Optimal Control

In contrast to state-feedback design based on the closed-loop pole assignment, state estimation and separation principle, linear quadratic regulation (LQR), linear quadratic estimation (LQE) and linear quadratic Gaussian (LQG) method solve the design problem, i.e. find a state-feedback controller as an optimisation problem by minimising a quadratic time-domain cost function. The solution of the optimisation problem requires the solution of algebraic Riccati equations (AREs). In the case a plant is subject to stochastic disturbances and measurement noise, the state estimator embedded in the state-feedback controller is a Kalman filter. The intensities of the stochastic signals are not needed if the LQG problem is formulated as an \(\mathrm {H}_{2}\) -optimal control problem, which also allows a frequency domain view. The objective of \(\mathrm {H}_{2}\) -optimal control as well as the one of \(\mathrm {H}_{\infty }\) -optimal control is to find an admissible controller that minimises the norm of the closed-loop transfer function matrix \(\mathbf {G}(s)\) from exogenous inputs \(\boldsymbol {w}\) to the control outputs \(\boldsymbol {z}\) . The \(\mathrm {H}_{2}\) -norm \(|| \mathbf {G}(s) ||{ }_2\) is a measure of the total energy corresponding to the impulse response. In contrast, \(|| \mathbf {G}(s) ||{ }_{\infty }\) indicates the maximum gain, i.e. the worst case of the unwanted transfer from \(\boldsymbol {w}\) to \(\boldsymbol {z}\) . Unlike the \(\mathrm {H}_{2}\) -norm \(|| \mathbf {G}(s) ||{ }_2\) , the \(\mathrm {H}_{\infty }\) -norm \(|| \mathbf {G}(s) ||{ }_{\infty }\) must be computed numerically by iteration, where, in practice, it is simpler to solve a suboptimal control problem, i.e. to find a controller which ensures that \(|| \mathbf {G}(s) ||{ }_{\infty }\) is below a pre-specified attenuation-level \(\gamma \) above the optimal level \(\gamma ^{*}\) . Both optimal control problems are supported by software.