Quantitative feedback design for sampled-data systems

In quantitative feedback theory (QFT) the plant uncertainty is defined by a set P = {P} ofpossible plants. The problem is to guarantee that the system response is in a specified acceptable set A, for all P in P. QFT has been developed for large classes of plants imbedded in continuous feedback structures. This paper extends QFT to sampled-data structures. A central problem is to find the minimum sampling frequency (ω_s)_min needed. The greater the plant uncertainty and the narrower the performance tolerances, the larger must ( ω_s)_min be. The detailed design procedure parallels very closely that for continuous systems, by using the complex variable w, which maps the unit circle in the z-domain to the imaginary axis in the w-domain.     

Controller reduction with closed loop H infinity performance constraints: A QFT perspective

Assume that an initial stabilizing controller K₀₍s₎, which satisfies various closed loop frequency domain specifications, has been a priori synthesized using, e.g. H infinity control, mu synthesis techniques or closed loop convex synthesis. Remembering that the order of K₀₍s₎ is typical y at least equal to the order of the plant to be controlled, the aim of this paper is to find a stabilizing reduced order controller, which also satisfies the performance specifications and which moreover minimizes the open loop bandwidth. To this aim, the structured singular value mu is used to translate at each frequency closed loop frequency domain specifications into requirements on the frequency response of the controller. The principle of our reduction method is thus very close to the original idea of the SISO QFT design approach, except that the problem of translating closed loop frequency domain specifications into open loop ones is much more complex in the MIMO case. The problem reduces to the issue of finding a controller, whose frequency response belongs at each frequency to a template. The convexity of these templates greatly facilitates the practical realization of the controller. As a final point, the method is successfully applied to a standard H problem, which is extracted from the mu Analysis and Synthesis Toolbox of Matlab, namely the synthesis of an autopilot for the space shuttle.     

On the stability radius of control systems with interval plants and first‐order controllers

This paper considers the stability radius problem of control systems with interval plants and first‐order controllers. The nominal plant P₀(s) used for the controller design is the centre of the interval plant. Under the condition that a lead controller C(s) stabilizes P₀(s), the stability radius of the closed‐loop system is determined in terms of the eigenvalues of four frequency‐independent matrices of the form H⁻¹_{β, i}H_{γ, i}, where both H_{β, i} and H_{γ, i} are Hurwitz‐like matrices. Copyright © 2000 John Wiley & Sons, Ltd.     

Direct connection between time-domain performance and frequency-domain characteristics

Presented are the relationships between the transient response characteristics of a closed-loop system due to a step input and the frequency response characteristics which influence this response. The class of transfer functions addressed include those which are RH ₂. The relationships developed are for the problem: Given: Y(s)=H(s)R(s) where H(s)∈ RH ₂ and R(s) is γs. Find:The relationship ∥y(t)∥_∞⩽Λγ∥H(jω)∥_∞ where the scaling factor Λ is a function of the frequency-domain characteristics of H(s). These relationships are useful for feedback control in that hard time-domain constraints can be enforced as amplitude and phase conditions on the open-loop transfer function. Two applications using these relationships are developed. The first application demonstrates controller design for an actuator saturation constraint. The second application is the development of a performance prediction technique for minimum phase, stable regulating systems. © 1998 John Wiley & Sons, Ltd.     

Unconstrained H∞ predictive control with H∞ prediction: single‐input–single‐output case

Does the replacement of the quadratic (H₂) predictor by the worst‐case (H_∞, or cumulative minimax) predictor robustify the predictive control laws? The present work provides a partial answer to this question, positive for the examples considered, representative of three broad classes of systems. The H_∞ prediction is demonstrated to be a powerful and convenient tool for frequency shaping of the gain of the closed‐loop complementary sensitivity function, capable of robustifying the closed loop for systems with different stability properties. The H_∞‐optimal k‐step ahead predictor is derived for an unstable single‐input–single‐ output CARMA model. A BIBO unstable filter for the disturbance rejection is obtained using the internal model principle and included into the closed loop, and the H_∞ predictor is applied to the combination of this filter with the plant. The sum over a finite horizon of the current and the predicted tracking error and control signal power spectral densities (PSDs) is decomposed into two parts, one induced by the worst‐case predicted disturbance and the other—by the known future reference input. A two degrees of freedom algorithm, referred to as the multi‐step closed‐loop polynomial H_∞ predictive control law, is obtained that minimizes the peaks of the PSD of the first part and the integral on the unit circle of the PSD of the second. It is demonstrated on several systems that H_∞ prediction introduces a very intuitive tuning knob in the form of the prediction horizon capable of setting a trade‐off between the steady‐state disturbance rejection perfor mance in terms of the output error variance and the closed‐loop robustness, however the efficacy of the knob strongly depends on the stability properties of the system and its inverse. The trade‐off becomes less pronounced or completely disappears when the H_∞ predictor is replaced by the quadratic one. Copyright © 2000 John Wiley & Sons, Ltd.     

H∞ feedback‐control theory in biochemical systems

In this paper we study the possible optimality of biochemical pathways in the H_∞ sense. We start by presenting simple linearized models of single enzymatic reaction systems, where we apply classical and modern tools of feedback‐control theory. We then apply the results obtained by our analysis to a linearly unbranched enzyme pathway system, where we explore the effect of a negative feedback loop internally exerted on the system by a self‐product of the pathway. We then probe the sensitivity of the enzymatic system to variations in certain variables and we deal with the problem of assessing the optimality of the static‐output feedback control, in the H_∞ sense, inherent to the closed‐loop system. In this point we demonstrate the applicability of our results via a theoretical example that provides an open‐loop and closed‐loop analysis of a four‐block enzymatic system. We then apply the various tools we developed to the optimal analysis of the Threonine synthesis pathway which is regulated by three feedback loops. We demonstrate that this pathway is optimal in the H_∞ sense, in the face of considerable uncertainties in the various enzyme concentrations of the pathway. Copyright © 2007 John Wiley & Sons, Ltd.     

PERFORMANCE FUNCTIONS AND SUBLEVEL SETS FOR FREQUENCY DOMAIN DESIGN OF SINGLE LOOP FEEDBACK CONTROL SYSTEMS WITH PLANT UNCERTAINTY

The frequency domain design of control systems involves the fitting of a designable complex function of frequency (e.g., loop transfer function, compensator transfer function) to specification derived constraints in the design space ℂ×ℝ. In the classical Bode approach and its recent modifications the designable function is the loop transfer function, the specifications lead to constraints on the magnitude of the loop transfer function and the constraints have circular cross-sections in ℂ. In more recent design procedures (e.g., H _∞ optimization or QFT) the specifications lead to more complex constraint surfaces in ℂ×ℝ and the design procedure must fit the designable function to these constraints. In this paper we develop explicit representations of some important ℂ×ℝ constraint surfaces encountered in the design of SISO control systems with both non-parametric (unstructured) and parametric plant uncertainty and study the characteristics of their frequency axis cross-sections or level sets: important information use in the fitting process. The results are presented in two systems of co-ordinates (i.e., designable functions): nominal closed loop transfer function T_o(jω) and and nominal open loop transfer function L_o(jω). While the results in T_o co-ordinates are more useful when using mathematical optimization (e.g., H _∞ optimization techniques), the results in L_o co-ordinates have significant advantages of insight and ease of graphical manipulation as demonstrated in QFT. The inclusion of results in both sets of co-ordinates increases their utility for workers in both areas and also reveals some links between these two different approaches. © 1997 by John Wiley & Sons, Ltd.     

Sampled‐data H∞ control for networked systems with random packet dropouts

This paper is concerned with the H_∞ control problem for networked control systems (NCSs) with random packet dropouts. The NCS is modeled as a sampled‐data system which involves a continuous plant, a digital controller, an event‐driven holder and network channels. In this model, two types of packet dropouts in the sensor‐to‐controller (S/C) side and controller‐to‐actuator (C/A) side are both considered, and are described by two mutually independent stochastic variables satisfying the Bernoulli binary distribution. By applying an input/output delay approach, the sampled‐data NCS is transformed into a continuous time‐delay system with stochastic parameters. An observer‐based control scheme is designed such that the closed‐loop NCS is stochastically exponentially mean‐square stable and the prescribed H_∞ disturbance attenuation level is also achieved. The controller design problem is transformed into a feasibility problem for a set of linear matrix inequalities (LMIs). A numerical example is given to illustrate the effectiveness of the proposed design method. Copyright © 2010 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Type-III closed loop control systems-Digital PID controller design

The problem of tuning digital PID controllers for type-III control loops is investigated in this work. Type-III control loops are capable of achieving perfect tracking of step, ramp and parabolic reference signals with zero steady state position, velocity and acceleration error. The proposed PID control law involves any dominant time constants of the process itself, and any parasitic dynamics introduced by both the process and the controller, i.e. time delays within the closed control system. The development of the proposed control law takes place in the frequency domain and basis of the theory is the principle of the Magnitude Optimum criterion. The final control law consists of closed form expressions which involve also the controller's sampling time T_s. The potential of the proposed theory is justified for the control of several benchmark process models throughout simulation examples. The affect of the choice of the controller's sampling time is investigated further to the step and frequency response of the control loop both for the output of the control loop and the controller's command signal.     

Efficient computational system reliability analysis of reinforced soil-retaining structures under seismic conditions including the effect of simulated noise

This article presents a computational reliability analysis of reinforced soil-retaining structures (RSRS) under seismic conditions. The internal stability of RSRS is evaluated using the horizontal slice method (HSM) with modified pseudo-dynamic seismic forces. Two different failure modes of RSRS are identified and their reliability indices are computed using the first-order reliability method (FORM). The critical probabilistic failure surface is identified using a three-tier optimization scheme. Reliability index of the system is computed by considering the modes of failure to be connected in series. The tension mode is found to be the most critical mode of failure. The present study identifies that the wall height (H), shear wave velocity of the soil (V_s), and predominant frequency of the input motion (ω) govern the response of RSRS. Reliability indices depend on a parameter termed as the normalized frequency (ωH/V_s) and their values decrease with an increase in the value of ωH/V_s. Increase in the damping ratio of soil, increases the value of reliability indices, especially for ωH/V_s values, which are close to π/2. The FORM suffers from few critical shortcomings such as linear assumption of limit state surface at the most probable point of failure and its ability to consider only the statistical uncertainties excluding the effect of epistemic uncertainties. This calls for sampling-based numerical techniques such as Monte-Carlo simulation (MCS) which gives more comprehensive understanding of the problem under consideration in a probabilistic framework. Thus, a computationally efficient surrogate-assisted MCS is carried out to validate the present formulation and provide numerical insights by capturing the system dynamics over the entire design domain. Adoption of the efficient surrogate-assisted approach allowed us to quantify the epistemic uncertainty associated with the system using Gaussian white noise (GWN). Subsequently, its effects on the system reliability index and probabilistic behavior of the critical parameters are presented. The numerical results clearly indicate that it is imperative to take into account the probabilistic deviations of the critical performance parameters for RSRS to ensure adequate safety and serviceability under operational condition while quantifying the reliability of such systems.

     

An Improved Anti‐Windup Bumpless Transfer Structures Design for Controllers Switching

In this paper, an anti‐windup bumpless transfer (AWBT) control structure combined with linear interpolation method is proposed for smooth switching control. By choosing an appropriate scheduling signal, different controllers can be switched smoothly under a unified framework. Meanwhile, some robust specifications including H₂/H_∞ performance, pole placement constraint, and passivity of the closed‐loop system can be preserved through controller switching. Furthermore, for the linear system subject to input saturation, the stability and L₂ gain of the closed‐loop system can be guaranteed. Finally, a cart‐spring pendulum system is simulated to demonstrate the effectiveness of the proposed scheme.     

Robust H∞ control for uncertain singular systems with state delay

This paper addresses the problem of robust H_∞ control for uncertain continuous singular systems with state delay. The singular system under consideration involves state time delay and time‐invariant norm‐bounded uncertainty. Based on the linear matrix inequality (LMI) approach, we design a memoryless state feedback controller law, which guarantees that, for all admissible uncertainties, the resulting closed‐loop system is not only regular, impulse free and stable, but also meets an H_∞‐norm bound constraint on disturbance attenuation. A numerical example is provided to demonstrate the applicability of the proposed method. Copyright © 2003 John Wiley & Sons, Ltd.     

Stability of nonlinear feedback systems in a Volterra representation

Presented in this paper is a stability condition for a class of nonlinear feedback systems where the plant dynamics can be represented by a finite series of Volterra kernels. The class of Volterra kernels are limited to p‐linear stable operators and may contain pure delays. The stability condition requires that the linear kernel is non‐zero and that the closed loop characteristic equation associated with the linearized system is stable. Next, a sufficient condition is developed to upper bound the infinity‐norm of an external disturbance signal thereby guaranteeing that the internal and output signals of the closed loop nonlinear system are contained in L_∞. These results are then demonstrated on a design example. A frequency domain controller design procedure is also developed using these results where the trade‐off between performance and stability are considered for this class of nonlinear feedback systems. Copyright © 2000 John Wiley & Sons, Ltd.     

Mixed H∞/H2 design of digital phase‐locked loops with polytopic‐type uncertainties

A robust H_∞ control method is applied to the design of loop filters for digital phase locked loop carrier phase tracking. The proposed method successfully copes with large S‐curve slope uncertainty and with a significant decision delay in the closed‐loop that may stem from the decoder and/or the equalizer there. The design problem is transformed into a state‐feedback control problem where phase and gain‐margins should be guaranteed in spite of the uncertainty. Of all the loop filters that achieve the required margins the one that minimizes an upper‐bound on the effect of the phase and the measurement noise signals is derived. Copyright © 2002 John Wiley & Sons, Ltd.     

Boundary control of linear stochastic reaction‐diffusion systems

This paper considers the boundary control problem for linear stochastic reaction‐diffusion systems with Neumann boundary conditions. First, when the full‐domain system states are accessible, a boundary control is designed, and a sufficient condition is established to ensure the mean‐square exponential stability of the resulting closed‐loop system. Next, when the full‐domain system states are not available, an observer‐based control is proposed such that the underlying closed‐loop system is stable. Furthermore, observer‐based controller is designed for the systems with an H_∞ performance. Simulation examples are given to demonstrate the effectiveness and potential of the new design techniques.     

L1‐Gain Analysis and Control for Switched Positive Systems with Dwell Time Constraint

This paper studies the problems of L₁‐gain analysis and control for switched positive systems with dwell time constraint. The state‐dependent switching satisfies a minimal dwell time constraint to avoid possible arbitrary fast switching. By constructing multiple linear co‐positive Lyapunov functions, sufficient conditions of stability and L₁‐gain property are derived under the proposed switching strategy. Then, an effective state feedback controller is designed to ensure the positivity and L₁‐gain property of the closed‐loop system. Finally, a simulation example is given to illustrate the effectiveness of the proposed method.     

Gain Scheduled LPV H∞ Control Based on LMI Approach for a Robotic Manipulator

A new approach to the design of a gain scheduled linear parameter‐varying (LPV) H_∞ controller, which places the closed‐loop poles in the region that satisfies the specified dynamic response, for an n‐joint rigid robotic manipulator, is presented. The nonlinear time‐varying robotic manipulator is modeled to be a LPV system with a convex polytopic structure with the use of the LPV convex decomposition technique in a filter introduced. State feedback controllers, which satisfy the H_∞ performance and the closed‐loop pole‐placement requirements, for each vertex of the convex polyhedron parameter space, are designed with the use of the linear matrix inequality (LMI) approach. Based on these designed feedback controllers for each vertex, a LPV controller with a smaller on‐line computation load and a convex polytopic structure is synthesized. Simulation and experiment results verify that the robotic manipulator with the LPV controller always has a good dynamic performance along with the variations of the joint positions. © 2002 Wiley Periodicals, Inc.     

A New Method for Getting Rational Approximation for Fractional Order Differintegrals

In this paper a new algorithm is presented to calculate the poles and zeros to approximate a fractional order (FO) differintegral (s^±α,α∈(0,1)) by a rational function on a finite frequency band ω∈(ω_l,ω_h). The constant phase property of the FO differintegral is the basis for development of the algorithm. Interlacing of real poles and zeros is used to achieve the constant phase. The calculations are done using the asymptotic Bode phase plot. A brief investigation is made to get a good approximation for the Bode phase plot. Two design parameters are introduced to keep the average phase close to the desired phase angle and to keep the error within the allowed bounds. A study is done to empirically understand the relationship between the error and the design parameters. The results thus obtained help in the further calculations. The algorithm is computationally simple and inexpensive, and gives a fairly good approximation of fractance frequency response on the specified frequency band.     

On Controlling the Transient Response of Linear Time Invariant Systems with Fixed Structure Controllers

The problem of controlling transient response is important in many industrial applications; for example, the speed and accuracy of motion control of robots directly relates to the productivity of the robot. The objective of transient control is to determine a feedback controller of a fixed structure that renders the closed loop response of a specified system to lie in a specified envelope. One may associate a set of errors which measures the deviation of the response from the envelope. The set of errors may be defined in such a way that all the errors are non‐negative if and only if the response does not deviate from the envelope at any time. The transient problem can be thus posed as the problem of determining a stabilizing controller that renders the set of all errors to be non‐negative at every time. One may associate a control parameter vector K with a controller of a specified structure. The main topic of investigation of this paper is to find a bound for the set of real control parameters, K, so that a rational, proper transfer function, has a decaying, non‐negative impulse response. It is assumed that the coefficients of the polynomials N(s, k) and D(s, K) are affine in K.