David Banjerdpongchai

A convex optimization approach to robust iterative learning control for linear systems with time-varying parametric uncertainties

In this paper, we present a new robust iterative learning control (ILC) design for a class of linear systems in the presence of time-varying parametric uncertainties and additive input/output disturbances. The system model is described by the Markov matrix as an affine function of parametric uncertainties. The robust ILC design is formulated as a min-max problem using a quadratic performance criterion subject to constraints of the control input update. Then, we propose a novel methodology to find a suboptimal solution of the min-max optimization problem. First, we derive an upper bound of the worst-case performance. As a result, the min-max problem is relaxed to become a minimization problem in the form of a quadratic program. Next, the robust ILC design is cast into a convex optimization over linear matrix inequalities (LMIs) which can be easily solved using off-the-shelf optimization solvers. The convergences of the control input and the error are proved. Finally, the robust ILC algorithm is applied to a physical model of a flexible link. The simulation results reveal the effectiveness of the proposed algorithm.

Parametric robust H2control design with generalized multipliers via LMI synthesis

A new combined analysis and synthesis procedure that provides a less conservative robust control design technique for systems with real parametric uncertainty is presented. The robust stability for these systems is analysed by the passivity theorem with generalized multipliers, and the worst case H2 performance is investigated using an upper bound on the total output energy. The dynamics of the multipliers are systematically chosen using knowledge from the linear part of the uncertain systems. This approach provides additional degrees of freedom in the synthesis that lead to a reduction of the conservatism in the worst-case H2 per formance and achieved robustness bounds. However, the formulation of the control design problem is very complicated and it is difficult to solve directly. This paper presents an iterative algorithm, which in an H2 equivalent of the D-K iteration for the u/Km synthesis, to account for the complicated couplings in the synthesis problem. We use a simple beam system with an uncertai...

Local control design for systems with saturating actuators using the Popov criteria

Details local control design approaches for systems with actuators that are subject to saturation. Three design algorithms are presented which produce output feedback controllers that either maximize regions of attraction, maximize disturbance rejection, or optimize an L/sub 2/-gain performance metric. In all cases, the stability analyses are based on the Popov stability criterion and the design techniques are given in terms of LMI/BMI algorithms that are readily solved with available software.

An LMI approach for robust Iterative Learning Control with Quadratic performance criterion

This paper presents the design of iterative learning control based on Quadratic performance criterion (Q-ILC) for linear systems subject to additive uncertainty. Robust Q-ILC design can be cast as a min-max problem. We propose a novel approach which employs an upper bound of the worst-case error, then formulates a nonconvex quadratic minimization problem to get the update of iterative control inputs. Applying Langrange duality, the Lagrange dual function of the nonconvex quadratic problem is equivalent to a convex optimization over linear matrix inequalities (LMIs). An LMI algorithm with convergence properties is then given for the robust Q-ILC. Finally, we provide a numerical example to illustrate the effectiveness of the proposed method.

On computing the worst-case norm of linear systems subject to inputs with magnitude bound and rate limit

In this paper, we propose a practical and effective approach to compute the worst-case norm of finite-dimensional convolution systems. System inputs are modelled to have bounded magnitude and rate limit. The computation of the worst-case norm is formulated as a fixed-terminal-time optimal control problem. Applying Pontryagins maximum principle with the generalized Karush–Kuhn–Tucker theorem, we obtain necessary conditions which are subsequently exploited to characterize the worst-case input. Furthermore, we develop a novel algorithm called successive pang interval search (SPIS) to construct the worst-case input for general finite-dimensional convolution systems. The algorithm is guaranteed to converge and give an accurate solution within a prescribed error bound. To verify the accuracy of the algorithm, we derive bounds on computational errors including the truncation error and the discretization error. Then, the bounds on the errors yielded by our algorithm are compared with those of a comparative discr...

Dynamic models of a rotary double inverted pendulum system

This paper describes the dynamic models of a double rotary inverted pendulum, which has been developed for the laboratory experiments. Two different-length rigid pendulums are connected to a horizontally rotating disc which is attached directly to a DC motor. The derivation of the dynamical equations and the linearized model are described. Finally, the time responses of open-loop system are shown and compared with the experimental data to verify the model validity.

Robust Iterative Learning Control for linear systems with time-varying parametric uncertainties

In this paper, we present a robust Iterative Learning Control (ILC) design for linear systems in the presence of time-varying parametric uncertainties. The robust ILC design is formulated as a min-max problem using a quadratic performance criterion subject to constraints of the control input update where the system model contains time-varying parametric uncertainties. An upper bound of the worst-case performance is employed in the min-max problem. Subsequently, applying Lagrangian duality to the min-max problem, we derive a dual problem which is reformulated as a convex optimization over linear matrix inequalities (LMIs). As a result, iterative input updates can be obtained by solving a series of LMI problems. We give an LMI algorithm for the robust ILC design and prove the convergence of the control input and the error. Finally, a numerical example is presented to illustrate the effectiveness of the proposed algorithm.

ON COMPUTING THE WORST-CASE NORM OF CONVOLUTION SYSTEMS: A COMPARISON OF CONTINUOUS-TIME AND DISCRETE-TIME APPROACHES

Abstract This paper compares two approaches to compute the worst-case norm of finite-dimensional convolution systems. All admissible inputs are defined to have bounded magnitude and limited rate of change. Due to physical and mathematical reasons, the inputs are also specified to start from zero. The first approach is based on continuous-time optimal control formulation. Necessary conditions obtained via the Pontryagins maximum principle provide a systematic means to characterize and construct the worst-case input. The second approach is based on discretization of the norm-computation problem which results in a large-scale finite-dimensional linear programming. We also investigate computational errors including truncation errors and discretization errors. Although the second approach seems to be simpler, the first approach is deemed to yield better accuracy.

Dynamic models of servo-driven conveyor system

This paper concerns dynamics of a servo-driven conveyor system, which has been developed for the laboratory-scale experiments. This system consists of a belt lying on the iron plate with two shafts at the end. A DC motor is used to drive the belt through the shaft. The belt is considered to behave like a nonlinear spring. We include the Lugre friction in this model. Then, the dynamics of the conveyor system are derived based on certain specified assumptions. At the same time, the linear and linearized model is used to analyze the behaviors for small signals. Finally, the time responses of these models are shown to verify the modelling analysis.

Robust Constrainned Model Predictive Control for Uncertain Linear Time-Varyilng Systems Using Multitple Lyapunov Functions

In this paper, we present a method for synthesizing a robust constrained model predictive control for uncertain time-varying systems. The goal is to design, at each sampling time, a state feedback control law that minimizes an upper bound of the worst-case objective function, subject to constraints of control inputs and plant outputs. The worst case performance is defined as an infinite-horizon quadratic function of states and control inputs. In order to guarantee robust performance, the method uses multiple Lyapunov functions each of which corresponds to a different vertex of the uncertainty polytope. The state feedback design problem is cast as convex optimization involving linear matrix inequalities which can be efficiently solved. The proposed technique yields an improved performance and less conservative than the robust MPC technique using a single Lyapunov function. Numerical examples based on a two-mass-spring system are given to illustrate the effectiveness of the control algorithm