Robust stabilization subject to structured uncertainties and mean power constraint

Automatica 92 (2018) 1–8

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Brief paper

Robust stabilization subject to structured uncertainties and mean power constraint✩ Yu Feng a , Xiang Chen b, *, Guoxiang Gu c a b c

College of Information Engineering, Zhejiang University of Technology, Hangzhou, 310032, Zhejiang, PR China Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, N9B 3P4, Canada Department of Electrical and Computer Engineering, Louisiana State University, Baton Rouge, LA, 70803-5901, USA

article

info

Article history: Received 27 July 2017 Received in revised form 30 October 2017 Accepted 8 January 2018

Keywords: Robust control Coupled algebraic Riccati equations Structured uncertainties Mean power constraint Mixed H2 /H∞ control

a b s t r a c t This paper deals with a robust stabilization problem for discrete-time systems subject to multiple disturbances occurring in controller and actuating channel, where both linear structured uncertainties and white Gaussian noises are included. The desired control law is aimed to robustly stabilize the system and to satisfy some pre-specified mean power constraint, simultaneously. By the philosophy of the mixed H2 /H∞ control, a solvability condition is first derived for single-input systems that reveals the intrinsic relation between the unstable poles of the plant and the disturbance parameters, together with two crosscoupled algebraic Riccati equations. The result is further generalized to multiple-input systems with a sufficient condition given again by the unstable poles of the plant. An example is included to illustrate the current results. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction In reality, signals are either exchanged by the use of medium with limited capacity or affected by diverse disturbances from internal components and external environment. This fact pushes the classic point-to-point control strategy to its limits. System analysis and design with limited information have thus received increasing attention from many communities over the recent decades (Murray, 2003). It is well known that analog communication systems are in general subject to power constraints, or so-called signal-to-noise ratio (SNR) constraints. In Braslavsky, Middleton, and Freudenberg (2007), control for single-input-single-output (SISO) systems over an SNR constrained channel is considered, which is closely related to the performance limitation problem in robust control (Chen, 1995; Su, Qiu, & Chen, 2009). The work of Silva, Goodwin, and Quevedo (2010) shows that it is possible to achieve mean square stability at SNRs arbitrarily close to the bound reported in ✩ The work was supported by the Natural Science Foundation of China under Grant 61573318, Zhejiang Provincial Natural Science Foundation of China for Distinguished Young Scholars under Grant LR17F030003, 111 Project (B12018) in China, and NSERC Discovery Grant in Canada. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Tongwen Chen under the direction of Editor Ian R. Petersen. Corresponding author. E-mail addresses: [email protected] (Y. Feng), [email protected] (X. Chen), [email protected] (G. Gu).

*

https://doi.org/10.1016/j.automatica.2018.02.005 0005-1098/© 2018 Elsevier Ltd. All rights reserved.

Braslavsky et al. (2007) by using linear time-invariant (LTI) controllers. Moreover, tracking problem over additive white Gaussian noise (AWGN) channels is studied in Li, Tuncel, Chen, and Su (2009) and an explicit expression of the minimal tracking error is given. By the sequential design idea originally reported in Wonham (1967), Qiu et al. introduce the technique of resource allocation and derive a solvability condition for multiple-input systems over AWGN channels in terms of the unstable poles of the plant in Qiu, Gu, and Chen (2013). Stabilization for two-input–two-output systems subject to both input and output SNR constraints is approached in Vargas, Silva, and Chen (2013). Recently, Song, Yang, and Zheng (2016) study the stabilizability and disturbance attenuation of Markov jump linear systems subject to white Gaussian noises through the concept of entropy power. Moreover, systems with multiplicative white Gaussian noises in the process/output channel have also been studied, and relevant control and filtering results have been reported in the literature. For instance, see Bouhtouri, Hinrichsen, and Pritchard (2000), Costa and Oliveira (2012), Feng, Chen, and Gu (2018), Mo and Sinopoli (2012), Su and Checi (2017), Su, Chen, Fu, and Qi (2017) and Xiao, Xie, and Qiu (2012) and the references therein. In a broader sense, both deterministic and stochastic errors are often involved in system modelling, signal treatment and transmission. Since no mathematical system can precisely model a physical system or procedure, it is of great importance to be aware of how modelling errors might adversely affect the performance of a control system (Doyle, Francis, & Tannenbaum, 2009). The concept of uncertainty and robustness is widely appealed to handle or

2

Y. Feng et al. / Automatica 92 (2018) 1–8

interpret issues with unknown disturbance, unmodelled dynamics, failures (Petersen & Tempo, 2014; Zhou, Doyle, & Glover, 1996), even cellular functions (Steilling, Sauer, Szallasi, Doyle, & Doyle, 2004) and human behaviours (Doyle & Csete, 2011) in certain level. Recently, deterministic uncertainties have also been used as a modelling tool for signal treatment and transmission, and relevant results have been reported in the literature (Fu & Xie, 2005). Stabilization for SISO systems with a norm bounded uncertainty and stochastic multiplicative noise is discussed in Feng, Chen, and Gu (2016) and a set of observer-based stabilizing controllers for such systems is further conducted by solving two algebraic Riccati equations (AREs) and an algebraic Riccati inequality. General dynamic output feedback controller design for such a problem is investigated in Feng, Chen, and Gu (2017) and the solvability condition reveals a trade-off between robust stability and robust performance. Motivated from these observations, in the current paper we attempt to explicitly characterize the solution to the robust stabilization problem subject to both LTI structured deterministic norm bounded uncertainty and mean power constraint for multipleinput discrete-time systems. Since norm bounded uncertainties and white noises have distinct feature, deterministic and stochastic models are commonly used to describe them, respectively. Considering these two factors at the same time in general makes the problem significantly complicated, and it yields naturally a multi-objective problem. The contributions of the current paper are threefold. First, we conduct a necessity and sufficient condition to such multi-objective problem by using mixed H2 /H∞ control (Chen & Zhou, 2001, 2002; Feng et al., 2018; Limebeer, Anderson, & Hendel, 1994), and show the mean power constraint is indeed a downgraded H2 performance due to the presence of the uncertainty. Second, a closed-form solvability condition is derived, which reveals the intrinsic relation between unstable poles of the plant and parameters of transmission disturbance and inaccuracy. Finally, the multiple-input case, which is an essential µ-type synthesis problem, is also analytically solved by adjusting disturbance parameters appropriately. A similar problem is addressed in Feng, Chen, and Gu (2013), with a sufficient condition reported for single-input systems. In the current paper, apart from a sufficient condition for the single-input case, we also conduct two crossedcoupled AREs for characterization of the necessity, and further generalize the sufficient condition to the multiple-input case. The remainder of the paper is organized as follows. Problem formulation is given in Section 2. In Section 3, a sufficient stabilizability condition is first derived for single-input systems in terms of the unstable poles of the plant, with two cross-coupled AREs characterizing of the necessity. Moreover, we further generalize the result to multiple-input systems and derive a solvability condition again by an inequality involving the unstable poles of the plant. A numerical example is included in Section 4 and concluding remarks are given in Section 5. The notation in this paper is fairly standard. The superscripts ‘T ’ and ‘∗’ represent the transpose and complex conjugate transpose, respectively. Rm represents m-dimension Euclidean space. For a real square matrix P, P ≥ 0 (P > 0) means that P is symmetric positive semidefinite (positive definite). The notations E {·}, ∥ · ∥, ρ (·) and λ(·) denote the standard expectation operator, Euclidean norm, spectrum radius and eigenvalue, respectively. Moreover, j = √ −1 is the imaginary number. In addition, for a matrix A ∈ Rn×n , its Mahler measure is denoted as M(A) :=

n ∏

i=1

max {1, |λi (A)|}.

A state-space realization of a] rational proper transfer function is [ A B denoted by G(z) = = D + C (zI − A)−1 B . C D

2. Problem formulation Consider a real discrete-time stochastic signal u(k) = u1 (k),

u2 (k),

[

··· ,

]T

um (k)

∈ Rm ,

where ui (k), i = 1, . . . , m, are random processes. We define the autocorrelation matrix and power spectral density (PSD) of u(k), if they exist, as follows N −1

} 1∑ { E u(k + τ )uT (k) , N →∞ N

Ru (τ ) := lim

k=0

Ψu ( ω ) =

∞ ∑

Ru (τ )e−jωτ ,

τ =−∞

where τ ∈ Z. Definition 1 (Zhou et al., 1996). A stochastic signal u(k) is said to have bounded power if both Ru (τ ) and Ψu (ω) exist. Let P be the space of all bounded power signals. Then the seminorm (mean power) can be defined on P as

  N −1  } √ 1∑ { E ∥u(k)∥2 = Tr{Ru (0)}. ∥u∥P = √ lim N →∞ N

k=0

∫π

One has Ru (τ ) = Ψ (ω)ejωτ dω. Thus, the power norm of u(k) −π u can also be computed from its PSD by 1 2π

√ ∥ u∥ P =

1

π

∫

2π

Tr{Ψu (ω)}dω.

−π

Note that the bounded power signals have been widely adopted in mixed H2 /H∞ control (Chen & Zhou, 2001; Zhou et al., 1996) and filtering problem (Chen & Zhou, 2002). Consider the following discrete-time system x(k + 1) = Ax(k) + Bu(k),

x(0) = x0 ,

n

(1)

m

where x(k) ∈ R and u(k) ∈ R are the state vector and control input, respectively. The matrices A and B are constant with consistent dimensions. Denote this system by [A|B] for simplicity. For the rest of this paper, we assume that [A|B] is stabilizable and A has no eigenvalue on the unit circle. Note that the latter condition is made to avoid a singular control problem. If it does not hold, we can let Aϵ = (1 + ϵ )A with ϵ > 0 such that the eigenvalues on the unit circle move outside the circle after perturbation. Then, by using the same arguments for [Aϵ |B] and taking the limit ϵ → 0, we can obtain the same result. In our setup, both computation in controller v (k) = Fx(k), F ∈ Rm×n , and transmission in actuating channels are nonideal, and the individual components vi (k), i = 1, . . . , m, of v (k) are transmitted independently. The overall model is depicted in Fig. 1, consisting of a received signal-to-error ratio (R-SER) model (Qiu et al., 2013) succeeded by an AWGN model with pre- and post-scaling factors, where Γi > 0, di (k) is a white Gaussian noise with variance σi2 and ∆i is a linear time-invariant norm bounded uncertainty. The induced norm of ∆i satisfies

∥wi ∥P ≤ δi , ∥qi ∥P ̸=0 ∥qi ∥P

∥∆i ∥∞ := sup

(2)

for some 0 < δi < 1. The upper bound of δi ensures that the feedback interconnection of 1 and ∆i is causal and bounded. Note that stability of ∆i does not depend on its initial condition. For simplicity, we assume zero initial condition for ∆i . Moreover, assume that the noises di (k), i = 1, . . . , m, are mutually uncorrelated

Y. Feng et al. / Automatica 92 (2018) 1–8

3

then, Problem 3 is solvable. Moreover, a desired feedback gain is given by F = −(1 + BT PB)−1 BT PA,

(5)

where P ≥ 0 is the stabilizing solution to the ARE AT P(I + BBT P)−1 A = P .

(6)

Conversely, if Problem 3 is solvable, then there exist stabilizing solutions P1 ≥ 0 and P2 ≥ 0 to the following two cross-coupled AREs

Fig. 1. Transmission model.

P1 = ATF P1 AF + F T F

(7)

+ F )ρ (B P1 AF + F ), P2 = A P2 A + (δρ )−2 F˜ T F˜ − [AT P2 B − (δρ )−2 F˜ T ] × [I + (δρ )2 BT P2 B]−1 [BT P2 A − (δρ )−2 F˜ ], +

(ATF P1 B

T

−2

T

T

(8)

such √ that 1 + BT P1 B < δ −2 and σ 2 BT P2 B < Φ , where AF = A + BF , ρ = δ −2 − 1 − BT P1 B, F = [I + (δρ )2 BT P2 B]−1 [(δρ )2 BT P2 A − F˜ ]

Fig. 2. Closed-loop system.

(9)

− δ B P1 A, 2 T

F˜ = ρ −2 BT P1 A + [(δρ )−2 − 1]F .

(Papoulis, 1965), i.e.

The following lemma is needed for the proof of Theorem 4.

E {di (k)dl (k)} = E {di (k)}E {dl (k)} = 0, ∀k ≥ 0, i ̸ = l, and the initial condition x(0) is independent of di (k) and ∆i . Remark 2. The model in Fig. 1 contains both deterministic and stochastic disturbances. Physically, the additive noises may arise from electronic components and amplifiers, which are also characterized as thermal noises. This type of noises is often characterized statistically as a Gaussian noise process (Proakis & Salehi, 2008). Moreover, the use of scaling factors around AWGN channels allows adjusting the transmission power (Silva et al., 2010). The R-SER model is appealed here for describing inherent uncertainty and numerical computation errors in the controller. Hence, the current model can be deemed as a general model to characterize this common feature to describe the mixed transmission disturbance and inaccuracy in controllers and channels. This AWGN model is required to satisfy the mean power constraint ∥si ∥2P < Φi for some pre-specified Φi > 0. Now, the closedloop system can be described by Fig. 2, where

∆ = diag [∆1 , . . . , ∆m ] , Γ = diag [Γ1 , . . . , Γm ] ,

(10)

∥ ∆ i ∥ ∞ ≤ δi ,

(3)

Lemma 5 (Inner Function Zhou et al., 1996). Let Ag be a stable matrix. Then G(z) = Dg + Cg (zI − Ag )−1 Bg is an inner, if there exists Xg ≥ 0 satisfying Xg = ATg Xg Ag + CgT Cg , DTg Cg + BTg Xg Ag = 0, DTg Dg + BTg Xg Bg = I . Now, we are in a position to prove Theorem 4. Proof. For the sufficiency, we show that under the condition (4), the feedback gain given by F = −(1 + BT PB)−1 BT PA with P ≥ 0 being the stabilizing solution to the ARE (6) solves Problem 3. Note that since [A|B] is stabilizable and A has no eigenvalue on the unit circle, the above ARE admits the stabilizing solution P ≥ 0. First, we prove that the closed-loop system is robustly stable with respect to the norm bounded uncertainty ∆, when d(k) = 0. The resulting closed-loop system is x(k + 1) = AF x(k) + Bw (k),

{

d(k) = [d1 (k), . . . , dm (k)] . T

Problem 3. The problem is to find a state feedback gain F such that, for any initial condition x(0) and LTI structured uncertainty ∆ in (3), the closed-loop system in Fig. 2 is robustly stable when d(k) = 0, and the mean power of si (k) satisfies ∥si ∥2P < Φi , with some Φi > 0, i = 1, . . . , m. 3. Main results

q(k) = Fx(k) + w (k), where AF = A + BF , w (k) = ∆(q)q(k) and q−1 is the shift operator. Justified by the well known small gain theorem (Zhou et al., 1996), the robust stability is achieved if and only if ∥1 + F (zI − AF )−1 B∥∞ < δ −1 . By the technique of completed square, we have ∥1 + F (zI − √ AF )−1 B∥∞ = 1 + BT PB. By Condition (i) of Lemma 10, there holds 1 + BT PB = M 2 (A) and the definition of the Mahler measure implies M(A) ≥ 1, which indicates δ 2 < 1. Hence,

3.1. Single-input systems

√ ∥1 + F (zI − AF ) B∥∞ < −1

In this subsection, we address Problem 3 for single-input systems, where the structured uncertainty ∆ in (3) is reduced to a scalar with ∥∆∥∞ ≤ δ . The scaling factor is given by Γ , the variance of the scalar white noise d(k) is σ 2 and the predetermined power level is given by Φ .

1 + Φ /σ 2 1 + δ 2 Φ /σ 2

< δ −1 ,

which indicates that the closed-loop system is robustly stable. Next, we prove that the mean power constraint is also satisfied. The closed-loop system depicted in Fig. 2 is written as x(k + 1) = AF x(k) + Bw (k) + BΓ −1 d(k),

{ Theorem 4. Consider the system in Fig. 2 with m = 1. If

√ M(A) <

1 + Φ /σ 2 1 + δ 2 Φ /σ 2

,

s(k) = Γ Fx(k) + Γ w (k). (4)

Since the Gaussian noise d(k) is power bounded, the states x(k) of the above closed-loop system are also power bounded. Note that

4

Y. Feng et al. / Automatica 92 (2018) 1–8

w(k) = 1−∆∆ Fx(k) with ∥∆∥∞ ≤ δ < 1. Hence, w (k) is a bounded power signal. By Condition (ii) of Lemma 10, the ARE (6) can be rewritten as the Lyapunov equation P = ATF PAF + F T F . Denoting V (k) = Γ 2 xT (k)Px(k), together with the above Lyapunov equation, we have

Since the closed-loop system is robustly stable, there holds |S(ejω )| < δ −1 ∀ω ∈ R. It follows from the bounded real lemma (de Souza & Xie, 1992) that there exists a stabilizing solution P1 ≥ 0 to ARE P1 = ATF P1 AF + F T F + (ATF P1 B + F T )ρ −2 (BT P1 AF + F ), satisfying 1 + BT P1 B < δ −2 . Denote

V (k + 1) − V (k)

F˜ = ρ −2 (BT P1 AF + F ) = ρ −2 BT P1 A + ρ −2 (1 + BT P1 B)F .

= Γ 2 xT (k)ATF PAF x(k) + 2Γ 2 xT (k)ATF PBw(k) + Γ 2 w(k)T BT PBw (k) + 2xT (k)Γ ATF PBd(k)

The above ARE can now be rewritten as

+ 2Γ w(k) B PBd(k) + d (k)B PBd(k)

P1 = ATF P1 AF + ρ 2 F˜ T F˜ + F T F .

T

T

T

T

− Γ x (k)Px(k) 2 T

In addition, the expressions of ρ and F˜ show that

= 2Γ 2 xT (k)ATF PBw(k) + Γ 2 w(k)T BT PBw(k)

FT : = F + F˜ = ρ −2 BT P1 A + [1 + ρ −2 (1 + BT P1 B)]F

+ 2Γ xT (k)ATF PBd(k) + 2Γ w(k)T BT PBd(k)

δ −2 − 1 − BT P1 B = ρ −2 BT P1 A + (δρ )−2 F ,

Hence,

∥s(k)∥2 + V (k + 1) − V (k) + 2Γ xT (k)ATF PBd(k) + Γ 2 w T (k)(BT PB + 1)w (k) + 2Γ wT (k)BT PBd(k). Since x(k) and w (k) are both independent of d(k), then

{ } E xT (k)ATF PBd(k) = 0, E w T (k)BT PBd(k) = 0.

[

]

S(z) G(z)

1

N →∞ N

E {V (N)}

≤ Γ 2 M 2 (A)∥w∥2P + σ 2 BT PB. ∥w∥P ≤ ∥∆∥∞ ∥q∥P ≤ δ∥q∥P = δ Γ

∥ s∥ P ,

∥s∥2P ≤ M 2 (A)δ 2 ∥s∥2P + σ 2 M 2 (A) − σ 2 . Note that the inequality (4) implies δ 2 M 2 (A) < 1. Hence,

σ M (A) − σ ∥s∥ ≤ 1 − δ 2 M 2 (A) σ 2 (δ 2 + σ 2 /Φ )−1 (1 + σ 2 /Φ ) − σ 2 = Φ. < 1 − δ 2 (δ 2 + σ 2 /Φ )−1 (1 + σ 2 /Φ ) 2

2

2 P

Conversely, we prove that if Problem 3 is solved, then there exist matrices P1 ≥ 0 and P2 ≥ 0 satisfying the cross-coupled AREs (7) and (8), respectively. For the system in Fig. 2, there holds s(k) = Sc (q)d(k) + S(q)Γ ∆(q)q(k),

= Sc (q)d(k) + S(q)∆(q)s(k), where Sc (z) = F (zI − A − BF )−1 B, S(z) = F (zI − A − BF )−1 B + 1, and q−1 is the shift operator. Note that since Problem 3 is solved for any norm bounded uncertainty ∆ satisfying ∥∆∥∞ ≤ δ , Problem 3 is also solved for the specific choice of ∆ such that |∆(ejω )| = δ for all ω. Under this circumstance, there holds 2

2

|Sc (e )| σ

2 2

1 − δ 2 |S(ejω )|

∥s∥ =

1 2π

∫

ρ

−ρ F˜

2

Π (z) =

[

−1

∀ ω ∈ R.

Sc (z)G(z)

A + BFT (δρ )−1 (FT − F˜ )

−1

. Then it is given by

B 0

]

. 2

It follows that Ψs (ω) = σ 2 |Π (ejω )| , and the mean power of s(k) is obtained as

∥s∥2P =

∫

1 2π

π

Ψs (ω) dω =

−π

σ2 2π

∫

π

2

|Π (ejω )| dω.

−π

Hence, minimization of ∥s∥2P over F with F := {F |ρ (A + BF ) < 1} is equivalent to minimization of ∥Π ∥2 that is a standard H2 optimal control problem under the state feedback control in light of the expression of Π (z)

Π (z) =

[

A + B2 FT C1 + D12 FT

B1 0

]

by taking B1 = B, B2 = B, C1 = −(δρ )−1 F˜ and D12 = (δρ )−1 . Its solution is given by Kwakernaak and Sivan (1972) FT = −[1 + (δρ )−2 BT P2 B]−1 [BT P2 A − (δρ )−2 F˜ ],

(14)

where P2 ≥ 0 is a stabilizing solution to the ARE P2 = AT P2 A + (δρ )−2 F˜ T F˜ − [AT P2 B − (δρ )−2 F˜ T ]

× [I + (δρ )2 BT P2 B]−1 [BT P2 A − (δρ )−2 F˜ ]. Hence, we have inf ∥s∥2P = σ 2 BT P2 B.

.

F ∈F

Hence, σ 2 BT P2 B < Φ . Finally, (12) and (14) lead to (9).

Hence, we have 2 P

⎤

B 1 ⎦

AF F

2

Ψs (ω) = |Sc (ejω )| σ 2 + δ 2 |S(ejω )| Ψs (ω), jω

(13) T

δ −2 − |S(ejω )| = |G(ejω )| ,

2

due to Ψd (ω) = σ 2 . It follows

2

is an inner function. Hence, the hypotheses on the robust stability and |S(ejω )| < δ −1 ∀ω ∈ R imply

which yields

Ψs ( ω ) =

= δ⎣

Define Π (z) := δ

Furthermore,

2

⎡

+ F = 0. Accordingly,

−1

(12)

By (11), (13) and δ + (δρ ) + B P1 B = 1, according to Lemma 5 one can claim that 2

T

∥s∥2P = Γ 2 (BT PB + 1)∥w∥2P + σ 2 BT PB − lim

F

F˜ = ρ −2 BT P1 A + [(δρ )−2 − 1]F .

δ

}

Moreover, it can be shown that

]

and

= 2Γ 2 xT (k)(ATF PB + F T )w(k) + dT (k)BT PBd(k)

ATF PB

1 + BT P1 B

[ = ρ −2 BT P1 A + 1 +

+ dT (k)BT PBd(k) − Γ 2 xT (k)F T Fx(k).

{

(11) 2

π

|Sc (ejω )| σ 2

−π

1 − δ 2 |S(ejω )|

2

2

dω.

The condition (4) reveals the intrinsic limit between unstable poles of the plant and parameters of the transmission inaccuracy. By this condition, the quality of AWGN model must satisfy σΦ2 >

Y. Feng et al. / Automatica 92 (2018) 1–8 M 2 (A)−1 1−δ 2 M 2 (A)

. If deterministic uncertainty is absent, i.e. δ = 0, then

design problem turns out to be a standard H2 control and the above inequality becomes σΦ2 > M 2 (A) − 1, or equivalently M(A) <

√

1+

Φ , σ2

which echoes the results in Braslavsky et al. (2007).

Accordingly, it can be concluded that the presence of uncertainty requires a better AWGN model to compensate the inaccuracy induced by the uncertainty. Moreover, if mean power constraint is removed, then the problem is a typical H∞ control and the solvability condition is M(A) < δ −1 , which is reported in Qiu et al. (2013). It is also observed that both of these single constraint cases allow a bigger upper bound for the Mahler measure. This fact implies that stabilization can be achieved for a larger set of systems, compared with the multiple constraints case. Remark 6. An iterative procedure can be employed to compute the respective stabilizing solutions to the cross-coupled AREs (7) and (8). First, we note that the ARE (8) can be rewritten into the following Lyapunov equation P2 = (A + BFT )T P2 (A + BFT ) + (δρ )−2 F T F ,

(15)

where F = FT − F˜ . Let P1 = 0 to begin with. Then (δρ ) = 1 and F˜ = 0 by the expressions of ρ 2 and F˜ in (13). Hence, the ARE (8) and the optimal FT in (13) reduce to the ARE (6) and (5), respectively. Substituting the above F into ARE (7) yields P1 = P, because the Lyapunov equation in (15) is the same as the ARE (7) and same as the ARE (6) under F˜ = 0 and (δρ )2 = 1. Further iteration shows that F˜ = 0 holds again and (δρ )2 P2 = P. Hence {P1 , P2 } = {P , (δρ )−2 P } is indeed a pair of the stabilizing solutions to the two cross-coupled AREs in (7) and (8), respectively, by F˜ = 0. However, it is still unknown whether there exist any other pair of stabilizing solutions {P1 , P2 } to the cross-coupled AREs. 2

For F˜ = 0, we have the following corollary. Corollary 7. Problem 3 is solvable by F in (5) where P is the stabilizing solution to the ARE (6), if and only if

√ M(A) <

1 + Φ /σ 2 1 + δ 2 Φ /σ 2

.

(16)

Proof. Sufficiency holds immediately by Theorem 4. For necessity, from the proof of Theorem 4, it is observed that

∥s∥2P =

1 2π

∫

π

|Sc (ejω )| σ 2

−π

1 − δ 2 |S(ejω )|

2

2

dω.

With the feedback gain F in (5), S(ejω ) is all pass and there holds 2 |S(ejω )| = M 2 (A). Hence,

∫ π σ2 2 |Sc (ejω )| dω 2π (1 − δ 2 M 2 (A)) −π σ 2 ∥Sc ∥22 σ 2 (M 2 (A) − 1) = = . 2 2 1 − δ M (A) 1 − δ 2 M 2 (A)

∥s∥2P =

Since ∥s∥2P < Φ , then

σ 2 (M 2 (A)−1) 1−δ 2 M 2 (A)

5

the sensitivity function, its H∞ norm is bounded by the inverse of uncertainty bound, which guarantees robust stability of the closedloop system. With the state feedback gain F in (5), there holds ∥1 + F (zI − A − BF )B∥∞ = M(A). As for the complementary sensitivity function, if uncertainty is removed, there holds ∥s∥P = σ ∥F (zI − A − BF )B∥2 , and the same feedback gain F in (5) leads to

√ ∥s∥P = σ M 2 (A) − 1. However, when the uncertainty is considered, √ the bound of the mean power of s(k) is increased with ∥s∥P ≤ 2 (A)−1 σ 1M , and 0 < 1 − δ 2 M 2 (A) < 1 due to the requirement of −δ 2 M 2 (A) robust stability, hence the warrantee of noise attenuation is clearly affected negatively by the uncertainty. Moreover, it is worth noting that the right side of (4) has a specific interpretation. In Fig. 1, w(k) and d(k) may be considered as input noises, and appear on the output side as w (k) and Γ −1 d(k), respectively. Consequently, w ¯ (k) := w(k) + Γ −1 d(k) is regarded as the output noise. Define the worst quasi-signal-to-noise ratio (W-qSNR) Υ of the transmission model as Υ := inf∆ ∥u∥P /∥w∥ ¯ P , with ∥w∥ ¯ P ̸= 0, which may be viewed as a measure of transmission efficiency in the mean power √ sense. Then, it follows Υ =

∥s∥2P /σ 2 +1 . δ 2 ∥s∥2P /σ 2 +1

Since 0 < δ < 1, it is

clear that the right side of (4) is actually the supremum of W-qSNR. Note that the sufficient part of Theorem 4 is conducted through signal characterization, which can also be easily extended to the nonlinear case. Hence, the sufficiency of Theorem 4 also holds for nonlinear norm bounded uncertainties. Here, we briefly discuss this extension. Given ∆ being a nonlinear operator, we assume that ∆(0) = 0 is the unique equilibrium point and the induced norm of ∆ satisfies

∥w∥P ≤ δ, ∥q∥P ̸=0 ∥q∥P

∥∆∥∞ = sup

for some 0 < δ < 1. We also assume that ∆ is independent of the initial condition x(0). The following corollary summarizes the result for nonlinear norm bounded uncertainty. Corollary 8. With ∆ being nonlinear and given above, the closed-loop system in Fig. 2 is robustly stable when d(k) = 0 and the mean power of s(k) satisfies ∥s∥2P < Φ , with some Φ > 0, if

√ M(A) <

1 + Φ /σ 2 1 + δ 2 Φ /σ 2

.

(17)

Moreover, a desired feedback gain F is given by (5) with P ≥ 0 being the stabilizing solution to the ARE (6). Proof. As Theorem 4, robust stability is achieved again by the small gain theorem. As for the mean power constraint, by the same thread, there holds ∥s∥2P ≤ Γ 2 M 2 (A)∥w∥2P + σ 2 BT PB. Note that

∥w∥P = ∥∆(q)∥P ≤ ∥∆∥∞ ∥q∥P ≤ δ∥q∥P = δ Γ −1 ∥s∥P . Hence, ∥s∥2P ≤

σ 2 M 2 (A)−σ 2 1−δ 2 M 2 (A)

< Φ.

< Φ . Note that the fact that the

closed-loop system is robustly stable implies 1 − δ 2 M 2 (A) > 0. Hence, we have σ 2 M 2 (A) − σ 2 < Φ − Φ δ 2 M 2 (A). Then, solving for M(A) leads to (16). The proof of Theorem 4 indicates that the mean power of s(k) is determined by not only the H2 performance of the complementary sensitivity function, which is related to the white noise, but also the H∞ performance of the sensitivity function, which is related to the deterministic uncertainty. This observation discloses the nature of the mixed H2 /H∞ property of Problem 3. More precisely, for

3.2. Multiple-input systems In this subsection, we generalize the previous result to the multiple-input case and give the following theorem. Theorem 9. Problem 3 is solvable, if M(A) <

m ∏ i=1

√

1 + Φi /σi2 1 + δi2 Φi /σi2

.

(18)

6

Y. Feng et al. / Automatica 92 (2018) 1–8

Fig. 3. Equivalence realization of Fig. 2.

Proof. The way to construct F and Γ is originally inspired by the sequential design idea used in the first multiple-input pole placement solution in Wonham (1967), and similar results can be found in Chen and Qiu (2013), Feng et al. (2017) and Qiu et al. (2013). Without loss of generality, we assume that the matrix pair [A|B] has the Wonham decomposition:

⎡

⋆

A1

⎢ ⎢0 A=⎢ ⎢. ⎣ ..

A2

..

. ···

0

··· .. . .. . 0

⋆ .. ⎥ . ⎥ ⎥, ⎥ ⋆⎦

b1

⋆

⎢ ⎢0 B=⎢ ⎢. ⎣ ..

..

⎡

⎤

0

Am

b2

. ···

··· .. . .. . 0

⋆ .. ⎥ .⎥ ⎥, ⎥ ⋆⎦ bm

m

where Ai ∈ Rni ×ni , bi ∈ Rni ×1 ,

∑

Fig. 4. State evolution with uncertainty (d(k) = 0).

⎤

ni = n, each matrix pair [Ai |bi ]

i=1

is stabilizable, and the symbol ⋆ represents the entry irrelevant to the discussion proceeded subsequently. Note that the closed-loop system in [ Fig. 2 can be equivalently ] transformed into Fig. 3 where −1 T (z) = Γ S(z)Γ −1 Γ Sc (z)Γ −1 , with [Sc (z) = F (zI − A ] − BF ) B −1 m−1 and S(z) = S (z) + I. Set Γ = diag 1 , ϵ, . . . , ϵ and U = c ] [ diag In1 , ϵ In2 , . . . , ϵ m−1 Inm , where ϵ > 0 is sufficiently small. Thus,

decomposition does not exist in general. In order to present an analytic solvability condition, some scholars impose an additional condition about triangular decomposition on systems in the literature, for example, (Xiao et al., 2012), where output feedback stabilization is addressed for systems subject to multiplicative random noises. Finding a less conservative analytic solution to controller synthesis for the MIMO case is still open and deserves further investigation. 4. Numerical illustration Here, we present a numerical example to illustrate the effectiveness of the current results. Let us consider the following twoinput system

[ ]−1 Γ Sc (z)Γ −1 = Γ FU zI − U −1 AU − U −1 BΓ −1 Γ FU Γ S(z)Γ

= I + Γ Sc (z)Γ

−1

,

where U −1 AU = diag[A1 , . . . , Am ] + o(ϵ ), U −1 BΓ −1 = diag[b1 , . . . , o(ϵ ) bm ]+ o(ϵ ), and ϵ approaches a finite constant when ϵ approaches zero. Hence, there holds T (z) = diag[T1 (z), . . . , Tm (z)] + o(ϵ, z),

[

Ti = Si (z)

Sci (z) ,

]

i = 1, . . . , m,

(

)−1

1+Φi /σi2

1+δi2 Φi /σi2

0 2 0

1 −1 , 2

]

[ B=

1 0 0

It is √ observed that M(A) = m ∏

1+Φi /σi2

1+δi2 Φi /σi2

i=1

0 1 . 1

]

16. By Theorem 9, the quantity √

must satisfy the relation

m ∏

such that (18) holds. By

Theorem 4, there exist feedback gains fi , i = 1, . . . , m, such that Problem 3 for m single-input systems is solvable and underlying feedback gains fi are given by fi = −(1 + bTi Pi bi )−1 bTi Pi Ai , where Pi ≥ 0 are stabilizing solutions to the AREs ATi Pi (I +bi bTi Pi )−1 Ai = Pi . Hence, the resulting feedback gain is given by F = diag [f1 , . . . , fm ]. Similarly, Theorem 9 is also true with regard to nonlinear structured norm bounded uncertainties, and the extension can be easily done. Note that the use of Wonham decomposition form does not bring conservatism, as long as the matrix pair [A|B] is stabilizable. However, for the multiple-input-multiple-output (MIMO) case, that is an output feedback controller is considered, a similar

1+Φi /σi2

1+δi2 Φi /σi2

i=1

> M(A) =

16. Let this quantity be slightly bigger than 16 and be given by 16.0333. Then, we specifically adjust the parameters as follows Φ1 = 1.5, σ12 = 0.36, ∥∆1 ∥∞ ≤ 0.2642, Φ2 = 1, σ22 = 0.16 and ∥∆2 ∥∞ ≤ 0.36. [ ] We choose the scaling matrix as Γ =

where Sci (z) = fi zIni − Ai − bi fi bi and Si (z) = 1 + Sci (z). Therefore, solving Problem 3 for a system with m inputs amounts to solving Problem 3 for m single-input systems. Note that we √ can always choose M(Ai ) <

0 0

A=

× U −1 BΓ −1 , −1

[ −4

1 0

0 10

. In order to find

the desired F , it suffices to find the state feedback gains for the following two single-input systems: A1 = −4,

b1 = 1 ,

[

2 A2 = 0

] −1 2

,

[ ] b2 =

1 . 1

By solving two underlying AREs, we have P1 = 15, f1 = 3.75, [ ] P2 = F =

[

27

−45

−45

78

3.75 0

0 2.25

, and f2 = 2.25

[

0 −5.25

]

] −5.25 . Hence, F is given by

.

Set the initial condition as x(0) = [0.1 − 0.5 0.3]T . When d(k) = 0, the closed-loop evolution of the plant states starting from the initial condition stimulated by an impulse is shown in Fig. 4. Clearly, the states converge to zero asymptotically. When uncertainty is absent, under the same initial condition, the closed-loop evolution of the plant states with noise is shown in Fig. 5. Moreover, when both uncertainty and noise are present, the closed-loop evolution of the plant states is in Fig. 6. Unlike the noise free case, the states in the last two figures do not converge to zero, but still

Y. Feng et al. / Automatica 92 (2018) 1–8

7

Lemma 10. Let the matrix P ≥ 0 be the stabilizing solution to AT P(In + BBT P)−1 A = P , such that Im + B PB < δ T

(A.1)

−2

T

Im . Then, the following two conditions hold:

2

(i) det(Im + B PB) = M (A); (ii) P = (A + BF )T P(A + BF ) + F T F , where F = −(Im + BT PB)−1 BT PA. Proof. (i) We can always find a nonsingular matrix Ξ such that A = Ξ −1

[

A11 0

where ρ (A11 )

]

0 Ξ, A22

<

B = Ξ −1

[ ]

B1 , B2

1 and A22 is anti-stable. Hence, M(A)

=

det(A [ 22 ). The ] stabilizing solution P to (A.1) can be written as P =

ΞT

Fig. 5. Noisy state evolution (∆ = 0).

0 0

0 P22

Ξ , where P22 > 0 is the stabilizing solution to AT22 P22 (In + B2 BT2 P22 )−1 A22 = P22 , such that Im + BT2 P22 B2 < δ −2 Im . Then, we have det(Im + BT PB)

= det(Im + BT2 P22 B2 ) = det(In + B2 BT2 P22 ) −1 T = det(A22 P22 A22 P22 ) = det(A22 ) det(AT22 )

= M 2 (A). (ii) In order to simplify the arguments, we assume with no loss of generality that the matrix A is anti-stable. Note that if it does not hold, the same treatment used to prove the condition (i) can be applied. Hence, now A is nonsingular. Note that with A anti-stable, the matrix P is positive definite. We have AT P(In + BBT P)−1 A − P

= A−1 (In + BBT P)P −1 A−T − P −1 = A−1 P −1 A−T + A−1 BBT A−T − P −1 = PA−1 P −1 A−T P + PA−1 BBT A−T P − P = 0. Fig. 6. Noisy State evolution with uncertainty.

Note that, by straightforward computation, there holds F = −(Im + BT PB)−1 BT PA

bounded, due to the noises introduced in the transmission models. Furthermore, it is also observed from Fig. 6 that the effect of noise is strengthened through the uncertainty, and the states vary more dramatically compared with the case without uncertainty. 5. Conclusion This paper is concerned with a robust control problem for discrete-time systems with linear norm bounded uncertainties and mean power constraint. By adopting the philosophy of mixed H2 /H∞ control, a solvability condition is derived for single-input system in terms of the Mahler measure of the plant and a necessary condition is given in terms of stabilizing solutions to the two crosscoupled AREs. This result is further extended to the multiple-input case, and the solution is given again by an inequality involving the Mahler measure of the plant. An extension to nonlinear norm bounded uncertainties is also discussed. Appendix The following lemma can be found in the literature, for instance Braslavsky et al. (2007) and Qiu et al. (2013). In order to be selfcontained, we prove this lemma in the paper.

= −BT P(In + BBT P)−1 A = −BT PP −1 A−T P . Moreover, we have A + BF = (In + BBT P)−1 A = P −1 A−T P. Hence, (A + BF )T P(A + BF ) + F T F

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Yu Feng received the B.E. degree in electrical engineering from Nanjing University of Science and Technology, China, in 2005, and the Diplôme d’Ingénieur and Ph.D. degree in electrical engineering from Ecole des Mines de Nantes, France, in 2007 and 2011, respectively. From 2012 to 2013, he held the postdoctoral position at University of Windsor, Canada. Since 2013, he has been a faculty member at the College of Information Engineering, Zhejiang University of Technology, Hangzhou, China, where he is currently an Associate Professor. His current research interests include networked control systems, distributed filtering, and robust analysis and control for uncertainty systems. He has served as an associate editor for the IEEE CSS Conference Editorial Board, and is member of the IEEE CSS technical committee on systems with uncertainty. He is presently the Qiangjiang distinguished professor of Zhejiang Province, China.

Xiang Chen received Ph.D. degree from Louisiana State University, Baton Rouge, LA, USA, in 1998 in Systems and Control. He is a Professor in the Department of Electrical and Computer Engineering, University of Windsor, Windsor, ON, Canada. His research interests include robust control, data-driven optimization, field sensor networks, and automotive control.

Guoxiang Gu received the Ph.D. degree in electrical engineering from the University of Minnesota, Minneapolis, in 1988. From 1988 to 1990, he was with the Department of Electrical Engineering, Wright State University, Dayton, Ohio, as a Visiting Assistant Professor. Since 1990, he joined Louisiana State University (LSU), Baton Rouge, where he is currently a Professor of Electrical and Computer Engineering. His research interests include networked control systems, system identification, and statistical signal processing. He authored two books, and published over 70 archive journal papers, plus numerous book chapters, and conference papers. He has held visiting positions at WrightPatterson Air Force Base, in Hong Kong University of Science and Technology, and in Southwest Jiaotong University of P.R. China. He served as an associate editor for the IEEE Transactions on Automatic Control from 1999 to 2001, SIAM Journal on Control and Optimization from 2006 to 2009, and Automatica from 2006 to 2012, and will serve again as an associate editor for IEEE Transactions on Automatic Control from 2018–2020. He is presently the F. Hugh Coughlin/CLECO distinguished professor of electrical engineering at LSU, and Fellow of IEEE.