Systems & Control Letters 58 (2009) 353–358 Contents lists available at ScienceDirect Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle Signal-to-noise ratio performance limitations for input disturbance rejection in output feedback control Alejandro J. Rojas ∗ ARC Centre of Excellence for Complex Dynamic Systems and Control, The University of Newcastle, Callaghan NSW 2308, Australia article info Article history: Received 22 July 2008 Received in revised form 10 December 2008 Accepted 5 January 2009 Available online 1 February 2009 Keywords: Signal to noise ratio Control over communication networks Performance limitations Input disturbance rejection Stability of linear systems a b s t r a c t Communication channels impose a number of obstacles to feedback control. One recent line of research considers the problem of feedback stabilization subject to a constraint on the channel signal-to-noise ratio (SNR). We use the spectral factorization induced by the optimal solution and quantify in closedform the infimal SNR required for both stabilization and input disturbance rejection for a minimum phase plant with relative degree one and memoryless additive white Gaussian noise (AWGN) channel. Finally we conclude by presenting a closed-form expression of the difference between the infimal AWGN channel capacity for input disturbance rejection and the infimal AWGN channel capacity required only for stabilizability. © 2009 Elsevier B.V. All rights reserved. 1. Introduction Fundamental limitations in control design have been an important area of research for many years, [1,2]. Recently, the study of fundamental limitations has been extended to problems of control over communication networks, see for example [3, Theorem 4.6], , as well as the special issue  and the recent survey by Nair et al. . Communication channels impose additional limitations to feedback, such as constraints in transmission data rate and bandwidth, and effects of noise and time-delay. One line of recent research introduced a framework to study stabilizability of a feedback loop over channels with a signal to noise ratio (SNR) constraint [7,8]. These papers obtained the infimal SNR required to stabilize an unstable linear time invariant (LTI) plant over an additive white Gaussian noise (AWGN) channel. A distinctive characteristic of the SNR approach is that it is a linear formulation, suited for the analysis of robustness using well-developed tools . For the case of LTI controllers and minimum phase plant models with no time delay, these conditions match those derived in  by application of Shannon’s theorem [11, Section 10.3]. Different techniques are used in [7,8] depending on whether stabilization is achieved by state feedback or by output feedback. A common framework for both state and output feedback cases is proposed in , where it is shown that both problems can ∗ Tel.: +61 02 4916023. E-mail address: [email protected]. 0167-6911/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2009.01.001 be solved as a linear quadratic Gaussian (LQG) optimization. Specifically, the optimal control problem arising from the infimal SNR constrained control problem may be posed as an LQG optimization with weights chosen as in the loop transfer recovery (LTR) technique (see [13–17]). Doing so not only allows a unified treatment of the state and output feedback cases, but also suggests how performance considerations may be analyzed in addition to stabilization. The first contribution of the present paper is to obtain the infimal SNR constrained solution in closed-form for stabilizability and input disturbance rejection in the case of memoryless AWGN channels. The second contribution is that by means of the infimal SNR constrained solution we quantify the difference between the infimal channel capacity Ĉ for performance and the infimal channel capacity required for stabilizability only, see . This difference between channel capacities has been identified in , based on information theoretic arguments, as a key element representing a fundamental limitation in control over network performance. To the best knowledge of the author the channel capacity difference imposed by the input disturbance rejection has not been quantified in closed-form before. Of the two possible configurations for the location of the idealized communication channel, we consider the case of an AWGN communication channel over the measurement link. Such a setting is common in practice and arises, for example, when sensors are far from the controller and have to communicate through a communication network. We neglect all pre- and post- signal processing involved in the communication link, which is then reduced only to the 354 A.J. Rojas / Systems & Control Letters 58 (2009) 353–358 Plant model assumptions: throughout the present work, if not stated otherwise, it is assumed that the plant model G(z ) is a strictly proper rational function with the following properties: – relative degree ng = 1, – all its zeros have moduli less than 1, – m unstable poles, |ρi | > 1, each with multiplicity ni , ∀i = 1, . . . , m. Channel additive noise process: the channel additive noise process is labeled n(k) and it is a zero-mean i.i.d. Gaussian white noise process with variance σ 2 . Input disturbance process: the input disturbance process is labeled d(k) and it is a zero-mean i.i.d. Gaussian white noise process with variance σd2 . Fig. 1. Feedback control stabilization of a discrete-time unstable plant subject to input disturbance over a discrete-time AWGN channel. Notice that if we lift the assumption of G(z ) minimum phase, then we would be required to invoke an inner factorization argument similar to the one presented in [16,19]. communication channel itself (see Fig. 1). The reason for this is motivated by the goal of quantifying the fundamental limitations imposed by the communication channel. The results in , which also do not consider explicitly encoder and decoder, give strength to the argument by agreeing with the results in , which considers explicitly an encoder and decoder. The paper is organized as follows: Section 2 introduces the paper assumptions and the problem definition. In Section 3 we proceed to quantify the infimal SNR and the channel capacity difference for stabilizability and input disturbance rejection in the case of memoryless AWGN channels, as in Fig. 1. In Section 4 we conclude with final remarks on the present work. A preliminary version of the present results has been communicated in . Terminology: Let D− , D̄− , D+ and D̄+ denote respectively the open unit-disk, closed unit-disk, open and closed unit-disk complements in the complex plane C, with ∂ D the unit-disk itself. Let R denote the set of real numbers, R+ the set of positive real − numbers, R+ o the set of non-negative real numbers and R the set + of real negative √ numbers. Let Z denote the set of positive integers. Define j = −1. A discrete-time signal is denoted by x(k), k = 0, 1, 2, . . ., and its Z-transform by X (z ), z ∈ C. The expectation operator is denoted by E . A rational transfer function of a discrete-time system is minimum phase if all its zeros lie in D̄− , and is non minimum phase if it has zeros in D+ . We define the H∞ space as a (closed) subspace of L∞ with functions that are analytic and bounded in D+ . The RH∞ space consists of all proper and real rational stable transfer functions. The norm of a system P (z ) in H∞ is given by kP k∞ = supθ ∈[−π,π) P (ejθ ) . In the discrete-time setting we define the L2 space of functions f : ej[−π,π) → C such R πas the jspace 1 2 θ 2 that kf k2 = 2π −π |f (e )| dθ < ∞. Define the H2 space as a (closed) subspace of L2 with functions f (z ) analytic in D+ . Finally, we define the H2⊥ space as the orthogonal complement of H2 in L2 , that is the (closed) subspace of functions in L2 that are analytic in D− . If a is in C, ā represents its complex conjugate and aH = āT the Hermitian (i.e., the transposed complex conjugate of a). By general convention we have 0! = 1. 2.2. Problem formulation 2. Assumptions and problem formulation Problem 1 (Infimal SNR for Stabilizability with Input Disturbance Rejection LTI Problem). Find a proper rational stabilizing controller C (z ) such that the feedback control loop is stable and the transfer functions in (2) achieve the infimum possible constraint (3) imposed on the admissible channel SNR. We consider the discrete-time feedback system depicted in Fig. 1. The AWGN channel is characterized by two parameters: the admissible input power level of the channel, P , and the channel additive noise process n(k). 2.1. Assumptions General assumptions involved in the present discussion, which will be in place unless stated otherwise, are We assume that C (z ) is such that the closed loop system is stable in the sense that, for any distribution of initial conditions, the distribution of all signals in the loop will converge exponentially rapidly to a stationary distribution. The channel input power, see Fig. 1, is required to satisfy an arbitrary imposed power constraint P > E y2 (1) for some predetermined power level P , where E y 2 stands for limk→∞ E y2 (k) and it is introduced to ease the notation. Under the assumption of stationarity, as in [20, Section 4.4], the power in the channel input may be computed as 2 = E y 1 π Z 2π jω |Tyn (e )| σ dω + 2 2 −π Z 1 2π π |Tyd (ejω )|2 σd2 dω, −π where Tyn (z ) = − C (z )G(z ) 1 + C (z )G(z ) , Tyd (z ) = G(z ) 1 + C (z )G(z ) , (2) are the transfer functions that relate y(k) with n(k) and y(k) with d(k). Since the feedback control system is stable and recalling the definition of the L2 norm, we have E y2 = Tyn 2 2 σ 2 + Tyd 2 2 σd2 . Thus, the power constraint (1) at the input of the channel translates into a channel SNR bound given by the H2 norm of Tyn (z ) and Tyd (z ) P σ2 > Tyn 2 + Tyd 2 2 2 σd2 . σ2 (3) An expression similar to Eq. (3) can be found in [21, Section VI] for output disturbance rejection. From (3) we observe that a fundamental limitation on the channel SNR will be given by the infimum of 2 Tyn 2 2 and Tyd 2 . This observation is the foundation of the infimal SNR for stabilizability with input disturbance rejection LTI problem. The search for the optimal stabilizing controller Ĉ (z ) that achieves the infimal H2 norm of Tyn (z ) and Tyd (z ) can be performed via the machinery of LQG estimation with LTR at the output. The procedure of LQG optimization with LTR at the output involves the solution of two Riccati equations, one associated A.J. Rojas / Systems & Control Letters 58 (2009) 353–358 with the design of the observer and another with the design of the regulator. If we were to perform the full design for the output feedback loop we would have to design two pairs of weighting matrices, one pair for the observer’s Riccati equation and a second pair for the regulator Riccati equation. The LTR procedure simplifies the LQG design by pre-assigning the weights for the regulator Riccati equation as a cheap control design. It is well known that as a result of the LQG/LTR approach, for a minimum phase plant model with relative degree one, the output feedback optimal sensitivity function Ŝ (z ) = 1/(1 + G(z )Ĉ (z )) recovers the observer’s design. That is, for a minimum phase plant model with relative degree one, we are able to recover the design for the observer at the output. This remark is very useful since a simple spectral factorization analysis can now be applied to obtain the closed loop characteristic polynomial whenever the infimal controller Ĉ (z ) is in place, see for example [22, Section 6.4.3] and also [18, Section V]. 3. Infimal channel signal to noise ratio for input disturbance rejection of the optimal Ŝ (z ) is given by Ŝ −1 (z ) ρi i=1 zi n i ! Ŝ −T (z −1 ) = 1 + G(z ) σ σ 2 d G 2 (z −1 ). G(z ) = q(z ) p(z ) = q(z ) m Q . (5) The polynomial q(z ) is assumed known, with degree n1 + n2 +· · ·+ nm − 1 and all its solutions are in D− . As we stated before, we are ultimately attempting to characterize the particular Ŝ (z ) that takes part into the infimal SNR for stabilizability with input disturbance rejection LTI solution. Notice that Ŝ (z ) must contain the m unstable plant poles ρi , including multiplicity, as non minimum phase (NMP) zeros to guarantee the internal stability of the closed loop. Thus, we only have to obtain from (4) the location of the poles of Ŝ (z ) Ŝ −1 (z ) m Y ρi ni i=1 = zi ! Theorem 2 (Infimal SNR for Stabilizability with Input Disturbance Rejection). Assume the plant to be as in (5) and the channel model to be a memoryless AWGN channel as in Fig. 1. Then, for the feedback loop to be stabilizable and ensure input disturbance rejection, the channel SNR must satisfy P σ2 p(z )p(z −1 ) + q(z ) p(z )p(z −1 ) q(z −1 ) . (6) From (6) we recognize that the poles of Ŝ (z ), labeledP zi , zi ∈ D− i = m 1, · · · m, with multiplicities n1 , n2 , · · · , nm , are the i=1 ni Schur’s solutions of p(z )p(z −1 ) + q(z ) > m Y (ρi )2ni − 1 + (ρi )2ni i =1 i =1 × m Y ni m X X i=1 l=1 nj m X X d l −1 mj,p z p−1 (l − 1)! j=1 p=1 dz l−1 (1 − z z̄j )p mi,l z =zi nj ni m X m X X d l −1 σ X gi,l gj,p z p−1 + σ i=1 l=1 (l − 1)! j=1 p=1 dz l−1 (1 − z z̄j )p 2 d 2 , z =zi (9) where mi,l = m Q (z − 1/ρj ) Q m nj (z − zj ) nj d n i −l j =1 1 (ni − l)! dz ni −l j=1 j6=i gi,l = d n i −l 1 (ni − l)! dz ni −l , (10) z =zi (q(z ))|z =zi . Proof. We proceed by considering the function spaces L2 , H2 , H2⊥ , and RH∞ , with the stability region given by the open unit disk in the complex plane. Introduce a coprime factorization such that G(z ) = N (z )/M (z ), where N (z ) = q(z ) m Q n i m Y z − ρi M (z ) = , 1 − z ρi i =1 , (1 − z ρi )ni i=1 and the parameterization of all stabilizing controllers (see [23, pp. 64–65]) C (z ) = (X (z ) + M (z )Q (z )) / (Y (z ) − N (z )Q (z )), Ŝ −T (z −1 ) σd2 σ2 (8) We stress that, although we do not have a closed-form for each zi , they can be computed by any of the many currently available algorithms for the purpose of finding the solutions of a polynomial, thus for all purposes we consider them as known quantities. Finally notice, also from (7), that as σd2 → 0, each zi will tend to one of the unstable plant poles mirrored images 1/ρi . By means of (8) we are able to quantify the infimal SNR for stabilizability with input disturbance rejection. (z − ρi )ni i =1 . z − zi i=1 (4) From Eq. (4) we have that the plant model G(z ), together with σ 2 and σd2 , will determine Ŝ (z ). Notice though that the stable poles of G(z ) will also play a role in (4), as different from the case of stabilizability with no input disturbance rejection where only the unstable poles of G(z ) played a role (see ). We now quantify the infimal SNR required for stabilizability and input disturbance rejection for the case of memoryless AWGN channels. To do so we use the spectral factorization result in (4) and specify the plant model to be n m Y z − ρi i Ŝ (z ) = Consider that Ĉ (z ), the controller that solves Problem 1, is in place. It is possible to verify that such controller will induce a spectral factorization given by m Y 355 σd2 −1 q(z ) = 0, σ2 Pm where X (z ) and Y (z ) satisfy the Bezout identity, N (z )X (z ) +M (z )Y (z ) = 1 and Q (z ) is the Youla parameter, see for example . From (2) and (3) we have P σ2 > kTyn k22 + σd2 kTyd k22 . σ2 Notice that for the infimal solution with input disturbance rejection S (z ) is equal to Ŝ (z ), thus 2 (7) a polynomial of degree 2 i=1 ni . It can be shown that the Pm other i=1 ni solutions of (7) are the reflections of each zi , that is 1/zi , 1/zi ∈ D+ i = 1, · · · m. We have then that a characterization P σ2 > kNX + NM Q̂ k22 + σ σ2 q(z ) 2 d m Q . (z − zi ) i =1 ni 2 (11) 356 A.J. Rojas / Systems & Control Letters 58 (2009) 353–358 We start by analyzing the term NX + NM Q̂ which can be factorized as kNX + NM Q̂ k22 = kM −1 NX + N Q̂ k22 , (12) since M (z ) is all-pass. Introduce now the following decomposition M −1 (z )N (z )X (z ) = Γ ⊥ (z ) + Γ (z ), where Γ ⊥ is in H2 and Γ is in H2 and replace in (12) kNX + NM Q̂ k22 = kΓ ⊥ k22 + kΓ + N Q̂ k22 . (13) The term kΓ ⊥ k22 is the infimal SNR for stabilizability with no input disturbance rejectionQ which, from applying Theorem III.1 in , m we know is equal to i=1 (ρi )2ni − 1. Also directly from (13) we can observe that the Youla parameter Q̃ (z ) achieving the infimal SNR for stabilizability with no input disturbance rejection would be given by −Γ (z )N −1 (z ), since this choice would ensure the squared H2 norm of Γ + N Q̃ to be zero. Therefore we can observe that m Y Reintroduce now M (z ), since it is all-pass, into the squared H2 norm term and add and subtract the term N (z )X (z ) m Y (ρi )2ni − 1 Rearrange terms and recognize T̃ (z ) = NX + NM Q̃ and T̂ (z ) = NX + NM Q̂ kNX + NM Q̂ k22 = Since T̂ (z ) = 1 − Ŝ (z ) and T̃ (z ) = 1 − S̃ (z ) and S̃ (z ) = Ŝ (z )|σ 2 =0 = i =1 z −ρi z −1/ρi ni d we have kNX + NM Q̂ k22 = m Y n m Y z − ρi i n i m Y z − ρi − z − 1/ρi i=1 z − zi i =1 (ρi )2ni − 1 + i=1 . 2 Q m Y (z − 1/ρi )ni − i =1 m Q 2 (z − zi )ni i =1 m Q . (14) (z − zi )ni 2 Applying partial fraction expansion on the term inside the RHS squared H2 norm and then invoking the Residue Theorem (see for example [25, pp. 169–172]) we obtain m Y (ρi )2ni − 1 + i =1 m × ni XX mi,l i =1 l =1 (l − 1)! ρ+ 1 ρ + K 2 σd2 ρσ 2 − r −ρ − 1 ρ − K 2 σd2 ρσ 2 2 −4 2 , (17) 1 ρ + K 2 σd2 ρσ 2 > 2, m Y (ρi )2ni i =1 m nj XX d j=1 p=1 l −1 dz l−1 mj,p z p−1 (1 − z z̄j )p , z =z i (15) > ρ2 − 1 + ρ2 (z1 − 1/ρ)2 σd2 K 2 + , σ 2 1 − z12 1 − z12 (18) from which we observe that as σd2 → 0, then z1 → 1/ρ and P (ρi )2ni i =1 kNX + NM Q̂ k22 = z1 = σ2 i=1 m × ρ+ σ2 2 By recognizing and extracting M (z ) we can claim that m Y K 2σ 2 z 2 + −ρ − ρ1 − ρσ 2d z + 1 σd2 1 + G(z ) 2 G(z −1 ) = , σ (z − ρ) z − ρ1 P i=1 kNX + NM Q̂ k22 = Theorem 2 quantifies the infimal SNR for stabilizability with input disturbance rejection. Notice how if σd2 = 0, then the last term in the RHS of (9) vanishes, whilst the second term vanishes as well since zi = 1/ρi in (14), and we are left with the infimal LTI SNR for stabilizability as in [8, Theorem III.1]. Let us introduce a simple example in which we can explicitly account for the pole of Ŝ (z ). as long as ρ ∈ R+ , which in turn allows us to claim that z1 < 1. With z1 known, applying Theorem 2 gives (ρi )2ni − 1 + with gi,l as in (10). Finally we observe that the result in (15), plus the result in (16), give the expression in (9) which concludes the proof. the only solution that satisfies z1 ∈ D− . This can be seen to be true from the fact that (ρi )2ni − 1 + kT̂ − T̃ k22 . i =1 Qm (16) therefore z1 , the solution for the numerator and pole of Ŝ (z ), is given by i=1 + k − NM Q̃ + NM Q̂ + NX − NX k22 . m Y , z =zi Example 3. Consider the plant to be G(z ) = K /(z − ρ), with ρ ∈ R+ , ρ > 1 and K ∈ R+ . From (4) we have that (ρi )2ni − 1 + k − N Q̃ + N Q̂ k22 . i =1 kNX + NM Q̂ k22 = Qm nj ni m X m X X gi,l σd2 X gj,p z p−1 d l −1 σ 2 i=1 l=1 (l − 1)! j=1 p=1 dz l−1 (1 − z z̄j )p ⊥ kNX + NM Q̂ k22 = ni with mi,l as in (10). Notice that the term in the i=1 (z − zi ) numerator of the squared H2 norm in (14) does not contribute to the residue since it and its derivatives in z are zero at each z = zi . We now focus on the second term in (11), for which applying partial fraction expansion and then invoking the Residue Theorem gives > ρ 2 − 1, the infimal LTI SNR for stabilizability with no input disturbance rejection result, see for example [8, Theorem III.2]. As a comparison with the LQG/LTR approach, consider now the input disturbance variance σd to be in [0, 10], the plant unstable pole ρ to be 2, the plant gain K to be 10 and the channel noise variance σ 2 to be 1. In Fig. 2 we can observe the difference between z1 obtained as in Eq. (17) and z1 obtained from equation an LQG/LTR approach. Since the expression in (17) for z1 in this example is exact, any difference shown in Fig. 2 can be in principal adjudicated to possible numerical issues involved in the LQG/LTR approach. As σd increases so does the numerical error value. Also, since there is a numerical error in z1 , there will be a numerical error in the infimal SNR expression as in (18), although in the present case we do not show it since it falls below the eps precision number of 2.2204 · 10−016 for Matlab. Remark 4. Notice from the previous example that, by extending the stabilizability problem to input disturbance rejection, the gain of the plant model K also plays a role in the performance limitation. A.J. Rojas / Systems & Control Letters 58 (2009) 353–358 357 capacity difference, which is then given by Ĉ − log2 ρ = 1 2 log2 1+ (z1 − 1/ρ)2 σ2 K2 + d2 2 σ (1 − z12 )ρ 2 1 − z1 . Observe that, as expected, when σd2 = 0 (that is no input disturbance process is present), the RHS of the above expression is zero and the channel capacity matches the infimal channel capacity for stabilizability of log2 ρ . 4. Conclusion and remarks Fig. 2. Difference between z1 obtained from Eq. (17) and z1 obtained from equation an LQG/LTR approach. The capacity of a communication channel C , defined as the maximum of the mutual information between the channel input and output (see [11, p. 241]), is also used to characterize a communication channel. For the case of a memoryless AWGN 1 2 log2 1 + σP2 bits per transmission, and is thus completely determined by its SNR. Notice that, as stated in , the presence of feedback does not increase the capacity of a memoryless AWGN channel. channel, the channel capacity is given by C = Corollary 5 (Channel Capacity Difference). Consider a plant model as in (5) and a memoryless AWGN channel as in Fig. 1. Then the infimal channel capacity Ĉ for stabilizability with input disturbance rejection must satisfy Ĉ − m X ni log2 ρi = i=1 1 2 log2 1 + γ1 + σd2 γ2 , m 2 σ Q (ρi )2ni (19) i=1 where γ1 = ni m X X i=1 l=1 mi,l d l −1 (l − 1)! dz l−1 nj m X X mj,p z p−1 (1 − z z̄j )p j=1 p=1 ! nj m X X gj,p z p−1 (1 − z z̄j )p j=1 p=1 ! , z =z i and γ2 = ni m X X i=1 l=1 gi,l d l −1 (l − 1)! dz l−1 , z =z i with mi,l and gi,l as in (10). Proof. Directly from Theorem 2, the definition of the capacity for an AWGN channel and the fact that it does not increase with feedback. The result in Corollary 5 quantifies the difference between the infimal channel capacity Ĉ for stabilizability with input disturbance rejection and the infimal channel capacity for Pm stabilizability i=1 ni log2 ρi . This difference has been shown in [4, Corollary 4.4] to represent a fundamental limitation in control over networks performance. Our present contribution is therefore to explicitly quantify in closed-form such performance limitation. Finally, we conclude by reprising Example 3 to compute with Corollary 5 the channel capacity difference. Example 6. In Example 3 we quantified the infimal SNR for stabilizability with input disturbance rejection. In the present example we apply the result from Corollary 5 to obtain the channel In the present paper we have addressed the infimal SNR for stabilizability and input disturbance rejection LTI problem for the case of AWGN channels. By studying the spectral factorization induced by the optimal solution to the SNR for stabilizability with input disturbance rejection LTI problem, we have quantified in closed-form the infimal LTI SNR for a class of minimum phase unstable plant models with relative degree one, a memoryless AWGN channel and direct feedthrough of the input disturbance process. 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