier. It follows from (13) that He∗ = sup {φ(α)} , (17) α>0
α σ 2 + (1 + α)η − αΓ. 1+α r
It is easy to verify that ∂φ(α) 1 2 = 2 σr + η − Γ, ∂α (1 + α)
Therefore, the channel is not exploited to the maximum extent. Thus, to improve the performance, in Section III-B we adopt a simple scaling scheme that guarantees the power of the channel input matches the constraint, and design jointly the scaling factor and the controller. Additionally, it is interesting to examine the power allocation among the parallel AWN channels under the optimal control scheme. The optimal allocation for general linear plants is difficult to calculate. Instead, we consider a special plant with only one real unstable pole: Corollary 1: Assume that the plant P has one real ¯ c and associated direction ω = unstable pole p ∈ D ¢ ¡ (1) T ω , . . . , ω (m) . Then,
and the power distributed to the kth channel is given by P −η σ 2 + (p2 −1)ω(k) , if η < Γ < η+σ 2 , rk r σr2 kΘ−1 ωk2 2 (k) 2 σ 2 + (p −1)ω if Γ ≥ η+σ . −1 2 ,
2
which is due to an appropriate selection of R ∈ RH∞ . Consequently, we arrive at
φ(α) =
Substituting α in φ(α), we obtain the best tracking performance. Remark 1: Theorem 1 reveals how the channel noise level and the power constraint may affect the tracking performance. It is clear that, when Γ is increasing in the interval (η, η +σr2 ), the tracking error is decreasing toward η. However, when Γ goes beyond that range, the tracking error can not be made smaller than η even though Γ is allowed to be infinite; in other words, it is no longer possible to recover the ideal zero tracking error. This performance downgrade is partly due to the insufficient use of the channel: the plant is sending the output signal directly. The following calculation supports this claim. From (11), (16) and (19), the power of the channel input is calculated to be ( Γ, if Γ < η+σr2 , 2 (20) E{kyk } = η + σr2 , if Γ ≥ η+σr2 .
η = (p2 − 1)/kΘ−1 ωk2
° ³ ´°2 ° ˜ −1 − B ˜ −1 (∞) ° = °Θ B ° =η Θ Θ
with
which is monotonically decreasing function with respect to α. When the system is stabilizable, by setting the derivative to 0, we obtain the optimal α (q 2 σr 2 Γ−η − 1 if η < Γ < η+σr , (19) α∗ = 0 if Γ ≥ η+σr2 .
(18)
In Section III-A, we have shown that the tracking performance is severely limited by the channel characteristics. A primary drawback with the configuration is that using channel with much weaker power constraint in the system may not upgrade the performance. Thus, to resolve this problem, in the configuration in Fig. 2, we shall include scaling factor, which offers one additional degree of freedom in design. Denote by Je∗ the minimal tracking error, the next theorem shows that the tracking performance is improved by taking this simple strategy: Theorem 2: Assume that P (s) has m outputs and that it has simple unstable poles pi ∈ C+ , i = 1, . . . , Np . The
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λ∗ = p
σr2 (Γ
− η)
.
(23)
Proof: The proof is analog to that of Theorem 1, therefore we shall mainly provide some key steps. The problem statement is given by (1). Because of the convexity of the optimization problem, we can minimize the tracking error by first minimizing over all stabilizing controllers K, and then minimizing over λ > 0 [15]. Thus, Je∗ = inf inf E{kek2 }. λ>0 K∈K
(24)
We first form the Lagrangian J(K, ², λ) , (1 − ²)E{kek2 } + ²(E{ky λ k2 } − P )
(25)
where 0 ≤ ² ≤ 1. Let Je (λ) denote the optimal tracking error for a given λ, then Je (λ) = sup
0≤²≤1
1 inf J(K, ², λ). 1 − ² K∈K
(26)
It follows from (24) that Je∗ = inf Je (λ). λ>0
The power of the error signal and channel input can be expressed as 1 2 kT Θk2 , λ2 2 2 E{kyλ k2 } = λ2 kN Qψr k2 + kT Θk2 . 2
E{kek2 } = k(I − N Q) ψr k2 +
(27) (28)
Then °·√ ¸ °2 ° ° 1 − ²(I − N Q) ° √ ψr ° inf J(K, ², λ) = inf ° ° λ ²N Q K²K Q²RH∞ 2 ¶ µ 1−² 2 + +² inf kT Λk2 − ²Γ. R∈RH∞ λ2
In addition, for single-input single-output (SISO) system, the tracking performance is inversely proportional to the channel SNR, as shown in the following corollary. Corollary 3: Let P (z) be a scalar transfer function. Under the assumptions in Theorem 2, the system is stabilizable if and only if Np Y P 2 > |pi | − 1 . Φ i=1 Under this condition, we have Np Y Φ 2 Je∗ = σr2 |pi | − 1 . P i=1
A further calculation reveals that µ ¶ λ2 ²(1 − ²) 2 1−² inf J = 2 σ + + ² η − ²Γ. K∈K λ ²+1−² r λ2
IV. C ONCLUSIONS
Define α , ²/(1 − ²) as the Lagrange multiplier. It follows from (26) that ¾ ½ 1 αλ2 2 σ + 2 η + α(η − Γ) . Je (λ) = sup 2 r λ α>0 1 + αλ And in fact
scaling this time. By designing the scaling factor, the power of the channel input is always equal to Γ. This leads to better performance than Theorem 1, as Je∗ < He∗ . As we have mentioned in the remark of previous section, the unstable poles do not affect the tracking performance of a unit step signal in a two-parameter configuration [9]. It is also a well-known fact that for minimum phase plants, use of two-parameter controllers will in general result in perfect tracking (i.e., zero tracking error) over noise-free feedback. Theorem 2 shows that only in the limit when Γ → ∞, i.e., when the channel has no input power constraint, the system achieves zero tracking error. Thus, in the presence of AWN channel, an additional limitation on the tracking performance results, and that the best tracking performance now depends explicitly on the power constraint, other than on the plant unstable poles and the noise strength. In particular, this additional limitation persists even when an additional design freedom is made available in selecting the scaling factor. We would like to examine the power allotment for the parallel AWN channels in the same spirit of Corollary 1. Corollary 2: For plant having one real unstable pole p ∈ ¡ ¢ ¯ c and associated direction ω = ω (1) , . . . , ω (m) T , η is D given by (21), and the power distributed to the kth channel is (p2 − 1)ω (k) P −η 2 σ + . σr2 rk kΘ−1 ωk2
p 2 (Γ − η)σr2 Γ Je (λ) = 2 − + σr2 , λ λ which achieves its minimum σr2 η/P at (23). The corresponding channel input power can be shown to be equal to Γ. Remark 2: Theorem 2 highlights again the important role played by the communication channel in tracking, with
In this paper we have investigated the best tracking performance of discrete-time MIMO control systems over an AWN channel with input power constraint. We have derived explicit expressions of the minimal tracking error for systems with or without scaling factors across the channel. The channel is not always fully exploited when the plant output signal is sent directly to the channel without scaling and thus severely limits the tracking performance. By an optimal design of the scaling factor and the controller, which always adjusts the power of the channel input to match the constraint, the performance can be significantly improved. The results also demonstrate that while the tracking error can be made as zero in a standard, noise-free setting for minimum phase plants, in the presence of noise and power constraint, an additional
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WeBIn6.13 limitation is imposed on the tracking performance, which is generally nonzero and depends on the plant unstable poles, the noise level and the channel power constraint. Finally, for the tracking problem under consideration, we derived explicit optimal power allocation strategy for a special plant. R EFERENCES [1] P. Antsaklis and J. Baillieul, “Special issue on networked control systems,” IEEE Trans. Autom. Control, vol. 49, no. 9, Sep. 2004. [2] S. Tatikonda and S. M. Mitter, “Control under communication constraints,” IEEE Trans. Autom. Control, vol. 49, no. 7, pp. 1056–1068, Jul. 2004. [3] G. N. Nair and R. J. Evans, “Exponential stabilisability of finitedimensional linear systems with limited data rates,” Automatica, vol. 39, no. 4, pp. 585–593, Apr. 2003. [4] ——, “Stabilizability of stochastic linear systems with finite feedback data rates,” SIAM J. Control Optim., vol. 43, no. 2, pp. 413–436, Jul. 2004. [5] L. Schenato, B. Sinopoli, M. Franceschetti, K. Poolla, and S. S. Sastry, “Foundations of control and estimation over lossy networks,” Proc. of the IEEE, vol. 95, no. 1, pp. 163–187, Jan 2007. [6] J. H. Braslavsky, R. H. Middleton, and J. S. Freudenberg, “Feedback stabilization over signal-to-noise ratio constrained channels,” in Proc. 2004 Amer. Control Conf., Boston, MA, Jun. 2004, pp. 4903–4908.
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