B. Ross Barmish Workshop 2009 1 Robust Control of Linear Quantum Systems Ian R. Petersen School of ITEE, University of New South Wales @ the Australian Defence Force Academy, Based on joint work with Matthew James, A.J. Shaiju, Hendra Nurdin, and Aline Maalouf.
B. Ross Barmish Workshop 2009 2 Introduction Recent developments in quantum and nano technology have provided a great impetus for research in quantum feedback control systems. In particular, it is being realized that robustness is critical in quantum feedback control systems, as it is in non-quantum feedback control systems. We present an overview of some recent results in the area of robust feedback control of linear quantum systems. We consider a class of quantum systems described by linear Heisenberg dynamics driven by quantum Gaussian noise processes, and controlled by a linear feedback controller which is also a quantum system.
B. Ross Barmish Workshop 2009 3 The most common area in which such systems arise is in the area of quantum optics; e.g., Photo courtesy of Elanor Huntington (UNSW@ADFA)
B. Ross Barmish Workshop 2009 4 Analysis of Linear Quantum Systems In the analysis of linear quantum systems, we consider noncommutative stochastic models of the form dx(t) = Ax(t)dt + [ B G ][ dw(t) T dv(t) T ] T ; dz(t) = Cx(t)dt + [ D H ][ dw(t) T dv(t) T ] T (1) where x(t) = [ x 1 (t)... x n (t) ] T is a vector of self-adjoint possibly non-commutative system variables. Also, A, B, G, C D are real matrices. The vector quantity w describes the input signals and is assumed to admit the decomposition dw(t) = β w (t)dt + d w(t) (2) where w(t) is the noise part of w(t) and β w (t) is a self adjoint process.
B. Ross Barmish Workshop 2009 5 In this quantum system, the input channel has two components, dw = β w dt + d w which represents disturbance signals, and dv, which represents additional noise sources. The process β w (t) serves to represent variables of other systems which may be passed to the system. This system may represent the closed loop quantum classical system in a quantum H control system.
B. Ross Barmish Workshop 2009 6 Definition. The above quantum stochastic system is said to be Strictly Bounded Real with disturbance attenuation g if there exists a positive operator valued quadratic form V (x) = x T Xx (where X is a real positive definite symmetric matrix) and constants λ > 0, ǫ > 0 such that V (x(t)) + t V (x(0)) + λt t > 0, 0 β T z β z (g 2 ǫ)β T wβ w + ǫx T x ds for all Gaussian states ρ for the initial variables x(0). Here we use the shorthand notation for quantum expectation over all initial variables and noises.
B. Ross Barmish Workshop 2009 7 Theorem. [Quantum Strict Bounded Real Lemma] The following statements are equivalent (i) The above quantum stochastic system is strictly bounded real with disturbance attenuation g. (ii) A is a stable matrix and C(sI A) 1 B + D < g. (iii) g 2 I D T D > 0 and there exists a positive definite matrix X > 0 : A T X + XA + C T C + ( XB + C T D) (g 2 I D T D) 1 (B T X + D T C) < 0. (iv) g 2 I D T D > 0 and the algebraic Riccati equation A T X + XA + C T C + (XB + C T D) (g 2 I D T D) 1 (B T X + D T C) = 0 has a stabilizing solution X 0. Furthermore, if these statements hold then X < X.
B. Ross Barmish Workshop 2009 8 H of linear quantum systems In the quantum H control problem we consider a plant which is described by noncommutative stochastic models of the form dx(t) = Ax(t)dt + B 0 dv(t) + B 1 dw(t) + B 2 du(t); x(0) = x; dz(t) = C 1 x(t)dt + D 12 du(t); dy(t) = C 2 x(t)dt + D 20 dv(t) + D 21 dw(t) We assume that the control input is of the form du(t) = β u (t)dt + dũ(t) where ũ(t) is the noise part of u(t).
B. Ross Barmish Workshop 2009 9 Assumptions 1. D T 12D 12 = E 1 > 0. 2. D 21 D21 T = E 2 > 0. [ ] A jωi B2 3. The matrix is full rank for all ω 0. C 1 D 12 [ ] A jωi B1 4. The matrix is full rank for all ω 0. C 2 D 21
B. Ross Barmish Workshop 2009 10 Riccati Equations Our quantum H result is stated in terms of the following pair of algebraic Riccati equations: (A B 2 E1 1 DT 12C 1 ) T X + X(A B 2 E1 1 DT 12C 1 ) +X(B 1 B1 T g 2 B 2 E1 1 B 2)X +g 2 C1 T (I D 12E1 1 DT 12 )C 1 = 0; (A B 1 D T 21E 1 2 C 2)Y + Y (A B 1 D T 21E 1 2 C 2) +Y (g 2 C T 1 C 1 C T 2 E 1 2 C 2)Y +B 1 (I D T 21E 1 2 D 21)B T 1 = 0.
B. Ross Barmish Workshop 2009 11 Controller Our quantum H control is defined by state space matrices constructed from the Riccati equations as follows: A K = A + B 2 C K B K C 2 + (B 1 B K D 21 )B T 1 X; B K = (I Y X) 1 (Y C T 2 + B 1 D T 21)E 1 2 ; C K = E 1 1 (g2 B T 2 X + D T 12C 1 ).
B. Ross Barmish Workshop 2009 12 Theorem. Necessity. If there exists a controller such that the resulting closed loop system is strictly bounded real with disturbance attenuation g, then the Riccati equations will have stabilizing solutions X 0 and Y 0 such that the matrix XY has a spectral radius strictly less than one. Sufficiency. Suppose the Riccati equations have stabilizing solutions X 0 and Y 0 such that the matrix XY has a spectral radius strictly less than one. If the controller is such that the matrices A K, B K, C K are as defined in terms of the Riccati solutions as above, then the resulting closed loop system will be strictly bounded real with disturbance attenuation g.
B. Ross Barmish Workshop 2009 13 Physical Realizability of Linear Quantum Systems In the quantum H control problem, the controller can be a classical system constructed from digital or analog electronics, a quantum system constructed from cavities, beamsplitters and phase shifters, or a combination of both. If the controller is to be a purely quantum system (coherent quantum feedback control), the question arises as to whether a given controller transfer function can be physically realized as a quantum system.
B. Ross Barmish Workshop 2009 14 Example Quantum H control system consisting of an optical cavity plant and an optical cavity controller: Photo courtesy of Hideo Mabuchi, Stanford.
B. Ross Barmish Workshop 2009 15 Physical Realizability of Quantum System Models To consider the issue of whether a state space quantum system model (controller) is physically realizable, it is most convenient to consider complex quantum systems of the form da = F 0 [ a a ] db = H 0 [ a a ] dt + G 0 [ dv dv ], dt + K 0 [ dv dv ]. We assume without loss of generality that the number of inputs equals the number of outputs.
B. Ross Barmish Workshop 2009 16 Also, we write F 0 = [ ] F 1 F 2 C n a 2n a, G 0 = [ ] G 1 G 2 C n a 2n v, H 0 = [ ] H 1 H 2 C n v 2n a, K 0 = [ ] K 1 K 2 C n v 2n v, [ ] [ ] F1 F F = 2 G1 G F2 F1, G = 2 G 2 G, 1 [ ] [ ] H1 H H = 2 K1 K, K = 2. H 2 H 1 K 2 K 1
B. Ross Barmish Workshop 2009 17 This complex linear quantum system is equivalent to a real linear quantum system of the form considered previously via the substitutions: x k = a k + a k, x n a +k = i(a k a k ). y l = b l + b l, y nv +l = i(b l b l), 1 l n v, w j = v j + v j, w nv +j = i(v j v j ), 1 j n v.
B. Ross Barmish Workshop 2009 18 This leads to the following system with self-adjoint variables and real coefficient matrices: dx = Ax dt + B dw, dy = Cx dt + D dw, where x = [x 1 x 2 x 2na ], y = [y 1 y 2 y 2nv ], A = 1 2 ΦFΦ, B = 1 2 ΦGΦ, C = 1 2 ΦHΦ, D = 1 2 ΦKΦ, [ ] [ ] I I I 0 Φ =, J =. ii ii 0 I
B. Ross Barmish Workshop 2009 19 Our notion of physical realizability is defined in terms of the preservation of certain quantum commutation relations and the equivalence of a quantum linear system to a quantum harmonic oscillator. An alternative algebraic condition is given in the following theorem. Theorem. The above complex linear quantum system is physically realizable if and only if there exists a hermitian matrix Ω = Ω such that FΩ + ΩF GJG = 0, G = ΩH J, K = I.
B. Ross Barmish Workshop 2009 20 Our main result on physical realizability relates this property to the system theory notion of a (J, J)-unitary transfer function. Consider the complex transfer function matrix corresponding to the above complex linear quantum system Ψ(s) = H(sI F) 1 G + K. Definition. A system with transfer function matrix Ψ(s) is said to be is said to be dual (J,J)-unitary if for s C : Re(s) 0. Ψ(s)JΨ ( s ) = J,
B. Ross Barmish Workshop 2009 21 Theorem. The above complex linear quantum system is physically realizable if and only if its transfer function matrix is dual (J,J)-unitary and K = I. This theorem shows the equivalence between the quantum mechanical notion of physical realizability and the frequency domain systems theory notion of dual (J, J)-unitary.