Feedback controllers with specific structure arise frequently in applications because they are easily apprehended by design engineers and facilitate on-board implementations and retuning. This work is dedicated to H ∞ -synthesis with structured controllers. In this context, straightforward application of traditional synthesis techniques fails, which explains why only a few ad-hoc methods have been developed over the years. In response, we propose a more systematic way to design H ∞ -optimal controllers with fixed structure using local optimization techniques. Our approach addresses in principle all those controller structures which can be built into mathematical programming contraints. We apply non-smooth optimization techniques to compute locally optimal solutions, and provide practical tests for descent and optimality. In the experimental part we apply our technique to H ∞ loop-shaping PID controllers for MIMO systems and demonstrate its use for PID control of a chemical process.

We discuss closed-loop stabilization of linear time-invariant dynamical systems, a problems which frequently arises in controller synthesis, either as a stand-alone task, or to initialize algorithms for H ∞ synthesis or related problems. Classical stabilization methods based on Lyapunov or Riccati equations appear to be inefficient for large systems. Recently, non-smooth optimization methods like gradient sampling [19] have been successfully used to minimize the spectral abscissa of the closed-loop state matrix (the largest real part of its eigenvalues) to solve the stabilization problem. These methods have to address the non-smooth and even non-Lipschitz character of the spectral abscissa function. In this work, we develop an alternative non-smooth technique for solving similar problems, with the option to incorporate second-order elements to speed-up convergence to local minima. Using several case studies, the proposed technique is compared to more conventional approaches including direct search methods and techniques where the spectral abscissa minimization problem is recast as a traditional smooth non-linear mathematical programming problem.

The standard way to compute H ∞ feedback controllers uses algebraic Riccati equations and is therefore of limited applicability. Here we present a new approach to the H ∞ output feedback control design problem, which is based on nonlinear and nonsmooth mathematical programming techniques. Our approach avoids the use of Lyapunov variables, and is therefore flexible in many practical situations.

Significant progress in control design has been achieved by the use of nonsmooth and semiinfinite mathematical programming techniques. In contrast with LMI or BMI approaches, these new methods avoid the use of Lyapunov variables, which gives them two major strategic advances over matrix inequality methods. Due to the much smaller number of decision variables, they do not suffer from size restrictions, and they are much easier to adapt to structural constraints on the controller. In this paper, we further develop this line and address both frequency-and time-domain design specifications by means of a nonsmooth algorithm general enough to handle both cases. Numerical experiments are presented for reliable or fault-tolerant control, and for time response shaping.