Coherent H ∞ Control for a Class of Linear Complex Quantum Systems. Aline I. Maalouf and Ian R. Petersen Abstract—This paper considers a coherent H ∞ control prob- lem for a class of linear quantum systems which can be defined by complex quantum stochastic differential equations in terms of annihilation operators only. For this class of quantum systems, a solution to the H ∞ control problem can be obtained in terms of a pair of complex Riccati equations. The paper also considers complex versions of the Bounded Real Lemma, the Strict Bounded Real Lemma and the Lossless Bounded Real Lemma. For the class of quantum systems under consideration, the question of physical realizability is related to the Bounded Real and Lossless Bounded Real properties. Index Terms—Quantum Feedback Control, H ∞ control, dis- sipativity, complex strict bounded real lemma, quantum optics, complex Riccati equations, quantum controller realization. I. I NTRODUCTION Robustness is an important issue in the control of quantum feedback systems; e.g. see [4], [5], [15], [13], [14] and [9]. In the recent paper [9], the problem of systematic robust control system design for quantum systems is tackled via an H ∞ approach. In [9], this problem was addressed by considering real and imaginary quadratures of the quantum system variables. This made the derivations quite complicated as the equations involving the complex annihilation operator were converted to a form involving the real quadratures. In this paper, to simplify the work of [9], we consider a class of linear quantum systems, which can be modeled purely in terms of the annihilation operator and not the creation operator. The class of quantum systems considered in this paper includes important ‘passive’ systems from the field of quantum optics such as interconnections of cavities, phase shifters and beam splitters; see [1]. For this class of quantum systems, the system can be described by complex linear quantum stochastic differential equations in terms of the annihilation operator. The dimension of this set of equations is half that which would be obtained by considering real quantum stochastic differential equations defined in terms of quadratures which was the case in [9]. Moreover, a solution to the quantum H ∞ control problem is obtained in terms of a pair of complex Riccati equations. The dimension of these Riccati equations is half that of the real Riccati equations obtained in [9]. In the quantum H ∞ control problem considered in this pa- per, we wish to construct a coherent quantum controller which This work was completed with the support of a University of New South Wales International Postgraduate Award and the Australian Research Council. Aline I. Maalouf and Ian R. Petersen are with the School of Infor- mation Technology and Electrical Engineering, University of New South Wales at the Australian Defence Force Academy, Canberra, ACT 2600 a.maalouf@adfa.edu.au; i.r.petersen@gmail.com is also in the class of quantum systems being considered. A connection is derived between the linear complex quantum stochastic differential equations (QSDEs) under consideration and a corresponding real QSDE defined in terms of quadrature variables. This connection is used in most of the proofs. In addition to simplifying the formulas, our derivation leads to more physical insight and better understanding of the underly- ing quantum systems. The paper presents complex versions of the Bounded Real, Strict Bounded Real and Lossless Bounded Real Lemmas. While the Strict Bounded Real Lemma plays an important role in the solution to the H ∞ control problem, the Bounded Real Lemma and the Lossless Bounded Real Lemma are found to be connected to the issue of physical realizability for the class of complex linear quantum systems under consideration. Also, this paper makes a contribution in considering a classical problem of complex H ∞ control which has not been investigated previously. Such a complex H ∞ control problem may have applications in other areas. This conference version of the paper only presents the results. All proofs will be presented in the full version of the paper. II. A CLASS OF LINEAR COMPLEX QUANTUM SYSTEMS The class of complex linear quantum systems under con- sideration can be described by using non-commutative or quantum probability theory [3]. In particular, the systems are described in terms of the complex annihilation operator by the quantum stochastic differential equations (QSDEs) da(t) = Fa(t)dt + Gdw(t); a(0) = a 0 dy(t) = Ha(t)dt + Jdw(t) (1) where F ∈ C n×n , G ∈ C n×nw , H ∈ C ny×n and J ∈ C ny×nw (n, n w , n y are positive integers). Here a(t) = [a 1 (t) ··· a n (t)] T is a vector of (linear combinations of) anni- hilation operators. The vector w represents the input signals and is assumed to admit the decomposition: dw(t)= β w (t)dt + d ˜ w(t) where ˜ w(t) is the noise part of w(t) and β w (t) is a self-adjoint adapted process (see [3], [11] and [8]). The noise ˜ w(t) is a vector of quantum noises with Ito table d ˜ w(t)d ˜ w † (t)= F ˜ w dt (see [2] and [11]) where F w is a non-negative definite Hermi- tian matrix. Here the notation † represents the adjoint transpose 2009 American Control Conference Hyatt Regency Riverfront, St. Louis, MO, USA June 10-12, 2009 WeC04.1 978-1-4244-4524-0/09/$25.00 ©2009 AACC 1472