Richard Conway

Solutions to the Inverse LQR Problem With Application to Biological Systems Analysis

In this brief, we present a set of techniques for finding a cost function to the time-invariant linear quadratic regulator (LQR) problem in both continuous- and discrete-time cases. Our methodology is based on the solution to the inverse LQR problem, which can be stated as: does a given controller K describe the solution to a time-invariant LQR problem, and if so, what weights Q and R produce K as the optimal solution? Our motivation for investigating this problem is the analysis of motion goals in biological systems. We first describe an efficient linear matrix inequality (LMI) method for determining a solution to the general case of this inverse LQR problem when both the weighting matrices Q and R are unknown. Our first LMI-based formulation provides a unique solution when it is feasible. In addition, we propose a gradient-based, least-squares minimization method that can be applied to approximate a solution in cases when the LMIs are infeasible. This new method is very useful in practice since the estimated gain matrix K from the noisy experimental data could be perturbed by the estimation error, which may result in the infeasibility of the LMIs. We also provide an LMI minimization problem to find a good initial point for the minimization using the proposed gradient descent algorithm. We then provide a set of examples to illustrate how to apply our approaches to several different types of problems. An important result is the application of the technique to human subject posture control when seated on a moving robot. Results show that we can recover a cost function which may provide a useful insight on the human motor control goal.

Robust Track-Following Controller Design in Hard Disk Drives Based on Parameter Dependent Lyapunov Functions

This paper presents a novel technique for designing robust track-following output-feedback controllers in hard disk drives (HDDs). In this paper, the manufacturing variations of HDDs are modeled as polytopic parametric uncertainties in linear time-invariant discrete-time systems. For this model, the robust track-following control problem is formulated as the worst-case H2 performance optimization. The optimization problem reduces to the one with bilinear matrix inequalities (BMIs), using the parameter dependent Lyapunov functions and the extended LMI condition introduced by de Oliveira. Although the formulated problem is nonconvex, and thus it is difficult to ensure global optimality, a numerical technique called ¿G-K iteration¿ is applied for optimization to guarantee monotonic non-increase of the worst-case performance during iterations. The proposed design technique will be useful in improving the track-following performance, and thus increasing the storage capacity of HDDs.

Optimal H ∞ Control for Linear Periodically Time-Varying Systems in Hard Disk Drives

Periodicity frequently occurs in hard disk drives (HDDs) whose servo systems with periodic phenomena can be usually modeled as linear periodically time-varying (LPTV) systems. This paper discusses optimal H∞ control synthesis for discrete-time LPTV systems via discrete Riccati equations. First, an explicit minimum entropy H∞ controller for general time-varying systems is obtained. Subsequently, the developed control synthesis algorithm is applied to LPTV systems and it is shown that the resulting controllers are periodic. The proposed control synthesis technique is evaluated through both single and multirate optimal H∞ track-following control designs. The single-rate servo design shows that our proposed control synthesis technique is more numerically robust in calculating optimal H∞ controllers for discrete-time linear time-invariant systems than the MATLAB function of “hinfsyn,” while the multirate servo design validates its ability of synthesizing multirate controllers to achieve the robust performance of a desired error-rejection function. Moreover, an experimental study-in which the developed control synthesis algorithm on a real HDD with missing position error signal sampling data is implemented-further demonstrates its effectiveness in handling LPTV systems with a large period and attaining desirable disturbance attenuation.

Model Reduction and Parametric Uncertainty Identification for Robust

This paper presents a systematic, semi-automated method for identifying parameters and parametric uncertainty for a set of dual-stage hard disk drives. A modal analysis technique is selected to extract parameters from a batch of frequency response data. In order to avoid redundancy in modal parameters, two methods are presented to reduce model order. One method combines experimental data to directly extract fewer parameters. The second method uses an optimized model truncation methodology. Finally, convex optimization and singular value decomposition are employed to obtain a minimally conservative, lower-order approximation of uncertain parameters. The result is a reduced-order state space model with parametric uncertainty to be used in robust H2 control synthesis for a track-following hard disk drive servo.

H_{2}

In this paper, we present a new algorithm for solving the LQG control problem with variance constraints which utilizes derivative information about the relevant ℋ2 costs to achieve quasi-Newton convergence. Using a lifting procedure, this algorithm is then generalized to work with linear periodically time-varying systems. This algorithm is then applied to the design of controllers for hard disk drives in order to assess the limits of performance of a particular setup. It is demonstrated that just by utilizing multirate sampling and actuation characteristics (i.e. without changing the hardware), the performance of this particular setup can improved by more than 39 %.Copyright

Control Synthesis for Dual-Stage Hard Disk Drives

This paper considers robust controller design for track-following in hard disk drives (HDD) with irregular sampling of the position error signal (PES) but regular (clock-driven) control updates. This sampling and actuation behavior is modeled by applying a novel discretization method to a continuous-time model of an HDD, resulting in a discrete-time linear periodically time-varying model. Then, the controller design is performed using optimal H∞ control for periodic systems and uses a generalization of the disk margin to quantify the robustness of the closed-loop system. To show the effectiveness of the proposed method, the design methodology is applied to a hard disk drive model and the resulting controller is validated by examining its nominal performance in terms of the root mean square of the standard deviation of the PES and robustness in terms of disk margin. Since the proposed controller has too many parameters to be implementable on an HDD due to memory limitations, we use a vector quantization method to approximate the entire parameter set of the designed controller by a smaller set of parameters.

A QUASI-NEWTON ALGORITHM FOR LQG CONTROLLER DESIGN WITH VARIANCE CONSTRAINTS

This paper presents a methodology for analyzing the H 2 guaranteed cost performance of a discrete-time LTI system with unstructured dynamic uncertainty. Using the methods of guaranteed cost control, an upper bound on H 2 guaranteed cost performance over unstructured parametric uncertainty is formulated in terms of feasibility of a linear matrix inequality. It is then shown that the feasibility of this inequality also guarantees the same level of performance also over unstructured dynamic uncertainty. This is then used to formulate the problem of finding the best upper bound on H 2 guaranteed cost performance over unstructured causal dynamic uncertainty as a semi-definite program. Finally, it is shown that this optimization problem can be solved efficiently and accurately using discrete algebraic Riccati equations.© 2009 ASME

Robust Track-Following Controller Design for Hard Disk Drives With Irregular Sampling

This paper presents, for discrete-time LTI systems with unstructured dynamic uncertainty, a methodology for designing full information controllers which minimize the upper bound on robust H2 performance given in [1]. It is first shown that this optimal control problem can be cast as a semi-definite program. Then, it is shown that this optimization problem can be solved efficiently and accurately using discrete algebraic Riccati equations.Copyright

Analysis of Discrete-Time H2 Guaranteed Cost Performance

This paper discusses optimal H∞ control synthesis via discrete Riccati equations for discrete linear periodically time-varying (LPTV) systems. Based on the results presented in [1], an explicit minimum entropy H∞ controller for general time-varying systems is obtained. The control synthesis technique is subsequently applied to LPTV systems and it is shown that the resulting controllers are also periodically time varying. In order to demonstrate the effectiveness of the proposed control synthesis technique, both single-rate and multi-rate discrete-time minimum entropy H∞ track-following control designs for hard disk drives are considered. It is shown, via a comprehensive simulation study, that track-following controllers designed using the H∞ synthesis technique proposed in this paper achieve the robust performance of a desired error rejection function. Moreover, as expected, multi-rate controllers has the ability of outperforming their single-rate counterparts.Copyright

Optimal Full Information H2 Guaranteed Cost Control of Discrete-Time Systems

Abstract This paper examines the limits of performance in systems with periodic irregular sampling rate when the actuation is not necessarily synchronized with the sampling. For such a system, three sampling and actuation schemes are considered: when the sampling and control rate are both regular, when they are both irregular, and when the sampling rate is irregular while the control rate is regular. To ascertain the limits of performance of this type of systems under each sampling and actuation scheme, the system is modeled as a linear periodically time-varying (LPTV) system; optimal LQG control design with a variance constraint is applied to find the smallest achievable mean variance of the performance signal subject to a constraint on the mean control effort variance. In addition, to deal with the computational delay of the controller, an innovative discretization method is proposed which does not introduce any extra states into the state space model. The proposed method is exploited to determine the performance of a hard disk drive (HDD) in track-following mode. A simulation study demonstrates that in the presence of 30% irregularity in sampling time, using the irregular sampling and regular control action scheme for the HDD achieves an RMS 3s value of the position error signal (PES) that is 40% smaller than the corresponding value achieved by using a controller provided by our industry partner, in which both the sampling and control rates are irregular.