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Convex optimization for finite-horizon robust covariance control of linear stochastic systems. (English) Zbl 1496.90048

In this paper the following finite horizon, discrete-time control problem is considered: \begin{gather*} x_{0}=z+s_{0},\quad x_{t+1}=A_{t}x_{t}+B_{t}u_{t}+B_{t}^{(d)}d_{t}+B_{t} ^{(s)}e_{t}\\ y_{t}=C_{t}x_{t}+D_{t}^{(d)}d_{t}+D_{t}^{(s)}e_{t},\qquad0\leq t\leq N-1 \end{gather*} where \(x_{t}\in \mathbb{R}^{n_{x}}\) are states, \(u_{t}\in \mathbb{R}^{n_{u}}\) are controls, \(y_{t}\in \mathbb{R}^{n_{y}}\) are observable outputs, \(z\in \mathbb{R}^{n_{x}}\) and \(d_{t}\in \mathbb{R}^{n_{d}}\) are deterministic factors, \(s_{0}\in \mathbb{R}^{n_{x}}\), \(e_{t}\in \mathbb{R}^{n_{e}}\) are stochastic factors. The deterministic disturbance vector \(\zeta=\left[ z;d_{0};\ldots;d_{N-1}\right] \) is assumed to lie in an ellitop (for example, a finite intersection of centered at the origin ellipsoids and elliptic cylinders). \(A_{t},B_{t},\ldots,D_{t}^{(s)}\) are known matrices.
For this problem a procedure for designing control policies that guarantee some performance specifications is developed. The parameters of the policy are obtained as solutions to an explicit convex program.

MSC:

90C17 Robustness in mathematical programming
90C47 Minimax problems in mathematical programming
90C22 Semidefinite programming
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
49M29 Numerical methods involving duality

Software:

CVX
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Full Text: DOI arXiv

References:

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