Reduced basis method for H 2 optimal feedback control problems

Transcription

1 A. Schmidt a B. Haasdonk b Reduced basis method for H 2 optimal feedback control problems Stuttgart, November 215 a Institute for Applied Analysis and Numerical Simulation, University of Stuttgart, Pfaffenwaldring 57, 7569 Stuttgart/ Germany {schmidta,haasdonk}@mathematik.uni-stuttgart.de Abstract In this paper we examine the application of known parametric model reduction techniques to the H 2 optimal feedback control problem. The H 2 control problem provides a realistic framework for control applications, since it considers disturbances in the system and in the measurement outputs. Furthermore it employs state-estimation to reconstruct the unknown state from the noisy measurements. It turns out, that the controller is a dynamical system and two solutions of algebraic Riccati equations (AREs) are required to form it. We apply parametric model order reduction techniques to the AREs and to the state equation of the observer and show by numerical examples, that this approach can yield a significant speed-up in multi-query scenarios for large scale parametric problems for the control of partial differential equations (PDE). Keywords Optimal Control parametric PDE Algebraic Riccati Equation State Estimation Stuttgart Research Centre for Simulation Technology (SRC SimTech) SimTech Cluster of Excellence Pfaffenwaldring 5a 7569 Stuttgart publications@simtech.uni-stuttgart.de

2 Reduced basis method for H 2 optimal feedback control problems A. Schmidt B. Haasdonk University of Stuttgart, Institute for Applied Analysis and Numerical Simulation, Pfaffenwaldring 57, 7569 Stuttgart - Germany {schmidta,haasdonk}@mathematik.uni-stuttgart.de Abstract: In this paper we examine the application of known parametric model reduction techniques to the H 2 optimal feedback control problem. The H 2 control problem provides a realistic framework for control applications, since it considers disturbances in the system and in the measurement outputs. Furthermore it employs state-estimation to reconstruct the unknown state from the noisy measurements. It turns out, that the controller is a dynamical system and two solutions of algebraic Riccati equations (AREs) are required to form it. We apply parametric model order reduction techniques to the AREs and to the state equation of the observer and show by numerical examples, that this approach can yield a significant speed-up in multi-query scenarios for large scale parametric problems for the control of partial differential equations (PDE). Keywords: Optimal control, feedback control, parametric PDEs, Riccati equation, state estimation, H 2 optimal control 1. INTRODUCTION The solution approaches to control problems can in general be divided into two main classes: Open-loop- and feedback control. In open-loop control, a control signal is calculated from the initial configuration and is then used to steer the system without getting back any information from the system. In contrast to that, feedback control uses measurements for the calculation of the control trajectory. It is an easy consequence of the conceptual designs, that only the latter method is able to cope with a-priori unknown disturbances. Realistic control problems should consider noise in the system dynamics and in the outputs. Furthermore, usually not the full state of the system is known, but only (partial) measurements, such as sensor information, can be used for control. A setup that takes into account all those design issues is the so called linear quadratic H 2 control problem, see for example Zhou et al. (1996). This particular problem aims at minimizing a certain norm in the frequency domain, which quantifies the influence of noise to the output of interest. The solution can be obtained by first finding an optimal state estimate, which is then used in a state-feedback controller to calculate the control signal. The main tool for both steps is the algebraic Riccati equation (ARE). The ARE is a matrix-valued nonlinear equation with many important applications in various areas of control theory, see Lancaster and Rodman (1995). We will consider systems that arise after spatial semidiscretization of parametric PDEs. The discretized problems are characterized by large dimensions, say in the order of The computational complexity can easily lead to impracticable calculation times for multi-query scenarios, for example in parameter studies or real-time applications. We thus propose to use recent model reduction techniques to obtain approximate controllers rapidly. The method of choice for the reduction of parametric problems is the reduced basis (RB) method, see Patera and Rozza (27). This method has already been successfully applied not only to the simulation of PDEs, but also for Kalmanfilter and the rapid approximation of solutions to the ARE, see Dihlmann and Haasdonk (215); Schmidt and Haasdonk (215); Schmidt et al. (214). We propose to use the RB method for the H 2 optimal control problem, where we reduce all high dimensional, and thus costly, operations. This paper is organized as follows. In Section 2 we give a brief introduction into the H 2 control setting and the control theory background. In Section 3, we first describe the reduction techniques for the ARE and for dynamical systems before we then apply them to the H 2 control problem. Numerical examples in Section 4 finally show the quality of the approximation. We conclude in Section 5 with final remarks. 2. H 2 OPTIMAL CONTROL Throughout this paper, we are discussing control problems that are modeled by parametric linear and time invariant (LTI) systems of the form ẋ(t) = A(µ)x(t) + B u (µ)u(t) + B w (µ)w(t) (1a) z(t) = C z (µ)x(t) + D zu (µ)u(t) (1b) y(t) = C y (µ)x(t) + D yw (µ)w(t) (1c) x() = x (µ). (1d) In the following the parameter µ is a vector from a bounded parameter set, denoted by P R ρ and all definitions and assumptions should be understood to hold

3 w(t) u(t) System Controller y(t) z(t) Fig. 1. Schematic representation of the control problem. for all parameters µ P. Furthermore, for the sake of brevity, we will omit the explicit dependence of the quantities on the parameter µ in the following. The dynamics of the state x(t) R n are modeled by the differential equation (1a) and can be influenced by the input function u(t) R mu, which will later be chosen as the control input. We consider two outputs of the system. The first output z(t) R pz is the performance output. This quantity should be rendered small by the controller. The second output y(t) R py describes the measurements from the system that are available as inputs for the controller. The function w(t) R mw models uncertain inputs to the system and enters in the state equation as well as in the measurement outputs. The dimensions of the matrices are A(µ) R n n, B u (µ) R n mu, B w (µ) R n mw, C z (µ) R pz n, D zu R pz mu, C y (µ) R py n and D yw (µ) R py mw. Remark 1. Discretizations like the finite element method usually lead to a mass matrix in front of the timederivative of the state: Eẋ(t) = Ax(t) + Bu(t). Formally, a multiplication with E 1 transforms the system to standard form. However, this should be avoided since it destroys important properties like sparsity or symmetry. A better approach is to work with the Cholesky factors (or the LU-decomposition in the general case), and to perform the inversion of E implicitly in the algorithms. The idea of feedback control is depicted in Figure 1: One tries to find a controller that transforms the measurement output signal y(t) to a control signal u(t) such that certain goals are achieved. Most importantly, the closed loop system should be stable, which means that if there is no disturbance w(t) entering the system, the state should decay to the origin x(t) for t. This is also called internal stability and is closely related to the stability of matrices: We say a matrix is stable, if all eigenvalues have negative real parts. We make a general ansatz for the controller, and chose an arbitrary dynamical system, where the measurements from the control system y(t) are inputs, and the control signal u(t) is the output: {ẋk (t) = A K x K (t) + B K y(t) (2a) u(t) = C K x K (t). (2b) Usually (but not necessarily), the dimension of x K (t) coincides with the state dimension n of the system. By closing the loop between y(t) and u(t), the interconnection of the system model with the controller can be written as an augmented LTI system of the form { ξ(t) = Aξ(t) + Bw(t) (3a) z(t) = Cξ(t), (3b) where ξ(t) R nk+n is an abbreviation for the combined vector (x(t) T, x K (t) T ) T and the system matrices are given as ( ) A B = C D B K C y A K B K D yw A B uc K B w C z D zu C K The closed loop system (3a)-(3b) thus provides a link between the disturbance input w(t) and the performance output z(t) and hides the interconnection of the system and the controller. For the purpose of controlling, the effect of the disturbance on the output should be minimized in some suitable sense. One way to measure the behavior of systems to noise, is to examine the transfer function G(s) of the system (3a)-(3b), which can be obtained by taking the Laplace transformation of the system equations (3a) (3b), see e.g. Kwakernaak and Sivan (1972). Z(s) = G(s)W (s), with G(s) = C(sI n+nk A) 1 B. Here, I n R n n is the n dimensional identity matrix and s is a complex number. We say that G(s) is stable, if all its poles have negative real parts, which is equivalent to the stability of A. The benefit of using the transfer function is, that it establishes a linear link between the input and the output of the system in the frequency domain. For such functions, the H 2 norm can be defined: G 2 2 := 1 tr ( G( jω) T G(jω) ) dω, 2π where j denotes the imaginary unit and tr ( ) is the trace operator. The functions W (s) and Z(s) are the Laplace transformations of w(t) respectively z(t). If not otherwise stated, the norm of a matrix G will always be the induced Euclidean 2-norm G = sup x =1 Gx. Before defining the H 2 optimal control problem and its solutions, we first recap some well known properties and intuitive interpretations of the above norms. Remark 2. (Facts about the H 2 norm). Given a stable transfer function G(s) = C(sI A) 1 B, the H 2 norm G 2 can be calculated by solving one of the following Lyapunov equations AL c + L C A T + BB T =, (4) A T L O + L O A + C T C =. (5) Then it holds G 2 2 = tr ( CL C C ) T = tr ( B T L O B ). The solutions to the above Lyapunov equations are also called observability and controllability Gramians of the system and play very important roles in different parts in linear systems theory, cf. Kwakernaak and Sivan (1972). The H 2 norm of a transfer function G(s) has an intuitive interpretation in the state space. Let z k (t) for k = 1,..., m u be the response of the following systems: ẋ k = Ax k, z k = Cx k, x k () = Be k, where e k R mu denotes the k-th unit input vector. Then the equality n H 2 2 = z k (t) T z k (t)dt k=1 holds, providing a relationship between the state space input-output behavior and the H 2 norm. Another useful interpretation is given, when the input w(t) is modeled as Gaussian white noise with unit covariance: In that case, the equality

4 [ 1 lim E t t t ] z(t) T z(t)dt = G 2 2 holds where E [ ] denotes the expected value. This establishes the link between linear Gaussian control and the H 2 optimal control problem, see also Remark 5. We now have all tools to formulate the H 2 optimal control problem: Definition 3. (H 2 optimal control problem). Given the system (1a) (1c), find a controller of the form (2a) (2b), such that the H 2 norm J := C(sI n+nk A) 1 B 2 2 of the closed loop system (3a) (3b) is minimized. Before we can give the solution for the H 2 and the H problem, we must make some rather technical assumptions on the structure of the problem: Assumption 4. (1) The pair (A, B u ) is stabilizable (i.e. there exists F R n mu such that (A + B u F ) is stable), and (A, C y ) is detectable (i.e. (A T, C T ) is stabilizable) (2) The pair (A, C z ) has no unobservable modes and (A, B w ) has no uncontrollable modes on the imaginary axis (3) DzuD T zu and D yw Dyw T are invertible (4) DzuC T z = and D yw Bw T = The first and second assumption are necessary in order to guarantee the existence of stabilizing solutions to the two AREs in Theorem 6. The third and fourth assumption can be relaxed when using advanced techniques like linear matrix inequalities, but they provide useful simplifcations in the solution structure. Remark 5. (Link to LQG-Control). It is worth to mention the relationship of the H 2 optimal control problem to the famous linear quadratic Gaussian (LQG) control problem, as it is merely a special kind of the general H 2 setting with a specific stochastic framework for the noise components. Assume that the disturbances w 1 (t) and w 2 (t) model independent Gaussian white noise with unit covariances. Consider the system ẋ(t) = Ax(t) + Bu(t) + B w w 1 (t), (6) y(t) = C y x(t) + D yw w 2 (t). (7) and the objective [ 1 t ] min lim E (x(s) T Qx(s) + u(s) T Ru(s)ds, (8) t t with Q and R being positive (semi)-definite matrices. This is a classical LQG problem, see also Athans (1971). The setup fits in the more general framework we stated above, when we set ẋ(t) = Ax(t) + (B w ) w + Bu(t) (9a) ( ) ( ) z(t) = Q 1/2 x(t) + R 1/2 u(t) (9b) y(t) = C y x(t) + ( D yw ) w(t) (9c) and abbreviate w(t) = (w 1 (t), w 2 (t)) T. Furthermore, it can be shown that the minimal value of (8) is the same as the H 2 norm of the closed loop system from the disturbances to the output z(t), see also Remark 2. Theorem 6. (Solution to the H 2 problem). Let P, Q R n n be the unique stabilizing solutions to the AREs A T P + P A P B u (D T zud zu ) 1 B T u P + C T z C z = (1) AQ + QA T QC T y (D yw D T yw) 1 C y Q + B w B T w =. (11) Define the control gain K := (D T zud zu ) 1 B T u P and the observer gain L := QC T y (D yw D T yw) 1. Then an H 2 optimal controller is given by u(t) := Kx K (t), (12) where x K (t) R n is the solution to the observer-equation: ẋ K (t) = (A BK)x K (t) + L[y(t) C y x K (t)]. (13) The optimal H 2 norm of the corresponding closed loop system is J = tr ( B T wp B w ) + tr ( B T u P QP V u ). Proof. See for example Zhou et al. (1996). The solution to the optimal output-feedback control problem has an interesting structure: If the full state x(t) is available for the control, then the controller u(t) = Kx(t) is optimal. This feedback controller is also called linear quadratic regulator (LQR). As the state x(t) is usually unknown, it is replaced by an optimal reconstruction in the sense of a Kalman filter, see Kalman and Bucy (1961); Luenberger (1964). The controller is hence an interconnection of an optimal state-feedback controller and an optimal state estimation. Remark 7. In Doyle (1978), it was shown that the LQG regulator, and thus also the H 2 feedback controller, lacks important robustness properties against uncertainties in the plant. Although we get very good results in the numerical examples, the robustness should be checked a- posteriori as it is not guaranteed by theory. 3. REDUCTION OF THE CONTROLLER We have seen in the last section that the determination of the optimal output based feedback controller requires the solution of two AREs (1) (11) and the integration of an n dimensional LTI system (13) for the state estimation. The latter must be performed in real-time, as the measurements from the true noisy output enter the controller, and the control signal generated by the controller must be present (almost) instantaneously to be fed back to the system. Since n can become very large in control problems for semidiscretized PDEs, the calculation of the control signal can get very expensive and easily impracticable for multi-query scenarios and real-time applications. Thus, we will show how existing parametric model reduction techniques for the ARE and for LTI systems can be applied simultaneously, in order to speed up the calculation of the controller and the control significantly. We begin with a brief review of existing reduction techniques for the ARE and dynamical systems. 3.1 Reduction of the ARE The key tool for calculating the control- and observation gain is the ARE, see Theorem 6: A T X + XA XBR 1 B T X + C T C =. (14) The ARE is of great importance in almost all fields of control theory, see for example Lancaster and Rodman (1995); Kwakernaak and Sivan (1972) and the references therein. Thus a lot of work has been done in the development of algorithms for solving large scale AREs, especially with

5 sparse matrices, as they appear for example in control problems for PDEs, see Benner and Saak (213). All of these algorithms work under the assumption that the solution matrices X R n n of the ARE can be factored in a so called low rank factorization X ZZ T with a thin rectangular matrix Z R n k. This is possible due to a fast decay in the singular values of X, which implies a low numerical rank of X. In a very recent paper, the singular value decay rate of solutions to the ARE has been studied, cf. Benner and Bujanović (214). In Schmidt and Haasdonk (215) it was shown, that the low rank structure furthermore allows the approximation of the solutions in the parametric case, by applying the reduced basis method. The calculation is divided in an offlineand an online stage. In the offline step, a low-dimensional subspace V R n with a basis matrix V R n N such that V = colspan V is constructed from the low rank factors Z of solutions to the ARE for carefully selected parameters from a training set P train P. This is done by using Greedy-type algorithms that make use of error estimators or error indicators to iteratively select the parameter which is approximated worst. Its corresponding solution to the full problem is then added to the basis. In the online phase, the full dimensional ARE is projected onto the low dimensional subspace V. Hence only a small ARE must be solved, which can be efficiently done by using standard methods, such as a Hamiltonian approach: A T N X N + X N A N XN T B N R 1 BN T X N + CNC T N = Once the N N matrix P N is calculated, the approximation to the full solution P can be defined as ˆP = V P N V T. The whole procedure can furthermore be rigorously validated by an a-posteriori error estimator P ˆP P. By using parameter separability (all data matrices have the form G(µ) = Q G l=1 Θ l(µ)g l ), the online calculation, including the error estimator, can be performed with a complexity independent of n. 3.2 Reduction of dynamical systems The state estimation requires the integration of an n dimensional differential equation, where the true measurements y(t) enter the observer. This should be performed in real-time, which is usually impossible due to the large dimension of the observer system. We thus propose to apply model order reduction techniques for the observer system, once the control- and observer gains are determined. Several methods exist for the reduction of dynamical systems, see Benner et al. (215) for a recent survey about parametric model reduction techniques. The reduction can be implemented to allow an efficient online calculation, cf. Haasdonk and Ohlberger (211). 3.3 Basis construction for the H 2 controller reduction Let P train P be a finite training set. We construct distinct bases for the two AREs and the observer, to achieve a flexible framework for the reduction. The pseudocode is given in Algorithm 1. We begin with the initialization of the empty basis matrices. In a loop over all training parameters, the snapshots for the AREs and the observer are calculated and concatenated in snapshot matrices. For the construction of the observer snapshots, we fix the input y (t) := sin(2πt) for t [, 1] and calculate the discretized solution for N T = 5 time steps. Once all snapshots are calculated, a singular value decomposition (SVD) is employed to extract the relevant data from the snapshot matrices, see Volkwein (211). The basis is then given by the left singular vectors V P, V Q and Vˆx of the corresponding SVDs. We take as many singular vectors, until we capture tol percent of the data. This can be done by finding the minimum number l such that l i=1 σ i(x) kx i=1 σ i(x) tol X, X {P, Q, ˆx} (15) is satisfied. Here the values σ i (X) are the singular values of the corresponding matrix X. Algorithm 1. Basis construction P [], Q [], ˆx [] for all µ P train do P = Z P ZP T Solution of (1) for µ P [ P, Z P ] Q = Z Q ZQ T Solution of (11) for µ Q [ Q, Z Q ] ˆx Solution of (2a) for µ with input y (t) ˆx [ ˆx, (x(t i )) N T i=1 ] end for [V P, S P ] SVD( P, tol P ) [V Q, S Q ] SVD( Q, tol Q ) [Vˆx, Sˆx ] SVD( ˆx, tolˆx ) 3.4 The reduced controller We assume that basis matrices V P R n N P, V Q R n N Q and Vˆx R n Nˆx for the AREs for P, Q and the observer ˆx are available. We define the reduced controller by using the reduced quantities in the equations for the feedback gain and the observer gain in the controller (12)-(13), and a subsequent projection of the reduced system for ˆx: Definition 8. (Reduced controller). Let ˆP = V P P NP VP T, ˆQ = VQ Q NQ VQ T be the approximate solutions to the ARE (1) and (11). Define the reduced gains ˆK := Bu T (DzuD T zu ) 1 ˆP, ˆL := T ˆQC y (D yw Dyw) T 1, and define the control signal û(t) := ˆK x(t), where x(t) R Nˆx satisfies the reduced observer equation x = Vx Ṱ (A B ˆK ˆLC y )Vˆx x + Vx Ṱ ˆLy (16) Remark 9. The above formulation captures the case where only a subset of the quantities P,Q and ˆx are reduced. This can be achieved by assuming V X = I n for any X equal to V P, V Q or Vˆx in Definition 8. Remark 1. Note that the whole procedure can be split in an offline/online efficient scheme, when parameter separability of the data matrices is assumed. This is possible due to an offline precalculation of matrix products that occur in the formulation of the controller. 4. NUMERICAL EXAMPLES We investigate the proposed procedure numerically. As a model we choose a damped wave equation on the 2D unit

6 1 µ 1 = 1 µ 1 = P Q ˆx Ω u y(t) 1 2 Ω y t Fig. 2. The setting for the numerical experiments (left) and the noisy uncontrolled output y for different damping parameters (right). square Ω = [, 1] 2. The PDE for the state f = f(t, ξ; µ), which models the wave propagation is given as f tt.2 f + µ 1 f t = χ Ωu u + B w w 1, ξ Ω, t > f(t, ξ; µ) =, ξ Ω, t f(, ξ; µ) = sin(ξ 1 ) sin(ξ 2 ), ξ Ω f t (, ξ; µ) =, ξ Ω. Here, χ Y = χ Y (ξ) is the characteristic function of the set Y at the point ξ and B w (ξ) := 2 sin(ξ 1 ) sin(ξ 2 ) describes the action of the noise w 1 (t) onto the system. The parameter µ 1 [1, 4] defines the damping of the wave. We define the measurement output s(t; µ) := 1 f(t, ξ; µ)dξ +.5w 2 (t), Ω y Ω y where w 2 (t) models additional disturbances. We choose two performance outputs: The undisturbed measurement 1 o 1 (t; µ) := µ 2 Ω y Ω y f(t, ξ; µ)dξ, weighted by the parameter µ 2 [1, 4] and the (scaled) control signal o 2 (t; µ) =.1u(t). A larger value for µ 2 results in a faster stabilization of the system. We chose independent Gaussian white noise for w 1 (t) and w 2 (t) with unit covariance. A picture of the setting and two example measurement outputs are given in Figure 2. We discretize the control model by applying a central finite difference (FD) scheme in space. The resulting n dimensional second order system can be rewritten as a n := 2n dimensional system with three outputs ẋ = A(µ)x + B w w + B u u y = C y x + D yw w z = C z (µ)x + D zu u, where we define the noise vector w(t) ( T ):= (w 1 (t), w 2 (t)) x1 and the augmented state x := R x 2n. The 2 vector z(t) is two-dimensional, where the first component represents the discretized version of o 1 (t) and the second component is the discrete weighted control o 2 (t). The system matrices are defined as ( ) ( ) ( In A(µ) =, B.2 h µ w =, B f B w u =, Bu) ( ) ( µ2 Cz D yw = ( 1), C z (µ) =, D zu (µ) =, µ3) where all matrices are the discrete counterparts of the continuous operators. h is the discrete FD-representation of the Laplace operator. The boundary values are not mapped in the vector x explicitly, but are considered in Fig. 3. Singular value decay for the snapshot matrices. The points and vertical lines indicate the basis size for which % of all snapshot data is covered. the discrete Laplace operator. The parameters vary in the parameter set P := [1, 4] [1, 5]. We chose a relative low dimension n = 4 since this already suffices to show the benefit of the proposed approach. The first order system is thus of dimension n = 8. All calculations were performed by using the Control Systems Toolbox in MATLAB. A full solution of one ARE takes about 1 seconds in MATLAB 215a with the function care. One timestep for the integration of the observer ODE takes seconds for the full 8 dimensional ODE. Note that both times can be significantly be sped up by using advanced solvers that make use of the sparsity and exploit the low rank structure. However, by increasing the dimension of the discretization, one can easily reach system sizes that make the procedure impracticable, even when using the latest solvers. We build the basis by applying Algorithm 1 and by using a finite training set with 5 elements, chosen randomly from the parameter domain P. In Figure 3 we plot the decay of the singular values for the snapshot matrices Q, P and ˆx. We see an exponential decay in all matrices, implying that the reduction will lead to good results. From the singular value decomposition, we extract a subset of the left singular vectors as basis for each of the components P, Q and ˆx. We do this by requiring the basis to capture % of the L 2 -energy of all snapshot data, see also Algorithm 1. The lines in the plot indicate this threshold. Tests with a larger basis introduced numerical inaccuracies that led to instabilities in the calculation of the solutions to the reduced AREs. The resulting bases (subsequently called full basis) have the sizes N P = 72, N Q = 2, and Nˆx = 67. Especially the low dimension of the observer allows a fast state estimation, which is the most crucial point for real-time applications. For testing the basis, we define a test set P test P with N test = 1 random parameters that were not contained in the training set P train for the basis building procedure. First, we compare the qualitative behavior by checking the control signals, see Figure 4. In this example, we used the full basis for the approximation of Q and ˆx. Already a small basis for the feedback matrix suffices to get a good control signal. For the full N P, the control signals are almost equal. A more quantitative comparison is given in Table 1. In this table, the relative error in the H 2 norm of the closed loop transfer functions from the disturbance input w(t) to the performance output z(t) is given

7 u(t) Table 1. Relative error in the transfer functions from w to z in the closed loop systems with zero initial condition. Using Nˆx = 4 leads to an unstable system. N P Nˆx Full N P =1 N P =2 N P = t Fig. 4. Control signal (second component of z(t)) for fixed µ = (1, 1), N Q = 2, Nˆx = 67 and increasing N P. G(s) G r (s) 2 G(s) 2. (17) The system behaves in an unstable way for Nˆx = 4, which could be due to the lack of robustness of the H 2 optimal controller. Uncertainties in the control loop can lead to instabilities. However, when using the full basis, we never experienced unstable behavior in all experiments that were performed. The good approximation behavior is also mirrored by the low relative error of.7%, when the full basis is used. One solution to the reduced ARE takes only 2ms and the speed-up for the integration of the observer is of factor 1, even in this small example. 5. CONCLUDING REMARKS In this article we considered the application of existing model order reduction techniques to the H 2 output feedback control problem. We showed how the controller can be reduced. Numerical examples proved that the procedure can lead to a large speed-up compared to a full calculation and simulation. Unfortunatelly, the H 2 control problem lacks robustness properties like the H robust control method. The numerical calculations showed instabilities, when too small basis sizes were employed. An application of the proposed framework to the H problem could lead to a more robust reduction, and maybe even allows to prove guaranteed bounds for ensuring the stability of the procedure. ACKNOWLEDGEMENTS The authors acknowledges funding by the Landesstiftung Baden Württemberg ggmbh and would like to thank the German Research Foundation (DFG) for financial support within the Cluster of Excellence in Simulation Technology (EXC 31/1) at the University of Stuttgart. REFERENCES Athans, M. (1971). The role and use of the stochastic linear-quadratic-gaussian problem in control system design. IEEE Transactions on Automatic Control, 16(6), doi:1.119/tac Benner, P. and Saak, J. (213). Numerical solution of large and sparse continuous time algebraic matrix Riccati and Lyapunov equations: a state of the art survey. GAMM-Mitteilungen, 36(1), doi: 1.12/gamm Benner, P. and Bujanović, Z. (214). On the solution of large-scale algebraic Riccati equations by using lowdimensional invariant subspaces. Preprint MPIMD/14-15, Max Planck Institute Magdeburg. Benner, P., Gugercin, S., and Willcox, K. (215). A survey of projection-based model reduction methods for parametric dynamical systems. SIAM Rev., 57(4), doi:1.1137/ Dihlmann, M. and Haasdonk, B. (215). A reduced basis Kalman filter for parametrized partial differential equations. ESAIM: Control, Optimisation and Calculus of Variations. doi:1.151/cocv/ Doyle, J. (1978). Guaranteed margins for LQG regulators. Automatic Control, IEEE Transactions on, 23(4), doi:1.119/tac Haasdonk, B. and Ohlberger, M. (211). Efficient reduced models and a posteriori error estimation for parametrized dynamical systems by offline/online decomposition. Math. Comp. Model Dyn., 17(2), doi:1.18/ Kalman, R.E. and Bucy, R.S. (1961). New results in linear filtering and prediction theory. Journal of Basic Engineering, 83(1), 95. doi:1.1115/ Kwakernaak, H. and Sivan, R. (1972). Linear optimal control systems, volume 1. Wiley-interscience New York. Lancaster, P. and Rodman, L. (1995). Algebraic Riccati equations. Oxford University Press. Luenberger, D. (1964). Observing the state of a linear system. Military Electronics, IEEE Transactions on, 8(2), doi:1.119/tme Patera, A. and Rozza, G. (27). Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations. To appear in (tentative) MIT Pappalardo Graduate Monographs in Mechanical Engineering. MIT. Schmidt, A., Dihlmann, M., and Haasdonk, B. (214). Basis generation approaches for a reduced basis linear quadratic regulator. In Proceedings of MATHMOD 215, doi:1.116/j.ifacol Schmidt, A. and Haasdonk, B. (215). Reduced basis approximation of large scale algebraic Riccati equations. Technical report, University of Stuttgart. Submitted. Volkwein, S. (211). Model reduction using proper orthogonal decomposition. Lecture notes, University of Konstanz. Zhou, K., Doyle, J., and Glover, K. (1996). Robust and Optimal Control. Prentice Hall.