Gijs Hilhorst

Sufficient LMI conditions for reduced-order multi-objective H 2 / H ∞ control of LTI systems

Abstract This paper presents a novel projection lemma based linear matrix inequality (LMI) framework to design reduced-order multi-objective H 2 / H ∞ controllers for linear time-invariant systems. This framework relies on a set of full-order H 2 / H ∞ controllers, which are used as parameters in sufficient LMIs for the reduced-order controller design. Continuous-time and discrete-time controller designs are treated in a unified fashion. It is theoretically and numerically demonstrated that the approach allows the computation of reduced-order controllers that are potentially less conservative than full-order designs resulting from well-known LMI approaches. Various comparisons with existing reduced-order controller design approaches illustrate the potential of the proposed framework of sufficient LMIs.

Fixed-Order Linear Parameter-Varying Feedback Control of a Lab-Scale Overhead Crane

This brief presents a numerically attractive approach to design fixed-order H2/H∞ controllers for discrete-time linear parameter-varying (LPV) systems. In this approach, the controller order that is completely determined by the number of states and the parameter dependence is selected in advance. For a prefixed controller order, parameter-dependent sufficient linear matrix inequalities (LMIs) are presented, relying on an a priori computed full-order LPV controller that stabilizes the LPV system for all the possible parameter trajectories. Pólyas theorem and polynomial approximations are used to obtain numerically tractable LMI problems that guarantee the feasibility of the parameter-dependent synthesis conditions. The practical viability of the approach is demonstrated by experimental validations on a lab-scale overhead crane with varying cable lengths.

An LMI approach for reduced-order ℋ 2 LTI controller synthesis

This paper presents a procedure to design reduced-order ℋ2 dynamic output feedback controllers for discrete-time linear time-invariant systems. Starting from a stabilizing full-order controller, novel sufficient linear matrix inequality conditions for the existence of reduced-order ℋ2 controllers are derived. The proposed approach can either be applied directly to compute a controller of desired order from the full-order result, or iteratively by reducing the controller order successively. Numerical experiments confirm the potential of the proposed controller design approach.

Approximate parametric cone programming with applications in control

Parametric programming analyzes the solution of parameter-dependent optimization problems as a function of the parameters. As the true parametrized solution is generally too complicated to be practicable in applications, research has turned to computing adequate approximate solutions. This paper presents a novel approach to compute such approximations for parametric cone programs with a polynomial parameter dependency. A piecewise polynomial parametrization is adopted for the optimizer function, and the coefficients are optimized to minimize the average suboptimality over the parameter domain. The resulting semi-infinite optimization problem is transformed into a tractable, yet conservative optimization problem by exploiting the positivity of the B-spline basis functions. Relying on duality, bounds on the suboptimality of the approximation are computed which can be used to locally refine the solution. The approach is implemented in an open source software tool and illustrated by three applications in control.

Reduced-order ℋ 2 /ℋ ∞ control of discrete-time LPV systems with experimental validation on an overhead crane test setup

This paper presents a numerically attractive approach to design reduced-order multi-objective ℋ₂/ℋ_∞ controllers for discrete-time linear parameter-varying (LPV) systems. The proposed controller synthesis approach relies on an a priori computed polynomially parameter-dependent full-order LPV controller that stabilizes the LPV system for all possible parameter trajectories. This full-order controller is subsequently used in a sufficient linear matrix inequality (LMI) optimization problem for reduced-order ℋ₂/ℋ_∞ LPV synthesis. Pólya relaxations are used to obtain tractable LMI formulations, and a simplicial subdivision of the parameter domain is applied to relieve the numerical burden. Experimental validations on a lab-scale overhead crane with varying cable length illustrate the practical viability of the approach.

Reduced-order multi-objective ℋ ∞ control of an overhead crane test setup

A novel convex approach for reduced-order multi-objective controller synthesis is presented, and experimentally validated on an overhead crane test setup. For this setup, a discrete-time linear time-invariant model is identified. Starting from a full-order controller for the identified model, reducedorder multi-objective ℋ∞ controllers are designed by successively solving convex optimization problems. Experimental validation results on the test setup sustain the practical potential of the proposed controller design approach.

Control of linear parameter-varying systems using B-splines

This paper presents a novel approach to efficiently solve parameter-dependent (PD) linear matrix inequality (LMI) problems for, amongst others, linear parameter-varying (LPV) control design. Typically, stability and performance is guaranteed by finding a PD Lyapunov function such that a PD LMI is feasible on a parameter domain. To solve the resulting semi-infinite problems, we propose a novel LMI relaxation technique relying on B-spline basis functions. This technique provides less conservative solutions and/or a reduced numerical burden compared to existing approaches. Moreover, an elegant generalization of worst-case optimization to the optimization of any signal norm is obtained by expressing performance bounds as a function of the system parameters. This generalization yields better performance bounds in a large part of the parameter domain. Numerical comparisons with the current state-of-the-art demonstrate the generality and effectiveness of our approach.

B-spline parametrized solution of robust PID control using the generalized KYP lemma

This paper presents a novel open-loop shaping approach for robust PID controller design, relying on polynomial spline parameterizations. The proposed approach exploits the generalized Kalman Yakubovic Popov (KYP) lemma to derive parameter-dependent linear matrix inequalities (LMIs) for robust PID synthesis. Multiple finite frequency domain specifications are taken into account to intuitively design practical controllers. By assuming piecewise polynomial parametrizations for the parameter-dependent optimization variables, and subsequently applying B-spline based relaxations, tractable conditions are derived that guarantee feasibility of the parameter-dependent LMIs for all uncertain parameter values. An elegant and effective approach results that solves the robust PID synthesis problem with limited conservatism. Numerical results demonstrate the potential of our approach.

Model Order Reduction for Time-Delay Systems, with Application to Fixed-Order \mathscr {H}_2 Optimal Controller Design

We review a model order reduction method for linear time-delay systems, which allows, in a dynamic way, the construction of a reduced, delay free model of a given dimension. The method builds on the equivalent representation of a delay differential equation as an ordinary differential equation over an infinite-dimensional function space. It combines ideas from a finite-dimensional approximation via a spectral discretization on the one hand, and a Krylov-Pade model reduction approach on the other hand. The method exhibits a good approximation of characteristic roots and it preserves moments of the transfer function (function value and derivatives) at zero and at infinity. A major advantage in the context of control design is that the reduction approach results in a linear time-invariant (LTI) system in a standard state-space representation, enabling a wide range of control design techniques. In the second part of the chapter, we illustrate this by designing low order \(\mathscr {H}_2\) optimal controllers. We employ a novel approach for designing reduced-order controllers for systems without delay, which is grounded in the Lyapunov framework and relies on solving linear matrix inequalities. We design low order controllers for a third order time-delay system, starting from a reduced LTI system and validate the obtained controller on the original delay system.

An iterative convex approach for fixed-order robust ℋ2/ℋ∞ control of discrete-time linear systems with parametric uncertainty

This paper presents a convex approach to design fixed-order robust ℋ2/ℋ∞ controllers for discrete-time linear time-invariant (LTI) systems affected by parametric uncertainty. Starting from an a priori computed stabilizing full-order parameter-dependent controller for the same system, which is designed under the assumption that the parameter is exactly known, parameter-dependent sufficient linear matrix inequalities (LMIs) for robust ℋ2/ℋ∞ analysis and synthesis are presented. A novel LMI procedure is proposed to iteratively compute less conservative robust controllers, utilizing a feasible solution of the fixed-order synthesis conditions as a starting point. Assuming polynomial parameter dependencies of all system matrices, tractable LMI formulations that guarantee feasibility of the parameter-dependent conditions are derived using well-known relaxations based on Pólyas theorem. Numerical comparisons with existing methods confirm the potential of the proposed robust controller design approach.