Randomized algorithms for robust stability and guaranteed cost control of stochastic jump parameter systems with uncertain switching policies
The author considers a linear jump control system described by the following differential equation: \[ \begin{aligned} \dot{x}(t) & = A( n(t)) x(t) + B (\eta(t)) u(t, \eta (t)) ,\\ x(0) & = x_0, \qquad \eta (0) = \eta_0 \end{aligned}\tag{1} \] where \(x(t) \in \mathbb R^n\) denotes the state of the controlled plant, \(u(t,i) \in \mathbb R^m\) denotes the control input relevant to the \(i\)th form of the plant, \(\{ \eta (t), t\geq 0\}\) is a continuous-time Markov process defined on the finite state space \(N = \{1, 2, \ldots, N\}.\) The transition probability matrix of this process \(P(t)\) is assumed to satisfy the Kolmogorov-Chapman equation \[ (d/ dt) P(t) = \Theta P(t) \tag{2} \] where the transition probability rate matrix \(\Theta\) satisfies the following conditions: \(q_{ij} \geq 0\), \(q_{ii} \leq 0.\) Let \(x(t)\) be a trajectory of the system (1) corresponding to an initial condition \(x(0) = x_0\), \(\eta (0) =i.\) The author considers state-feedback control policies \[ u(t) = K(t, x(t), \eta (t)), \quad K\: [0, \infty) \times \mathbb R^n \times N \to \mathbb R^m \tag{3} \] and presents the definition the robustly stabilizing control system of the form (1) with the control law (3). The problem is to find a controller that attains the minimum of the cost functional \(\inf_n J (u, x_0, i)\) where \[ J(u, x_0, i) = {\mathcal E}^{x_0, i} \biggl\{ \int_0^\infty [ x(t)' R(\eta(t)) x(t) + u(t)' G(\eta(t)) u(t)]\biggr\}\,dt. \] The author presents a decision of this problem in particular cases. An example is presented that illustrates the application of the results to design of stabilizing controllers for unstable power systems.
scientific article; zbMATH DE number 2162408