Optimal Control of Linear Systems with Stochastic Parameters for Variance Suppression

Transcription

1 Optimal Control of inear Systems with Stochastic Parameters for Variance Suppression Kenji Fujimoto, Yuhei Ota and Makishi Nakayama Abstract In this paper, we consider an optimal control problem for a linear discrete time system with stochastic parameters in the infinite time horizon case. This paper focuses on optimal control for systems with stochastic parameters whereas the traditional stochastic optimal control theory mainly considers systems with deterministic parameters with stochastic noises. This paper extends the authors former result on the same subject in the finite time horizon case to the infinite time horizon case. The main result is to provide a feedback controller suppressing the variation of the state and to prove stochastic stability of the corresponding feedback system by taking care of both the average and the variance of the state transient. Furthermore, a numerical simulations demonstrate the effectiveness of the proposed method. I. INTRODUCTION Stochastic and statistical methods are used in several problems in control systems theory. In particular, system identification and estimation methods rely on them. Recently, it is reported that Bayesian inference and the related estimation tools in machine learning [1], [2] are applied to system identification of state-space models [3], [4], [5], [6]. Consequently, statistical information of the system parameters become available for control. There also exists a result reporting that the quality of the manufactured products are estimated by those methods in process control [7]. So called randomized approach is also proposed to apply statistical tools to controller design problems [8], [9]. On the other hand, optimal control is an important and established control method. QG inear Quadratic Gaussian method can take care of optimal control problems with stochastic disturbances [10], [11]. Stochastic control theory has been developed by many authors based on it which mainly focuses on systems with deterministic parameters and stochastic noises. In stochastic control theory, MCV Minimum Cost Variance control [12], [13] and RS Risk Sensitive control [14] are proposed which can suppress the variance of the cost function with respect to the stochastic disturbance. There is a paper comparing those method as well [15]. Those existing results focus on the systems with deterministic parameters with stochastic disturbances. In order to utilize the Bayesian estimation results for state space K. Fujimoto is with Department of Mechanical Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, , Japan fujimoto@nagoya-u.jp Y. Ota is with Department of Mechanical Science and Engineering, Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, , Japan ota@haya.nuem.nagoya-u.ac.jp M. Nakayama is with Kobe Steel, TD., Kobe, , Japan nakayama.makishi@kobelco.com models, controller design methods for state space systems with stochastic parameters are needed. For this problem, De Koning proposed a controller design method for state space systems with stochastic system parameters [16], [17], [18] which employs standard QG type cost function. The results in the paper, however, cannot suppress the variance of the state transient of the resulting control system caused by the variation of the system parameters. The authors proposed a generalized version of [16] to take the variation of the state into account by adopting a novel cost function including the covariance of the states [19]. But the result in [19] only considers an optimal control problem in the finite time horizon case. The present paper extends the authors former result to the infinite time horizon case. The optimal state feedback controller suppressing the variation of the transient is derived. Furthermore, stability of the corresponding feedback system is proved using the analysis tools for stochastic systems provided by De Koning [16]. Furthermore, numerical simulations demonstrate the effectiveness of the proposed method. The proposed method can provide a new framework to stochastic control which can be used together with Bayesian inference of the system parameters. II. PREIMINARIES This section gives notations and some preliminary results according to [1], [16]. The symbol N denotes the space of natural numbers. R n denotes the n-dimensional Euclidean space. M mn and M n denote the space of m n real valued rectangular matrices and that of n n real valued square matrices, respectively. S n denotes the space of n n real valued symmetric matrices. The symbol vec denotes a function satisfying a 1 a 1i a 2 veca =. a 2i Rmn, a i :=. Rm. a n a mi where a ij is the i, j element of the matrix A M mn. The expectation of a time-varying stochastic parameter a t is denoted without the time parameter t as E[a] instead of E[a t ], if its statistics is time invariant. We use several transformations for symmetric matrices. et us define a monotonic transformation. Definition 1: [16] A transformation for symmetric matrices A : S n S n is said to be monotonic if it satisfies

2 AX AY, for any symmetric matrices X, Y S n satisfying 0 X Y. The following lemma holds for monotonic transformations. emma 1: [16] Consider the following equation. X = AX + B, X, B S n 1 Suppose that the transformation A is linear, monotonic and stable and that B 0. Then there exists a solution X 0 satisfying 1. Further, the next lemma holds. emma 2: Suppose that the transformation A i : S n S n is monotonic and X 0 A i X 0 holds for any integer i. Then a composition of the transformations A i s { A i:j Aj A := j 1 A i i j 2 id i > j is monotonic and X 0 A i:j X 0 holds for any integers i and j. Proof: emma is proved by induction. In the case j i 0, the claim holds obviously. Suppose that it holds when j i = k for a non-negative integer k. Then A i:j is monotonic. Since A j+1 is also monotonic, we have X 0 A i:j X 0 A j+1 A i:j X A j+1 0 A i:j+1 X 0. Further, the monotonicity of A i:j and A j+1 implies 0 X Y A i:j X A i:j Y A j+1 A i:j X A j+1 A i:j Y 0 A i:j+1 X A i:j+1 Y. Therefore, A i:j+1 is monotonic. This implies that the claim holds for the case j i = k + 1. Hence the claim holds for any integers i and j, which proves the lemma. Next, let us consider a discrete time system x t+1 = A t x t + B t u t 3 where x t R n is the state, u t R m is the control input, A t M n and B t M nm are the system matrices. Suppose that the system matrices A t and B t consist of stochastic variables with time invariant statistics. Assume also that the initial state x 0 is deterministic. Apply a state feedback u t = x t 4 to the system 3 with a feedback gain M mn. Then the resulting feedback system is described by x t+1 = Ψ,t x t 5 with a new system matrix Ψ,t := A t B t M n. Stability of this feedback system is defined as follows. Definition 2: [16] The feedback system 5 is said to be m-stable if lim t E[x t x 0 ] = 0 holds for x 0 R n. It is said to be ms-stable if lim t E[ x t 2 x 0 ] = 0 holds for x 0 R n. In order to describe the behavior of the state x t with its variance, let us define a transformation A : S n S n describing the expectation of a quadratic function along the feedback system 5 as follows. A X := E[Ψ T XΨ ], X S n 6 The transformation A thus defined satisfies the following lemma. emma 3: [16] a E[x T i Xx i x 0 ] = x T 0 A i Xx 0 holds for any natural number i N and any symmetric matrix X S n. b The transformation A i is linear and monotonic for any natural number i N. c The feedback system 5 is m-stable if and only if E[Ψ ] is Hurwitz asymptotically stable. Furthermore, it is msstable if and only if the transformation A is stable. Stabilizability of the system 3 with A t, B t is defined as follows. Definition 3: [16] The pair A t, B t is said to be m- stabilizable if there exists a feedback gain M mn in such a way that the feedback system 5 is m-stable. It is said to be ms-stable if if there exists such that 5 is ms-stable. emma 3 c and Definition 3 imply the following lemma. emma 4: [16] The pair A t, B t is m-stabilizable if and only if there exists a feedback gain in such a way that E[Ψ ] is asymptotically stable. Furthermore, it is msstabilizable if and only if there exists such that A is stable. III. OPTIMA CONTRO FOR VARIANCE SUPPRESSION This section is devoted to optimal control for variance suppression. A. The finite time horizon case In the authors former result [19], optimal control for variance suppression in the finite time case is presented. This subsection briefly reviews this result. Consider a linear discrete-time system with stochastic parameters x t+1 = A t x t + B t u t + G t ɛ t 7 where ɛ t R n is a stochastic external noise. Matrices A t M n, B t M nm, and G t M n are stochastic parameters defined by time invariant statistics. For this system, let us consider the following cost function. [ N 1 { J N U N, x 0 = E x T t Qx t + u T t Ru t + trs cov[x t+1 x t ] } + x TN F x N x 0 ] Here the matrices Q 0 S n, R 0 S m, F 0 S n, and S 0 S n are design parameters and U N = {u t, 0 t N 1} denotes the collection of the input. The third term in the right hand side of Equation 8 reduces to tr S cov[x t+1 x t ] = E [ x T t+1sx t+1 x t ] E[xt+1 x t ] T S E[x t+1 x t ] which is the weighted sum of the covariance of the states. The coefficient matrix S can be used to select the weights between the variation and the average of the state. For this system, the optimal control problem for variance suppression is defined as follows. 8

3 Definition 4: Consider the system 7 and the cost function 8. For a given initial condition x 0 R n, find an input sequence U N = {u t, 0 t N 1} minimizing the cost function J N U N, x 0 and the corresponding minimum value of the cost function J N x 0. We call this control problem as a finite time optimal control for variance suppression. Before solving the variance suppression problem, let us consider the relationship between the value of the cost function at the time t with the state x t and that at the time t 1 with the state x t 1. Suppose that a state feedback u t = x t is employed, then the term to evaluate the variance of the state E[trS cov[x t+1 x t ] x 0 ] in the cost function can be calculated as E[trS cov[x t+1 x t ] x 0 ] E [trs cov[x t+1 x t ] x 0 ] = E [ E[x T t+1sx t+1 x t ] E[x t+1 x t ] T ] S E[x t+1 x t ] x 0 = E [ x T t E[Ψ T t SΨ t ] E[Ψ T t ]S E [Ψ t ] x t x 0 = E [ x T t At S E[Ψ T t ]S E[Ψ t ] ] x t x 0 = E [ x T t 1A t At S E[Ψ T t ]S E[Ψ t ] ] x t 1 x 0 = E [ x T { 0 A 0:t S A 1:t E[Ψ T 0 ]S E[Ψ 0 ] } ] x 0 x 0 = x T { 0 A 0:t S A 1:t E[Ψ T 0 ]S E[Ψ 0 ] } x 0 9 Here we use the notation A i:j = A j A j 1 A i as in Equation 2. Now the function B : S n S n is defined by B X := A X + Q + T R + A S E[Ψ T ]S E[Ψ ], X S n. 10 The function β : S n R is defined by βx := E[ɛ T G T XGɛ]. 11 Then the value of the cost function 8 becomes J N U N, x 0 = E = x T 0 [ N 1 { x T t Q + T t R t x t } ] + trs cov[x t+1 x t ] + x TN F x N x 0 { N 1 A 0:t Q + T 0 R 0 + A 0 S E[Ψ T 0 ]S E[Ψ 0 ] + β N 1 { A 1:t j=0 + A 0:N F } x 0 t 1 F + A 1:j Q + T 0 R 0 + A 0 S E[Ψ T 0 ]S E[Ψ 0 ] } + NS = x T 0 B 0:N 1 F x 0 + β NS + N 1 B 1:t F 12 We can prove a property of the transformation B i:j as follows. emma 5: The transformation B i:j is monotonic and X X 0 holds for any non-negative integers i and j. 0 B i:j Proof: Proof is omitted for the limitation of space. Using this lemma, we can prove the main theorem which provides the solution to the optimal control for variance suppression. Theorem 1: [19] The optimal control law to minimize the cost function 8 and the minimum value of the cost function J N x 0 are given as follows. u t = B N t 1 F x t, t = 0,..., N 1 13 J Nx 0 = x T 0 B N F x 0 + β NS + N 1 B t F Here the function B : S n S n is defined by, x 0 14 B X := B X X, X S n. 15 The function B X is defined in Equation 10 and the gain matrix X is defined by X := E[B T XB] + Σ BB + R 1 E[B T XA] + Σ BA, X S n 16 where Σ XY := E[X T SY ] E[X] T S E[Y ]. et us denote the value of the cost function at the time t by V t = x T 0 Π t x 0. Then Equation 15 reduces to the following recursive equation similar to the Riccati equation. Π t 1 = Q + Σ AA + E[A T Π t A] E[A T Π t B] + Σ AB E[B T Π t B] + Σ BB + R 1 E[B T Π t A] + Σ BA Here Π N = F and B k F = Π N k, namely, u t = Πt x t. 17 Furthermore, Equation 17 reduces to a Riccati equation for the conventional QG problem if the parameters A t and B t are deterministic. This implies that the proposed method is a natural generalization of the conventional QG method. Note that the computation of Π t using Equation 17 needs the 2nd order statistics of the stochastic variables A t and B t. B. The infinite time horizon case This subsection derives the optimal control law for the infinite time horizon which is the main result of the present paper. et us consider the system 7 with G t 0, that is, the system 3 without the noise ɛ t is considered. For this system, the following cost function with the infinite time horizon is adopted [ { J U, x 0 = E x T t Qx t + u T t Ru t where U = {u 0, u 1,...}. } ] + trs cov[x t+1 x t ] x 0 18

4 The optimal control problem considered here is formalized as follows. Definition 5: Consider the system 3 and the cost function 18. For a given initial condition x 0, find the control input U = {u t } minimizing the cost function J U, x 0. The corresponding minimum value is denoted by J x 0. This problem is called an infinite time optimal control problem for variance suppression. Before stating the main result, we need to clarify some properties of the optimal control for variance suppression in the finite time horizon case. The transformation B : S n S n in Equation 15 is to convert the matrix Π t characterizing the cost V t x t = x T t Π t x t to another Π t 1 corresponding the cost V t 1 x t 1 = x T t 1Π t 1 x t 1. The difference between B and B defined in Equation 10 is explained as follows. Since B corresponds to the constant feedback gain, the resulting cost function is Jx 0 = x T 0 B N F x 0. On the other hand, since B := B X corresponds to the timevarying feedback gain X, the resulting cost function satisfies JN x 0 x T 0 B N X F x 0 whereas we can still use the expression JN x 0 = x T 0 B N F x 0. The following lemma on the transformation B is useful in proving the main theorem. emma 6: a An inequality 0 B N F B 0:N 1 F holds for any natural number N N, any matrix M n m, and any symmetric matrix F 0, F S n. b The transformation B N is monotonic for any natural number N N. Proof: a Due to the optimality of the solution, we have 0 J Nx 0 J N U N, x 0 0 x T 0 B N F x 0 x T 0 B 0:N 1 F x 0, x 0 Hence the claim of the lemma holds. b The part a implies that B X 0 holds for any X 0. emma 5 suggests that B X = B X X B Y Y = B Y holds for any X and Y satisfying 0 X Y. Hence B is monotonic. It follows from emma 2 a that B N is monotonic for any natural number N N. Now we are ready to prove the main result. Theorem 2: Suppose that the system A t, B t is msstabilizable. Then Π := lim N B N 0 exists and Π is the minimum positive semi-definite solution to Π = B Π = A Π Π + Q + T ΠR Π + A Π S E[Ψ T Π ]S E[Ψ Π ] = Q + Σ AA + E[A T ΠA] E[A T ΠB] + Σ AB E[B T ΠB] + Σ BB + R 1 E[B T ΠA] + Σ BA. 19 Furthermore, the optimal feedback input is given by u t = Π x t = E[B T ΠB] + R + Σ BB 1 E[B T ΠA] + Σ BA x t 20 for which the cost function 18 takes its minimum value J x 0 = x T 0 Πx 0. Proof: First of all, we prove that the function J N x 0 is monotonically non-decreasing with respect to the terminal time N. Select N 1 and N 2 satisfying 0 < N 1 < N 2, then the following equations hold for any initial state x 0. J N 1 x 0 = x T 0 B N 1 0x 0 J N 2 x 0 = x T 0 B N 2 0x 0. The definition 15 of the transformation B implies that B 0 0. Furthermore, it follows from emma 6 b that B N is monotonic hence B N 2 0 = B N 2 1 B 0 B N 2 1 B N1+1 0 = B N1 0 B 0 B N1 0. This means J N 1 x 0 J N 2 x 0 and, consequently, the cost function J N x 0 is monotonically non-decreasing with respect to the terminal time N. Therefore, there exists a lower bond of J N x 0. Next we prove the existence of Π = lim N B X. Since the system A t, B t is ms-stabilizable, it follows from emma 5 that there exists a feedback gain such that A is stable. Hence emma 1 implies that the equation Π = B Π= A Π + Q + T R + A S E[Ψ T ]S E[Ψ ] has a solution Π 0. Further, by induction, we have B N Π = Π. emma 6 suggests Π = B N Π B N Π B N So the sequence {B N 0} has an upper bound and is monotonically non-decreasing. Therefore, the limit Π = lim N B N 0 exists. The equation B N+1 0 = B B N 0 implies Π = B Π 22 by taking the limit N. Next we prove that Π = lim N B N 0 is the minimum solution to 22. Suppose that Π 0 is another solution of 22. Then emma 6 b implies that B N is monotonic for any natural number N N. Hence B N 0 B N Π = Π holds. Taking the limit N, we obtain Π Π which means that Π is the minimum solution. Finally, let us prove that the feedback 20 is the optimal input. et Û := {û t }, t 0 denote the sequence of the feedback input û t = Π x t. et J N U N, x 0, X denote the cost 8 where the terminal time is N and the terminal cost F = X. Then we have J N Û, x 0, 0 J N Û, x 0, Π = x T 0 B N Π Πx 0 = x T 0 Πx 0 for any N N. Taking the limit N, On the other hand, J Û, x 0 x T 0 Πx x T 0 B N 0x 0 = J N x 0, 0 J x 0 J Û, x 0 24 also holds. Inequalities 23 and 24 suggest x T 0 B N 0x 0 J x 0 J Û, x 0 x T 0 Πx 0.

5 Again, taking the limit N, J x 0 = J Û, x 0 = x T 0 Πx which implies U = Û. Therefore u t = û t = Π x t for t 0. This completes the proof. Equation 19 reduces to the algebraic Riccati equation for the conventional discrete time optimal control if the system parameters A t and B t are constant deterministic. However, in the stochastic case, Equation 19 is not a Riccati equation anymore and it cannot be solved with the conventional technique in the linear control systems theory. For instance, although E[A T ΠB] is linear in Π, there does not exist constant deterministic matrices X and Y satisfying E[A T ΠB] = XΠY in general because the system parameters A t and B t are stochastic variables. Therefore we need to employ nonlinear optimization to solve Equation 19. Here we use the following procedure to obtain the solution Π. First of all, define the cost function for nonlinear optimization ΓΠ as follows. ΓΠ : = Π + Q + Σ AA + E[A T ΠA] E[A T ΠB] + Σ AB E[B T ΠB] + Σ BB + R 1 E[B T ΠA] + Σ BA Since Equation ΓΠ = 0 is equivalent to Equation 19, we compute Π numerically by minimizing ΓΠ F = vecγπ where F is the Frobenius norm. Here we select the initial value Π = Π 0 the solution to the conventional Riccati equation in the deterministic case Π 0 =Q + Σ AA + E[A] T Π 0 E[A] E[A] T Π 0 E[B] + Σ AB E[B] T Π 0 E[B] + Σ BB + R 1 E[B] T Π 0 E[A] + Σ BA. This procedure often gives us the true solution when the variances of the system parameters are small. When the nonlinear optimization does not give the global optimal the solution to Equation 19, then applying the nonlinear optimization procedure recursively by changing the variances of the system parameters from 0 to the true value would give us the global optimal. IV. NUMERICA EXAMPE This section gives a numerical example to demonstrate the effectiveness of the proposed method in comparison to the conventional QG method. et us consider the plant described by Equation 3 with the state x t R 2 and the input u t R. [ ] E[A] = [ ] 0 E[B] = 0.01 cov [vec[a, B]] = [ ] 10 0 Q = 0 10 R = 1 [ ] 10 x 0 = 10 The covariance cov [vec[a, B]] are selected in such a way that the parameters A 21, A 22 and B 2 have 20% standard deviations. Fig.1 shows the time responses of 100 random samples of the feedback system with the conventional QG controller. Fig.2 shows the time responses by the proposed method with the design parameter S = 1000I. In those figures, the plus signs + denote the upper and lower bounds of the 1σ deviation from the average and the solid lines denote the sampled responses. Figures show that both the conventional QG controller and the proposed controller achieve both m-stability and msstability. The response with the proposed controller in Fig.2 achieves smaller variance than that with the QG controller in Fig.1 which indicates the effect of the proposed method. i State transition of x 1 Fig. 1. i State transition of x 1 Fig. 2. ii State transition of x 2 State transition QG ii State transition of x 2 State transition proposed method Next, Figs.3 4 show the case in which the variances of the parameters are bigger than the case in Figs.1 2. The covariance of the system parameters A and B is selected as follows in such a way that the parameters A 21, A 22 and B 2

6 have 40% standard deviations. cov [vec[a, B]] = Figs.3 4 show the time responses of 100 random samples by the QG method and the proposed method with the design parameter S = 1000I, respectively. In those figures, the plus signs + denote the upper and lower bounds of the 1σ deviation from the average and the solid lines denote the sampled responses as in Figs.1 2. i State transition of x 1 Fig. 3. i State transition of x 1 Fig. 4. ii State transition of x 2 State transition QG ii State transition of x 2 State transition proposed method Fig.3 shows the time responses of the state of the feedback system with the QG controller. In the figure the states diverge, since the QG controller does not take care of the variance of the system parameters. Fig.4 shows the time responses of the state with the proposed controller for the parameter S = 1000I. This result shows that the proposed method can stabilize the variance of the states in the sense of ms-stability, whereas the variance of the feedback system with the OG controller becomes unstable in the sense of ms-stability. Those results confirms the effectiveness of the proposed method. V. CONCUSION This paper generalizes the authors former result on optimal control with variance suppression to the infinite time horizon case. We have derived an algebraic equation similar to the algebraic Riccati equation to characterize the solution to the optimal control problem and proved stochastic stability of the corresponding feedback system by taking care of the average and the variance of the state transient. Furthermore, some numerical simulations exhibit the effectiveness of the proposed method. We believe that the proposed method provides a new stochastic control framework to work with the Bayesian inference methods. VI. ACKNOWEDGMENTS The authors greatly appreciate Prof. Fujisaki of Osaka University for his valuable suggestion. REFERENCES [1] C. M. Bishop, Pattern Recognition and Machine earning, Springer, New York, [2] H. Attias, Inferring parameters and structure of latent variable models by variational bayes, in Proc. 15th Conf. on Uncertainty in Artificial Intelligence, 1999, pp [3] M. J. Beal, Variational Algorithms for Approximate Bayesian inference, Ph.D. thesis, University of ondong, ondon, UK, [4] D. Barber and S. Chiappa, Unified inference for variational Bayesian linear Gaussian state-space models, in Advances in Neural Information Processing Systems 19 NIPS 20, pp The MIT Press, [5] S. Fukunaga, Y. Ishihara, and K. Fujimoto, Byesian estimation methods for state-space models, in Proc. SICE 8th Annual Conference on Control Systems, 2008, in Japanese. [6] K. Fujimoto, A. Satoh, and S. Fukunaga, System identification based on variational bayes method and the inavriance under coordinate transformations, Submitted, [7] M. Kano, Modeling from process data, Measurement and Control, vol. 49, no. 2, pp , 2010, in Japanese. [8] G. Calafiore, R. Tempo, and F. Dabbene, Randomized Algorithms for Analysis and Control of Uncertain Systems, Springer, [9] Y. Fujisaki and Y. Kozawa, Probabilistic robust controller design: probable near minimax value and randomized algorithms, in Probabilistic and Randomized Methods for Design under Uncertainty, G. Calafiore and F. Dabbene, Eds., pp Springer-Verlag, [10] J. M. Mendel and D.. Gieseking, Bibliography on the inear- Quadratic-Gaussian problem, IEEE Trans. Autom. Contr., vol. 6, pp , [11] K. J. Aström, Introduction to Stochastic Control Theory, Dover Publications, [12] M. K. Sain, Control of linear systems according to the minimal variance criterion: A new approach to the disturbance problem, IEEE Trans. Autom. Contr., vol. 11, no. 1, pp , [13] M. K. Sain, C. H. Won, and B. F. Spencer Jr, Cumulants in risk-sensitive contro: The full-state-feedback cost variance case risksensitive and MCV stochastic control, in Proc. 34th IEEE Conf. on Decision and Control, 1995, pp [14] P. Whittle, Risk-sensitive inear/quadratic/gaussian control, Advances in Applied Probability, vol. 13, pp , [15] C.-H. Won and K. T. Gunaratne, Performance study of QG, MCV, and risk-sensitive control methods for satellite structure control, in Proc. American Control Conference, 2002, pp [16] W.. De Koning, Infinite horizon optimal control of linear discrete time systems with stochastic parameters, Automatica, vol. 18, no. 4, pp , [17] W.. De Koning, Optimal estimation of linear discrete-time systems with stochastic parameters, Automatica, vol. 20, no. 1, pp , [18] W. De Koning and. G. van Willigenburg, Randomized digital optimal control, in Non Uniform Sampling Theory and Practice / Farokh Marvasti, pp Kluwer Academic/Plenum Publishers, [19] K. Fujimoto, S. Ogawa, Y. Ota, and M. Nakayama, Optimal control of linear systems with stochastic parameters for variance suppression: The finite time horizon case, in Proceedings of the 18th IFAC World Congress, 2011.