<p>In order to design robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the plant, the problem is no longer convex and it is then relatively hard to solve. These problems become very complex, even when the order of the system to be controlled is low.</p><p>The approach used in the thesis is based on formulating the constraint on the maximum order of the plant as a polynomial equation. By using the fact that the polynomial is non-negative on the feasible set, the problem is reformulated as an optimization problem where the nonconvex polynomial function is to be minimized over a convex set defined by linear matrix inequalities.</p><p>To solve this optimization problem, two methods have been proposed. The first method is a barrier method and the second one is a method based on a primal-dual framework. These methods have been evaluated on several problems and compared with a well-known method found in the literature. To motivate this choice of method, we have made a brief survey of available methods available for solving the same or related problems.</p><p>The proposed methods emerged as the best methods among the three for finding lower order controllers with the same or similar performance as the full order controller. When the aim is to find the lowest order controller with no worse than +50% increase in the closed loop H-infinity norm, then the three compared methods perform equally well.</p>

In order to design robust controllers, H-infinity synthesis is a common tool to use. The controllers that result from these algorithms are typically of very high order, which complicates implementation. However, if a constraint on the maximum order of the controller is set, that is lower than the order of the plant, the problem is no longer convex and it is then relatively hard to solve. These problems become very complex, even when the order of the system to be controlled is low. The approach used in the thesis is based on formulating the constraint on the maximum order of the plant as a polynomial equation. By using the fact that the polynomial is non-negative on the feasible set, the problem is reformulated as an optimization problem where the nonconvex polynomial function is to be minimized over a convex set defined by linear matrix inequalities. To solve this optimization problem, two methods have been proposed. The first method is a barrier method and the second one is a method based on a primal-dual framework. These methods have been evaluated on several problems and compared with a well-known method found in the literature. To motivate this choice of method, we have made a brief survey of available methods available for solving the same or related problems. The proposed methods emerged as the best methods among the three for finding lower order controllers with the same or similar performance as the full order controller. When the aim is to find the lowest order controller with no worse than +50% increase in the closed loop H-infinity norm, then the three compared methods perform equally well.

When designing controllers with robust performance and stabilization requirements, H-infinity synthesis is a common tool to use. These controllers are often obtained by solving mathematical optimization problems. The controllers that result from these algorithms are typically of very high order, which complicates implementation. Low order controllers are usually desired, since they are considered more reliable than high order controllers. However, if a constraint on the maximum order of the controller is set that is lower than the order of the so-called augmented system, the optimization problem becomes nonconvex and it is relatively difficult to solve. This is true even when the order of the augmented system is low. In this thesis, optimization methods for solving these problems are considered. In contrast to other methods in the literature, the approach used in this thesis is based on formulating the constraint on the maximum order of the controller as a rational function in an equality constraint. Three methods are then suggested for solving this smooth nonconvex optimization problem. The first two methods use the fact that the rational function is nonnegative. The problem is then reformulated as an optimization problem where the rational function is to be minimized over a convex set defined by linear matrix inequalities (LMIs). This problem is then solved using two different interior point methods. In the third method the problem is solved by using a partially augmented Lagrangian formulation where the equality constraint is relaxed and incorporated into the objective function, but where the LMIs are kept as constraints. Again, the feasible set is convex and the objective function is nonconvex. The proposed methods are evaluated and compared with two well-known methods from the literature. The results indicate that the first two suggested methods perform well especially when the number of states in the augmented system is less than 10 and 20, respectively. The third method has comparable performance with two methods from literature when the number of states in the augmented system is less than 25.