Controller Synthesis for Maximizing Disturbance Bounds in Linear Uncertain Systems

Structural Synthesis of Multivariable Controllers

For track-following control in an optical disk drive, a compensator should be designed to satisfy disturbance attenuation property, overall system stability, and so on. In order to ease the compensator design procedure by reducing trial and error, a disturbance observer is added on to a conventional single loop track-following control system, which may cause control input saturation due to excessive loop gain. Because the track-following control system is conditionally stable, control input saturation leads to actual reduction of loop gain whereby the system becomes unstable. To preserve the overall stability during transient response, where control input saturation possibly occurs, the circle criterion based on the pull-in condition is utilized to check the stability under control input saturation. The constraints from the circle criterion and robust stability condition are converted into frequency domain bounds, which are represented on a Nichols chart. Then a design method using quantitative feedback theory (QFT) is applied to satisfy the constraints shown on the Nichols chart. To show the feasibility, some experiments are conducted, and the results are presented.

A graphical method is introduced that solves the robust infinite horizon H/sub /spl infin// multiple-objective control problem for single measurement, single control input systems. The solution is obtained by describing boundaries on the Nichols chart. Each boundary defines the set of all admissible gain and phase values for the loop transmission at a given frequency. These boundaries are obtained by using the well-known parameterization of all the solutions for a single objective H/sub /spl infin// control problem. The new method links between the theories of H/sub /spl infin// and quantitative feedback theory (QFT). It can be used to design robust H/sub /spl infin// controllers with almost no overdesign, and it provides a convenient solution of H/sub /spl infin// multiple-objective problems that are difficult to solve by the standard four-block setting. It also extends the methods of SISO QFT to deal with a vector of disturbances. The latter may affect the controlled plant through any input coupling matrix and not necessarily through the controller input, or the measurement output.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

There is given a minimum-phase plant transfer function, with prescribed bounds on its parameter values. The plant is imbedded in a two-degree-of-freedom feedback system, which is to be designed such that the system time response to a deterministic input lies within specified boundaries. Subject to the above, the design should be such as to minimize the effect of sensor white noise at the input to the plant. This report presents a design procedure for this purpose, based on frequency response concepts. The time-domain tolerances are translated into equivalent frequency response tolerances. The latter lead to bounds on the loop-transmission function L(jω), in the form of continuous curves on the Nichols chart. Properties of L(jω) which satisfy these bounds with minimum effect of sensor white noise are derived. The design procedure is quite transparent, providing the designer with the insight to make necessary tradeoffs, at every step in the design process. The same design philosophy may be used to attenuate the effect of disturbances on plants with parameter ignorance.

Open-loop solution to the output control problem in SISO systems by means of feedforward and cascade controllers is investigated in this paper. A simple characterization of feedforward controllers which achieve steady-state disturbance rejection is given in a transfer-function setting. Cascade controllers which cause steady-state command tracking are also characterized. Furthermore, disturbance decoupling and command matching controllers are identified. Conditions for existence of feedforward and cascade controllers are also given. For unstable systems, it is shown that a stabilizing feedback controller can be employed without affecting the feedforward and cascade controllers used for output control; hence, the three controllers can be designed independently. Finally, output control by a combination of feedforward and feedback is discussed briefly.

This paper presents a linear quadratic regulator (LQR) for robust closed-loop pole-placement within a vertical strip, and disturbance rejection with an H ∞-norm constraint for the uncertain linear systems. The concerned systems cover both matched and mismatched uncertain linear systems with unstructured or structured uncertainties existing in both the system and input matrices. A set of tuning parameters is incorporated for some flexibility in finding a solution to the algebraic Riccati equation, and a controller gain parameter is selected for robust pole clustering. Also, a constraint is established to verify whether the proposed robust LQRs preserve H 2 optimality with respect to a specific quadratic cost function.

Event-triggered robust cooperative output regulation for a class of linear multi-agent systems with an unknown exosystem

This paper deals with the robust passivity synthesis problem for a class of uncertain linear systems with time-varying delay in state and control input. The parameter uncertainties are norm-bounded and allowed to appear in all matrices of the model. The problem aims at designing an observer-based dynamic output-feedback controller that robustly stabilizes the uncertain systems and achieves the strict passivity of closed-loop systems for all admissible uncertainties. By converting the problem at hand into a class of strictly passive control problem for a parameterized system, the explicit solution is established and expressed in terms of a linear matrix inequality. A numerical example is provided to demonstrate the validity of the proposed approach.

Conventional deterministic set-membership (SM) estimation is limited to unknown-but-bounded uncertainties. In order to exploit distributional information of probabilistic uncertainties, a probability-guaranteed SM state estimation approach is proposed for uncertain linear time-invariant systems. This approach takes into account polynomial dependence on probabilistic uncertain parameters as well as additive stochastic noises. The purpose is to compute, at each time instant, a bounded set that contains the actual state with a guaranteed probability. The proposed approach relies on the extended form of an observer representation over a sliding window. For the offline observer synthesis, a polynomial-chaos-based method is proposed to minimize the averaged H2 estimation performance with respect to probabilistic uncertain parameters. It explicitly accounts for the polynomial uncertainty structure, whilst most literature relies on conservative affine or polytopic overbounding. Online state estimation restructures the extended observer form, and constructs a Gaussian mixture model to approximate the state distribution. This enables computationally efficient ellipsoidal calculus to derive SM estimates with a predefined confidence level. The proposed approach preserves time invariance of the uncertain parameters and fully exploits the polynomial uncertainty structure, to achieve tighter SM bounds. This improvement is illustrated by a numerical example with a comparison to a deterministic zonotopic method.

This note addresses disturbance suppression problem for uncertain plant systems using prior disturbance data which contain some measurement errors, and gives a further result which attains lower computational complexity than our previously proposed method. Under the conditions that the plant uncertainties are assumed to be expressed by bounded but time-invariant uncertain delays at the control input and the measurement errors are expressed as affine with respect to some bounded constant uncertain vector, we have already proposed a MPC-based controller which achieves disturbance suppression with robustness against the plant uncertainties and the measurement errors. In this paper, we tackle the same problem but under the condition that the implemented computing power is limited. For this problem, we propose a simple methodology, i.e. constant inputs are supposed for several steps. We apply our proposed method to flight controller design problem to suppress vertical acceleration driven by turbulence, i.e. Gust Alleviation (GA) flight controller design problem, and demonstrate the applicability with numerical simulations.

Sliding mode control is known for its robustness against plant uncertainties and disturbance rejection. An extension of this technique is the addition of super twisting algorithm that suppresses the chattering effect commonly observed with sliding mode control application. This paper elaborates on the design and analysis of the super twisting algorithm with focus on optimizing the gain parameters. Two main components of the sliding mode super twisting controller were designed and validated; that is, the switching function and the control laws. The traditional sliding surface was used as the switching function while control laws with continuous control action were applied. Two gain parameters of the control laws were designed and analyzed in terms of tracking error reduction and chattering effect. A Kalman-Bucy continuous filter was applied for velocity signal estimation. Fast Fourier Transform was applied on the control inputs to investigate the chattering amplitude and root mean square of the tracking error was used as performance indicator for tracking error reduction. Analyses were performed on the effect of gains variation to control performance. Experimental results showed that gain L was sensitive to chattering amplitude while gain W provided wider range of tuning. The control performance was compared to the traditional pseudo-sliding mode controller in terms of tracking error reduction and chattering effect. Results showed that the super twisting algorithm produced superior results in tracking error reduction and was effective in suppressing chattering effect compared to the pseudo-sliding mode controller.

The problem of robust H/spl infin/ filtering for continuous-time uncertain linear systems with multiple time-varying delays in the state variables is investigated. The uncertain parameters are supposed to belong to a given convex bounded polyhedral domain. The aim is to design a stable linear filter assuring asymptotic stability and a prescribed H/spl infin/ performance level for the filtering error system, irrespective of the uncertainties and the time delays. Sufficient conditions for the existence of such a filter are established in terms of linear matrix inequalities, which can be efficiently solved by means of powerful convex programming tools with global convergence assured. An example illustrates the proposed methodology.

H∞ control of discrete-time uncertain linear systems with quantized feedback

In this paper, the controllability and observability of linear time-invariant uncertain systems are investigated. The systems under consideration contain time-invariant uncertain parameters that may take arbitrarily large values. In such a situation, the locations of uncertain parameters in system matrices play an important role. We examine the permissible locations of uncertain parameters in system matrices for a linear uncertain system to be controllable and observable independently of the bounds of the uncertain parameters. The objective of this paper is to show that a linear uncertain system is controllable and observable, irrespective of the bounds of uncertain parameters, if and only if the system has a particular configuration called a complete generalized antisymmetric stepwise configuration (CGASC). Furthermore, the dual configuration of a CGASC is introduced and studied here.

This paper presents quantitative feedback theory (QFT) based control and tracking of ball and beam system. QFT is a frequency domain approach using Nichols chart. This technique achieves desired robust design over a specified range of plant uncertainty. QFT is used for the design of robust controllers for the plant with uncertainties in the parameters, input/output disturbances and noises. Ball and beam system with parametric uncertainty is defined in state space form using QFT control toolbox and templates are generated. Then, based upon the performance and tracking specifications, bounds are determined. After determining the bounds, controller and prefilter are designed which fulfill the necessary control and tracking requirements. Additionally, a first order sliding mode controller (SMC) is intended for the comparison purpose.