Gain gradients applied to optimization of distributed‐parameter matching circuits for a microwave transistor subject to its potential performance

In this article, a rigorous design procedure is carried out for a microwave amplifier by employing the Feasible Design Space and simple analytical gain gradients of the matching circuits. Physical lengths and characteristic impedances of the transmission lines used in the matching circuits are chosen as the design variables and their lower and upper limits are bounded by the limits of the planar transmission line technology so that resulted microwave amplifier can be realized by this technology. Feasible Design Target Space is determined by the compatible performance [noise (F), input VSWR (V_i), gain (G_T)] triplets and their source Z_S(ω_i) and load Z_L(ω_i) terminations resulted from the performance characterization of the active device. These triplets take into account the physical limitations of the device and realization conditions so that F_req ≥ F_min, V_ireq ≥ 1, G_{T min} ≤ G_{T req} ≤ G_{T max}; and Z_S(ω_i) and Z_L(ω_i) terminations be taken place within the “Unconditionally Stable Working Area”. Design of the amplifier for the compatible performance triplets is reduced to the design of the Z_S(ω_i) and Z_L(ω_i), i = 1…N terminations, which is achieved by the gain optimization of the two passive, reciprocal matching two‐ports using the Darlington theorem. Analytical expressions of the gain gradients of the matching circuits are obtained by the two different methods: (i) chain sensitivity matrix approach; (ii) adjoint network approach. Gain gradients of the L‐, T‐, and Π‐types of distributed‐parameter matching circuits are obtained as worked examples. Then typical design examples are given with together the synthesized, target, simulated characteristics. © 2008 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2008.     

NARMAX modelling and robust control of internal combustion engines

Detailed in this paper is a SISO non-linear modelling and robust controller design methodology experimentally verified on an internal combustion engine. The methodology begins with the identification of a NARMAX model that captures the non-linear dynamics relating the input to the output of a system. This model is converted to a describing function representation for the purpose of robust feedback controller design. The ideology for the describing function recovery is developed in the form of an algorithm which can be extended to other NARMAX model structures not considered here. The controller design is executed in the frequency domain where the output performance specification is |y(t)|≤β?t>0 and the actuator saturation constraint is |u(t)|≤K?t>0. For the engine idle speed control application of this study, a SISO NARMAX model of the engine is developed between the by-pass idle air valve (BPAV) and engine speed. The performance objective for the controller design is the time domain tolerance of |Δ rpm| ≤ 100 rpm on idle speed perturbations despite a non-measurable 20 N m external torque disturbance. The controller is validated through numerical simulations as well as experimental verification.     

Convergence and computation of describing functions using a recursive Volterra series

Presented in this paper is a comparison of algorithms for computing an approximation to the sinusoidal input describing function (SIDF) for the nonlinear differential equation ?(t)+b₁y(t)+b₂u²(t)y(t) = K(u?(t)+b₃u(t)) The importance of this nonlinear differential equation comes from the context of nonlinear feedback controller design. Specifically, this equation is either a linear lead or lag controller (depending on the coefficient values) augmented with a nonlinear, polynomial type term. Consequently, obtaining a SIDF representation of this nonlinear differential equation or creating a process to obtain SIDFs for other similar differential equations, will facilitate nonlinear controller design using classical loop shaping tools. The two SIDF approximations studied include the well‐established harmonic balance method and a Volterra series based algorithm. In applying the Volterra series, several theoretical issues were addressed including the development of a recursive solution that calculates high order Volterra transfer functions, and the guarantee of convergence to an arbitrary accuracy. Throughout the paper, case studies are presented. Copyright © 2004 John Wiley & Sons, Ltd.     

Practical multi-constraint H∞ controller synthesis from time-domain data

This paper presents a practical algorithm for MIMO controller design with multiple H^∞ norm constraints. The plant is described by its sampled-data impulse response matrix, which could be determined directly from measurements; the controller design parameters are the tap weights of an FIR discretization of the Q-(or Youla-) parameter. We approximate the H^∞ norm by the maximum transfer matrix 2-norm on an even frequency sampling over θ = [0, π]. The control design algorithm uses the Newton method to minimize two cost functions sequentially: the first determines multiple H^∞ feasibility, and the second minimizes a generalized entropy. As a design example, we control the acoustic radiation from a (mathematically modelled) submerged spherical shell. The plant-model impulse response matrix has McMillan degree 8 800. We specify and synthesize a controller that simultaneously achieves ten H^∞ constraints: one on the radiation reduction, one for stability robustness, and eight on actuator authority.     

On a problem of analytic design of an optimal controller

This article is devoted to the further development of the technique of analytic design of an optimal controller for the case of functionals with an integrand of the form x′Qx+x′Pu+ u′ Ru. In this case the problems of analytic design are considered both without disturbances and at constantly acting disturbances. The statement of the problem for optimal control of a vibration-protective system (VPS) is given, for the solution of which the technique outlined in the article can be used for the solution of the problem of analytic design of the optimal controller.     

Robustness optimization in LQG multivariable systems with time-varying non-linear perturbations

This paper presents an algorithm to synthesize a controller in order to treat the problem of robustness optimization in LQG (linear-quadratic-gaussian) multi-variable systems subjected to time-varying non-linear perturbations. The controller not only minimizes the cost functional in LQG problems but also maximizes the excess robustness with respect to the Hardy H^∞-norm criterion by selecting two frequency-dependent weighting matrices Q(s) and R(s). Our approach is based on the Wiener-Hopf technique (frequency domain approach) and two weighting matrices in the cost functional are shaped by the inverse LQG method to achieve the robustness optimization. Furthermore, the plant of the system has no stable, proper, square, and minimum phase constraints. An example is given to illustrate our result.     

A -based optimal finite-word-length controller design

A novel optimal Finite Word Length (FWL) controller design is proposed in the framework of μ theory. A computationally tractable close-loop stability measure with FWL implementation considerations of the controller is derived based on the μ theory, and the optimal FWL controller realizations are obtained by solving the resulting optimal FWL realization problem using linear matrix inequality techniques.     

Resilient H∞ Control Design for Discrete‐Time Uncertain Linear Systems: An Auxiliary System Transformation Approach

This paper studies the resilient (non‐fragile) H∞ output‐feedback control design for discrete‐time uncertain linear systems with controller uncertainty. The design considers parametric norm‐bounded uncertainty in all state‐space matrices of the system, output and controller equations. The paper shows that the resilient H∞ output‐feedback control problem is equivalent to a scaled H∞ output‐feedback control problem of an auxiliary system without any system or controller uncertainty. Using the existing optimal H∞ design to solve the auxiliary system, the design guarantees that the resultant closed‐loop systems are quadratically stable with disturbance attenuation γ for all admissible system and controller uncertainties. A numerical example is given to illustrate the design method and its benefits.     

Sliding‐mode control of a three‐degrees‐of‐freedom nanopositioner

This paper presents the sliding‐mode control of a three‐degrees‐of‐freedom nanopositioner (Z, θ_x, θ_y). This nanopositioner is actuated by piezoelectric actuators. Capacitive gap sensors are used for position feedback. In order to design the feedback controller, the open‐loop characteristics of this nanopositioner are investigated. Based on the results of the investigation, each pair of piezoelectric actuators and corresponding gap sensors is treated as an independent system and modeled as a first‐order linear model coupled with hysteresis. When the model is identified and the hysteresis nonlinearity is linearized, a linear system model with uncertainty is used to design the controller. When designing the controller, the sliding‐mode disturbance (uncertainty) estimation and compensation scheme is used. The structure of the proposed controller is similar to that of a proportional integral derivative controller. Thus, it can be easily implemented. Experimental results show that 3‐nm tracking resolution can be obtained. Copyright © 2008 John Wiley and Sons Asia Pte Ltd and Chinese Automatic Control Society     

Robust FOPID controller design for fractional‐order delay systems using positive stability region analysis

In this paper, a robust fractional‐order PID (FOPID) controller design method for fractional‐order delay systems is proposed based on positive stability region (PSR) analysis. Firstly, the PSR is presented to improve the existing stability region (SR) in D‐decomposition method. Then, the optimal fractional orders λ and μ of FOPID controller are achieved at the biggest three‐dimensional PSR, which means the best robustness. Given the optimal λ and μ, the other FOPID controller parameters k_p, k_i, k_d can be solved under the control specifications, including gain crossover frequency, phase margin, and an extended flat phase constraint. In addition, the steps of the proposed robust FOPID controller design process are listed at length, and an example is given to illustrate the corresponding steps. At last, the control performances of the obtained robust FOPID controller are compared with some other controllers (PID and FOPI). The simulation results illustrate the superior robustness as well as the transient performance of the proposed control algorithm.     

Optimal hold functions for MDCS sampled‐data problems

In this paper the H² and H^∞ sampled‐data problems with mixed discrete/continuous specifications (MDCS) are considered. The hold function is not fixed a priory but rather is the design parameter. The (sub)optimal controller obtained is of the form of a serial interconnection of a (sub)optimal digital controller and a (sub)optimal hold function. It is demonstrated that the incorporation of the hold function into the sampled‐data design with MDCS extends considerably the possibility to reach a required trade‐off between discrete and continuous specifications in comparison with designs with a fixed hold function. Copyright © 2003 John Wiley & Sons, Ltd.     

Cross standard form: a solution to improve a given controller with H 2 or H ∞ specifications

This paper introduces in cross standard form (CSF) as a solution to the inverse optimal control problem. That is, the CSF is a canonical standard problem whose unique H _∞ or H ₂ optimal controller is a given controller. From the control design point of view, the general idea is to apply the CSF to a given controller in order to set up a standard problem which can be completed to handle frequency domain H ₂ or H _∞ specification. The analytical formulation of the CSF proposed in this paper can be applied to reduced-, full- or augmented-order compensators or two-degree of freedom compensations. Numerical and academic examples are given.     

New results for near-optimal control of linear multiparameter singularly perturbed systems

In this paper, we consider the linear quadratic optimal control problem for multiparameter singularly perturbed systems in which N lower-level fast subsystems are interconnected through a higher-level slow subsystem. Different from the existing methods, a new method is developed to design a near-optimal controller which does not depend on the unknown small parameters. It is shown that the resulting controller in fact achieves an O(||μ||²) approximation to the optimal cost of the original optimal control problem.     

Controller reduction via stable factorization and balancing

A controller reduction procedure based on a representation of a controller as a matrix function defined using stable proper transfer functions and employing a balancing technique is studied in this paper. For a certain right coprime factorization of an LQG designed controller K(s) = N(s)D^-1(s), we approximate using a balancing technique the pair [D(s), N(s)]^T by a low-order pair [D₁(s), N₁(s)]^T defining a factorization of the reduced-order controller K ₁(s) = N₁(s)D₁ ^-1(s). We show that reducing the controller order in this way is motivated in a natural way, which leads to the expectation of both good stability properties and good accuracy of approximation of closed-loop behaviour. This is also demonstrated in some examples.     

Process control using VSI cause selecting control charts

The article considers the variable process control scheme for two dependent process steps with incorrect adjustment. Incorrect adjustment of a process may result in shifts in process mean, process variance, or both, ultimately affecting the quality of products. We construct the variable sampling interval (VSI) Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub>{Z_{\overline{X}}-Z_{S_X^2}} and Z<sub>[`(e)]}-Z_Se²{Z_{\bar{{e}}}-Z_{S_e^2}} control charts to effectively monitor the quality variable produced by the first process step with incorrect adjustment and the quality variable produced by the second process step with incorrect adjustment, respectively. The performance of the proposed VSI control charts is measured by the adjusted average time to signal derived using a Markov chain approach. An example of the cotton yarn producing system shows the application and performance of the proposed joint VSI Z_{[`(X)]</sub> -Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}} -Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts in detecting shifts in mean and variance for the two dependent process steps with incorrect adjustment. Furthermore, the performance of the VSI Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}}-Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts and the fixed sampling interval Z_{[`(X)]</sub> -Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}} -Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts are compared by numerical analysis results. These demonstrate that the former is much faster in detecting small and median shifts in mean and variance. When quality engineers cannot specify the values of variable sampling intervals, the optimum VSI Z_{[`(X)]</sub>-Z<sub>SX<sup>2</sup></sub> {Z_{\overline{X}}-Z_{S_X^2 }} and Z<sub>[`(e)]} -Z_Se² {Z_{\bar{{e}}} -Z_{S_e^2 }} control charts are also proposed by using the Quasi-Newton optimization technique.     

Adaptive control with optimal robust tracking

The problem of optimal robust tracking in two-parameter adaptive control systems under non-linear time-varying unmodelled dynamics is examined. A new robust stability criterion is derived for analysing the robustness of adaptive control systems with non-linear time-varying model errors. Based on the concept of excess robustness and the theory of the minimum H^∞norm, a simple and feasible design algorithm is presented to synthesize a two-parameter adaptive controller which ensures that adaptive control systems can achieve the object of optimal robust tracking in the presence of non-linear time-varying unmodelled dynamics. Simulation results that demonstrate features of the two-parameter adaptive controller with optimal robust tracking in the light of the design algorithm are included.     

Adaptive control of nonlinear systems: A case study of underwater robotic systems

The problem of controlling underwater mobile robots in 6 degrees of freedom (DOF) is addressed. Underwater mobile robots where the number of thrusters and control surfaces exceeds the number of controllable DOF are considered in detail. Unlike robotic manipulators underwater mobile robots should include a velocity dependent thruster configuration matrix B( q ), which modifies the standard manipulator equation to: Mq + C( q ) q + g(x) = B( q ) u where x = J( x ) q . Uncertainties in the thruster configuration matrix due to unmodeled nonlinearities and partly known thruster characteristics are modeled as multiplicative input uncertainty. This article proposes two methods to compensate for the model uncertainties: (1) an adaptive passivity-based control scheme and (2) deriving a hybrid (adaptive and sliding) controller. The hybrid controller combines the adaptive scheme where M, C, and g are estimated on-line with a switching term added to the controller to compensate for uncertainties in the input matrix B. Global stability is ensured by applying Barbalat's Lyapunov-like lemma. The hybrid controller is simulated for the horizontal motion of the Norwegian Experimental Remotely Operated Vehicle (NEROV).     

Stochastic nonlinear stabilization — II: Inverse optimality

After considering the stabilization of a specific class of stochastic nonlinear systems in a companion paper, in this second part, we address the classical question of when is a stabilizing (in probability) controller optimal and show that for every system with a stochastic control Lyapunov function it is possible to construct a controller which is optimal with respect to a meaningful cost functional. Then we return to the problem from Part I and design an optimal backstepping controller whose cost functional includes penalty on control effort and which has an infinite gain margin.     

On mean‐square H∞ control for discrete‐time nonlinear stochastic systems with (x,u, v)‐dependent noises

In this paper, the H_∞ control problem is investigated for a general class of discrete‐time nonlinear stochastic systems with state‐, control‐, and disturbance‐dependent noises (also called (x, u, v)‐dependent noises). In the system under study, the system state, the control input, and the disturbance input are all coupled with white noises, and this gives rise to considerable difficulties in the stability and H_∞ performance analysis. By using the inequality techniques, a sufficient condition is established for the existence of the desired controller such that the closed‐loop system is mean‐square asymptotically stable and also satisfies H_∞ performance constraint for all nonzero exogenous disturbances under the zero‐initial condition. The completing square technique is used to design the H_∞ controller with hope to reduce the resulting conservatism, and a special algebraic identity is employed to deal with the cross‐terms induced by (x, u, v)‐dependent noises. Several corollaries with simplified conditions are presented to facilitate the controller design. The effectiveness of the developed methods is demonstrated by two numerical examples with one concerning the multiplier‐accelerator macroeconomic system.