Juanyu Bu

An exact solution to continuous-time mixed H/sub 2//H/sub /spl infin// control problems

Multiobjective control problems have been the object of much attention in the past few years, since they allow for handling multiple, perhaps conflicting, performance specifications and model uncertainty. One of the earliest multiobjective problems is the mixed H/sub 2//H/sub /spl infin// control problem, which can be motivated as a nominal LQG optimal control problem subject to robust stability constraints. This problem has proven to be surprisingly difficult to solve, and at this time no closed-form solutions are available. Moreover, it has been shown that except in some trivial cases, the optimal controller is infinite-dimensional. In this paper, we propose a solution to general continuous-time mixed H/sub 2//H/sub /spl infin// problems, based upon constructing a family of approximating problems, obtained by solving an equivalent discrete-time problem. Each of these approximations can be solved efficiently, and the resulting controllers converge strongly in the H/sub 2/ topology to the optimal solution.

A linear matrix inequality approach to synthesizing low-order suboptimal mixed ℓ1/Hp controllers

Abstract Mixed objective control problems have attracted much attention lately since they allow for capturing different performance specifications. However, optimal multiobjective controllers may exhibit some undesirable properties such as arbitrarily high order. This paper addresses the problem of designing stabilizing controllers that minimize an upper bound of the l 1 norm of a certain closed-loop transfer function, while maintaining the H 2 norm (mixed l 1 / H 2 ), or the H ∞ norm (mixed l 1 / H ∞ ), of a different transfer function below a prespecified level. The main results show that these suboptimal controllers have the same order as the generalized plant and can be synthesized by a two-stage process, involving an LMI optimization problem and a line search over (0, 1) .

A linear matrix inequality approach to synthesizing low order l/sup 1/ controllers

The l/sup 1/ control theory is appealing, since it allows for directly incorporating time-domain specifications into the controller synthesis procedure and furnishes a complete solution to the robust performance problem. Moreover, in the SISO case, the synthesis procedure can be recast into a finite-dimensional linear programming problem and solved efficiently. The MIMO case can be solved iteratively by adding fictitious inputs and outputs to recast the problem into an one-block form. However, it is well known that, in contrast to the H/sub 2/ and H/sub /spl infin// cases, optimal l/sup 1/ controllers can have arbitrarily high order, even when the states of the plant are available for feedback. In this paper, we address the problem of designing low order suboptimal l/sup 1/ controllers using a linear matrix inequality (LMI) optimization approach. The main results show that, in the state-feedback case, the suboptimal controller is static, while in the output-feedback case it has the same order as that of the plant. In both cases the synthesis process involves solving an LMI feasibility problem and a scalar minimization over (0,1).

A linear matrix inequality approach to synthesizing low order mixed l/sub 1//H/sub p/ controllers

Mixed objective control problems have attracted much attention since they allow for capturing different performance specifications without resorting to approximations or the use of weighting functions, thus eliminating the need for trial and error type iterations. The paper addresses the problem of designing stabilizing controllers that minimize the l/sub 1/ norm of a certain closed-loop transfer function, while maintaining the H/sub 2/ norm (mixed l/sub 1//H/sub 2/), or the H/sub /spl infin// norm (mixed l/sub 1//H/sub /spl infin//), of a different transfer function below a prespecified level. Based on a linear matrix inequality approach, the main results of this paper show that, suboptimal controllers can be synthesized by a two-stage process, involving an LMI optimization problem and a line search over (0, 1). Furthermore, this approach also provides an LMI-based parameterization of all suboptimal-output feedback controllers, including reduced order ones, for mixed l/sub 1//H/sub /spl infin// and l/sub 1//H/sub 2/ problems.

A solution to MIMO 4-block l/sup 1/ optimal control problems via convex optimization

Proposes an alternative solution to 4-block l/sup 1/ problems. This alternative is based upon the idea of transforming the l/sup 1/ problem into an equivalent (in the sense of having the same solution) mixed l/sup 1//H/sub /spl infin// problem that can be solved using convex optimization techniques. The proposed algorithm has the advantage of generating, at each step, an upper bound of the cost that converges uniformly to the optimal cost. Moreover, it allows for easily incorporating frequency and regional pole placement constraints. Finally, it does not require either solving large LP problems or obtaining the zero structure of the plant and computing the so-called zero interpolation and the rank interpolation conditions. The main drawback of this method is that it may suffer from order inflation. However, consistent numerical experience shows that the controllers obtained, albeit of high order, are amenable to model reduction by standard methods, with virtually no loss of performance.

Mixed H 2 /L 1 control with low order controllers: a linear matrix inequality approach

This paper addresses the problem of designing stabilizing controllers that minimize the /spl Hscr//sub 2/ norm of a certain closed-loop transfer function while maintaining the /spl Lscr//sub 1/ norm of a different transfer function below a prespecified level. This problem arises in the context of rejecting both stochastic as well as bounded persistent disturbances. Alternatively, in a robust control framework it can be thought as the problem of designing a controller that achieves good nominal /spl Hscr//sub 2/ performance, while at the same time, guaranteeing stability against unmodeled dynamics with bounded induced /spl Lscr//sub /spl infin// norm. The main result of this paper shows that, for the state feedback case, a suboptimal static feedback controller can be synthesized by a two stage process involving a finite-dimensional convex optimization problem and a line-search.

On the Properties of the Solutions to Mixed l 1 / H ∞ Control Problems

Abstract Mixed performance control problems have been the object of much attention lately. These problems allow for capturing different performance specifications without resorting to approximations or the use of weighting functions, thus eliminating the need for trial and error type iterations. However, it has been recently shown that in some cases the resulted closed-loop system may exhibit some undesirable properties. In this paper the discrete-time mixed l 1 / H ∞ control problem will be investigated. The main result of the paper shows that i) this problem admits a minimizing solution in l 1 ii) a recently proposed method produces sequences of rational controllers and closed-loop systems that converge to the optimum in the l 1 topology.

Robustness analysis with mixed performance objectives

Mixed performance control problems have been the object of much attention lately. These problems allow for capturing different performance specifications without resorting to approximations or the use of weighting functions. However, up to date most of the work concerning multiobjective control is limited to guaranteeing nominal performance and robust stability. In this paper we analyze robust performance for a class of mixed problems. The main results of the paper furnish sufficient conditions for guaranteeing performance (in the l/sub /spl infin// sense) under model perturbations having an l/sub 2/ to l/sub 2/ bounded norm. These condition can be combined with previously proposed multiobjective control synthesis techniques to obtain controllers guaranteeing robust performance.

Mixed L/sub 1//H/sub 2/ controllers for continuous time systems

We consider the problem of minimizing the L/sub 1/ norm of a closed-loop transfer function while keeping its H/sub 2/ norm under at a specified level. It is shown that the optimal closed-loop impulse response has finite support, and thus a non-rational Laplace transform. To solve this difficulty we propose a method for synthesizing rational controllers with performance arbitrarily close to optimal.

Mixed /spl Hscr//sub 2///spl Lscr//sub 1/ control with low order controllers: a linear matrix inequality approach

This paper addresses the problem of designing stabilizing controllers that minimize the /spl Hscr//sub 2/ norm of a certain closed-loop transfer function while maintaining the /spl Lscr//sub 1/ norm of a different transfer function below a prespecified level. This problem arises in the context of rejecting both stochastic as well as bounded persistent disturbances. Alternatively, in a robust control framework it can be thought as the problem of designing a controller that achieves good nominal /spl Hscr//sub 2/ performance, while at the same time, guaranteeing stability against unmodeled dynamics with bounded induced /spl Lscr//sub /spl infin// norm. The main result of this paper shows that, for the state feedback case, a suboptimal static feedback controller can be synthesized by a two stage process involving a finite-dimensional convex optimization problem and a line-search.