A robust synthesis methodology for neutrally stable uncertain SISO plants under input amplitude saturation

In this article, a control system design methodology for neutrally stable, uncertain, single-input single-output plants under input amplitude saturation is presented. Based on Horowitz's original three degree of freedom design and extensions developed afterwards, this approach concentrates on neutrally stable, higher type, uncertain plants. A three degree of freedom non-interfering loop structure is used for the synthesis, along with the structure of the additional, independent loop transmission around the saturating element proposed for designing the third degree of freedom H(s). Robust stability and performance are established. The circle criterion, the describing function and non-overshooting conditions are utilised to obtain design constraints. Finally, all these design constraints are expressed in frequency domain bounds and synthesis follows from loop shaping methods such as quantitative feedback theory.     

On a performance-robustness trade-off intrinsic to the natural anti-windup problem

Any natural definition of the anti-windup (AW) control problem requires the design of an add-on compensator which, connected to a saturating closed loop system (which would be well-behaved in the absence of saturation), guarantees stability and, as long as the saturation limits are never exceeded, preservation of the original linear behaviour. Three main contributions are given in this paper. First, it is shown that the “model-matching” requirement implied by the preservation of the linear response can be incompatible with the achievement of robust stability in the presence of large uncertainties, even if the controlled plant is robustly open loop stable. Then, a reasonable “weakened” AW problem is introduced, in which the “model-matching” requirement is considered just as a performance requirement (instead of a hard constraint) whose relaxation can be traded off with robustness to larger uncertainties. Finally, the approach proposed in Teel and Kapoor [(1997a). In Proceedings of the fourth European control conference] is extended to deal with the new problem, leading to a family of state feedback compensators parameterized in terms of a state feedback gain (already present in the original approach) and a stable linear time invariant filter. A detailed design procedure for determining suitable values of the parameters is also described.     

A combined QFT/H ∞ design technique for TDOF uncertain feedback systems

A new way of incorporating QFT principles into H X -control design techniques for solving the two-degrees of freedom feedback problem with highly uncertain plants is developed. The proposed practical design approach consists of two stages. In the first stage, the robustness problem, due to plant uncertainties, is solved by H X -norm optimization. In this stage, the controller inside the loop (the first degree of freedom) is designed, with the ultimate goal of minimizing the cost of feedback. Minimization of the sensor white noise amplification at the input to the plant is also performed using QFT principles. In the second stage of the design, the prefilter outside the loop (the second degree of freedom), is used to achieve the tracking specifications by conventional classical control theory, as practiced by the QFT design procedure. The combined QFT/H X design procedure for single input-single output (SISO) feedback systems is directly applicable to multi input-multi output (MIMO) feedback uncertain systems. The efficiency of the proposed technique is demonstrated with SISO and MIMO design examples for higly uncertain plants.     

基于QFT的无人机纵向飞行控制系统设计

介绍了定量反馈理论(QFT)的基本原理和设计步骤;定量反馈理论作为一种新颖的频率域鲁棒控制技术,综合考虑了对象的不确定性范围和系统的性能指标要求,以定量方式进行分析设计,从而保证了设计结果具有稳定鲁棒性和性能鲁棒性;无人机飞行过程中具有较强的不确定性,气动参数会不断发生变化,运用QFT对无人机纵向飞行控制系统进行设计,可以很好解决飞行控制系统中的不确定性问题;仿真结果显示,QFT设计的控制器能够很好地满足无人机鲁棒稳定性指标和跟踪性能,符合纵向控制的要求。     

Design of rate and amplitude saturation compensators using three degrees of freedom controllers

This paper proposes two frequency-domain design techniques for rate and amplitude saturation compensators formulated as three degrees of freedom controllers. For single constraint systems, it shows an equivalence between the a posteriori compensator design and the a priori controller synthesis proposed by Horowitz. For rate and amplitude constrained systems, the three degrees of freedom controller is a special case of the general saturation compensator. Through multivariable frequency domain analysis of the compensators, hidden properties of the three degrees of freedom controller are revealed. These inherent properties are utilized in compensator designs by techniques shaping the characteristic loci and specification of singular values. Simulated examples illustrate the success of the design methods.     

Simultaneous quadratic stabilization for a class of non-linear systems with input saturation using dynamic surface control

In this paper, a new method to estimate the initial condition set which guarantees the quadratic stability for a class of non-linear systems via dynamic surface control (DSC) in the presence of input saturation has been proposed. The estimated set is enlarged to allow some degree of input saturation, while achieving the simultaneous quadratic stability of the system in the domain. A convex optimization problem to obtain ellipsoidal initial condition sets via a linear matrix inequality (LMI) approach as well as an illustrative example will be presented.     

Stability Analysis and Design of Uncertain Discrete‐time Switched Systems with Actuator Saturation Using Antiwindup and Multiple Lyapunov Functions Approach

The stability analysis and anti‐windup design problem is investigated for a class of discrete‐time switched systems with saturating actuators by using the multiple Lyapunov functions approach. Firstly, we suppose that a set of linear dynamic output controllers have been designed to stabilize the switched system without input saturation. Then, we design anti‐windup compensation gains and a switching law in order to enlarge the domain of attraction of the closed‐loop system. Finally, the anti‐windup compensation gains and the estimation of domain of attraction are presented by solving a convex optimization problem with linear matrix inequality (LMI) constraints. A numerical example is given to demonstrate the effectiveness of the proposed design method.     

A new model reference control architecture: Stability,performance, and robustness

In this paper, we develop a new model reference control architecture to effectively suppress system uncertainties and achieve a guaranteed transient and steady‐state system performance. Unlike traditional robust control frameworks, only a parameterization of the system uncertainty given by unknown weights with known conservative bounds is needed to stabilize uncertain dynamical systems with predictable system performance. In addition, the proposed architecture's performance is not dependent on the level of conservatism of the bounds of system uncertainty. Following the same train of thought as adaptive controllers that modify a given reference system to improve system performance, the proposed method is inspired by a recently developed command governor theory that minimizes the effect of system uncertainty by augmenting the input signal of the uncertain dynamical and reference systems. Specifically, a dynamical system, called a command governor, is designed such that its output is used to modify the input of both the controlled uncertain dynamical and reference systems. It is theoretically shown that if the command governor design parameter is judiciously selected, then the controlled system approximates the given original, unmodified reference system. The proposed approach is advantageous over model reference adaptive control approaches because linearity of the uncertain dynamical system is preserved through linear control laws, and hence, the closed‐loop performance is predictable for different command spectrums. Additionally, it is shown that the architecture can be modified for robustness improvements with respect to high frequency content due to, for example, measurement noise. Modifications can also be made in order to accommodate actuator dynamics and retain closed‐loop stability and predictable performance. The main contribution of this paper is the rigorous analysis of the stability and performance of a system utilizing the command governor framework. A numerical example is provided to illustrate the effectiveness of the proposed architecture. Copyright © 2015 John Wiley & Sons, Ltd.     

Non‐diagonal MIMO QFT controller design reformulation

This paper presents a reformulation of the full‐matrix quantitative feedback theory (QFT) robust control methodology for multiple‐input–multiple‐output (MIMO) plants with uncertainty. The new methodology includes a generalization of previous non‐diagonal MIMO QFT techniques; avoiding former hypotheses of diagonal dominance; simplifying the calculations for the off‐diagonal elements, and then the method itself; reformulating the classical matrix definition of MIMO specifications by designing a new set of loop‐by‐loop QFT bounds on the Nichols Chart, which establish necessary and sufficient conditions; giving explicit expressions to share the load among the loops of the MIMO system to achieve the matrix specifications; and all for stability, reference tracking, disturbance rejection at plant input and output, and noise attenuation problems. The new methodology is applied to the design of a MIMO controller for a spacecraft flying in formation in a low Earth orbit. Copyright © 2008 John Wiley & Sons, Ltd.     

A robust constrained reference governor approach using linear matrix inequalities

The purpose of this paper is to examine and provide a solution to the output reference tracking problem for uncertain systems subject to input saturation. As well-known, input saturation and modelling errors are very common problems at industry, where control schemes are implemented without accounting for such problems. In many cases, it is sometimes difficult to modify the existing implemented control schemes being necessary to provide them with external supervisory control approaches in order to tackle problems with constraints and modelling errors. In this way, a cascade structure is proposed, combining an inner loop containing any proper controller with an outer loop where a generalized predictive controller (GPC) provides adequate references for the inner loop considering input saturations and uncertainties. Therefore, the contribution of this paper consists in providing a state space representation for the inner loop and using linear matrix inequalities (LMI) to obtain a predictive state-vector feedback in such a way that the input reference for the inner loop is calculated to satisfy robust tracking specifications considering input saturations. Hence, the final proposed solution consists in solving a regulation problem to a fixed reference value subjected to a set of constraints described by several LMI and bilinear matrix inequalities (BMI). The main contribution of the paper is that the proposed solution is a non-linear setpoint tracking approach, that is, it is allowed that the system goes into saturation facing the problem of setpoint tracking instead of regulating to the origin. An illustrative numerical example is presented.     

On characterizing sensitivity-based and traditional formulations for quantitative feedback theory

Recent developments in quantitative feedback theory include the 'new formulation' approach in which a robust performance and robust stability problem, similar to Horowitz's traditional QFT formulation, is developed in terms of sensitivity function bounds. The motivation for this approach was to provide the basis for a more rigorous treatment of nonminimum phase systems and/or plants characterized by mixed parametric and non-parametric uncertainty models. However, it has been found in practice that the sensitivity-based formulation exhibits some unique behaviour, i.e. in terms of the open loop design bounds obtained for various choices of nominal plant. Experience has shown that these bounds will dominate (i.e. are more conservative than) the corresponding traditional QFT bounds for the same problem; it has also been observed that the degree to which this occurs varies with choice of the nominal plant. Further, it has been found that the choice of nominal, in certain cases, can lead to a problem which is infeasible with respect to Bode sensitivity (i.e. requiring S(jomega) < 1 as omega infinity), while the traditional QFT problem remains feasible. Heretofore, this behaviour has not been fully explained. In this paper, these issues are characterized in the simplest possible setting, focusing primarily on the behaviour at zero phase angle. A 'modified' sensitivity-based QFT formulation is proposed here in which limitations on the choice of nominal plant are clearly delineated; this formulation results in open loop design bounds which are equivalent to the traditional QFT problem at zero phase angle, while over-bounding them elsewhere. The modified formulation is also shown to meet the same necessary condition for Bode feasibility as traditional QFT. In conclusion, these issues are demonstrated by means of a basic example.     

MIMO QFT CAD PACKAGE (VERSION 3)

The MIMO/QFT CAD package is capable of carrying through a multivariable control system robust design, for tracking and the rejection of external disturbance signals, from problem set-up to a frequency domain analysis of the compensated system. The package automates: the selection of the weighting matrix; the discretization of the plants; the formation of the square effective plants; the polynomial matrix inverse required to form the equivalent plants; the generation of templates; the selection of a nominal plant; the generation of stability, tracking, cross-coupling effects, external disturbance rejection, gamma and composite bounds; the loop shaping; and the design of the prefilter elements. This is followed by a frequency-domain analysis of the completed design, and export of this design to the MATLAB SIMULINK® toolbox in order to validate the frequency domain design by time-domain simulations. © 1997 by John Wiley & Sons, Ltd. This paper was produced under the auspices of the US Government and it is therefore not subject to copyright in the US.     

Model recovery anti-windup for continuous-time rate and magnitude saturated linear plants

In this paper two approaches are given for anti-windup design for nonlinear control systems with linear plants subject to limitations both in the magnitude and the rate of variation of the control input. Both approaches are based on the so-called Model Recovery Anti-Windup (MRAW) framework. The first approach is built by treating the rate + magnitude saturation as a single dynamic nonlinearity, while in the second one, the dynamic compensator dynamics is extended with extra states to treat the two saturations separately. Both approaches lead to global stability with exponentially stable plants and local stability in all other cases. For both approaches, stability and performance guarantees are proven, numerical recipes are given and the relative merits are comparatively highlighted on a simulation example.     

Dynamic output feedback robust MPC for LPV systems subject to input saturation and bounded disturbance

For linear parameter varying (LPV) systems with unknown scheduling parameters and bounded disturbance, a synthesis approach of dynamic output feedback robust model predictive control (OFRMPC) with input saturation is investigated. By pre-specifying partial controller parameters, a main optimization problem is solved by convex optimization to reduce the on-line computational burden. The main optimization problem guarantees that the estimated state and estimation error converge within the corresponding invariant sets such that recursive feasibility and robust stability are guaranteed. The consideration of input saturation in the main optimization problem improves the control performance. Two numerical examples are given to illustrate the effectiveness of the approach.     

Control of discrete‐time periodic linear systems with input saturation via multi‐step periodic invariant sets

This paper studies local control of discrete‐time periodic linear systems subject to input saturation by using the multi‐step periodic invariant set approach. A multi‐step periodic invariant set refers to a set from which all trajectories will enter a periodic invariant set after finite steps, remain there forever, and eventually converge to the origin as time approaches infinity. The problems of (robust) estimation of the domain of attraction, (robust) local stabilization (with bounded uncertainties), and disturbance rejection are considered. Compared with the conventional periodic invariant set approach, which has been used in the literature for local stability analysis and stabilization of discrete‐time periodic linear systems subject to input saturation, this new invariant set approach is capable of significantly reducing the conservatism by introducing additional auxiliary variables in the set invariance conditions. Moreover, the new approach allows to design (robust) stabilizing periodic controller, in the presence of norm bounded uncertainties, whose period is the same as the open‐loop system and is different from the existing periodic enhancement approach by which the period of the controller is multiple times of the period of the open‐loop system. Several numerical examples are worked out to show the effectiveness of the proposed approach. Copyright © 2013 John Wiley & Sons, Ltd.     

Regulation in bimodal systems

This paper considers the regulation problem for bimodal systems against known disturbance and reference signals. Switching between the two plant models as well as between the disturbance and reference signals is defined according to a switching surface. The design of the proposed regulators involves three main steps. First, a set of observer‐based Q‐parameterized stabilizing controllers for the switched system is constructed. The stability and the input/output properties for the resulting closed‐loop switched system with the Q‐parameterized controllers are analysed. Second, regulation conditions for each of the two subsystems in the resulting bimodal switched closed‐loop system are presented. In the third step, regulation conditions for the switched closed‐loop system are developed using two approaches. In the first approach, sufficient regulation conditions are derived based on the closed‐loop system's input–output properties. In the second approach, the forced switched closed‐loop system is transformed into an unforced impulsive switched system using an appropriate coordinate transformation. Hence, the regulation problem for the switched closed‐loop system is transformed into a stability analysis problem for the origin of an impulsive switched system. A regulator synthesis method based on solving some linear matrix inequalities is proposed. Finally, a numerical example is presented to illustrate the effectiveness of the proposed method. Copyright © 2007 John Wiley & Sons, Ltd.     

Fast reference governors for systems with state and control constraints and disturbance inputs

Reference governors are applied to closed-loop tracking systems that are linear and discrete time and have constraints on state and control variables. Earlier results are extended in significant ways. Disturbance inputs, whose values belong to a specified set, are allowed and a general class of reference governors is introduced. Each governor in the class guarantees constraint satisfaction for all reference and disturbance inputs. Moreover, if the reference input is ultimately confined to a neighbourhood of a constraint-admissible constant input, the eventual action of the reference governor reduces to a unit delay. By appropriately selecting reference governors from the allowed class it is possible to simplify significantly their implementation. The increase in on-line speed of operation overcomes prior limits on the practical application of reference governors. Algorithmic procedures are described which facilitate design of the reference governors. Several examples are presented. They illustrate the design process and the excellence of response to large inputs. Copyright © 1999 John Wiley & Sons, Ltd.     

Fixed-structure controller design for systems with actuator amplitude and rate non-linearities

In this paper we develop output feedback controllers and fixed-order (i.e., full- and reduced-order) dynamic compensators for systems with actuator amplitude and rate saturation constraints. The proposed design methodology employs a rate limiter as part of the controller architecture. The problem of simultaneous control amplitude and rate saturation is embedded within an optimization problem by constructing a Riccati equation whose solution guarantees closed-loop global/local asymptotic stability in the face of sector bounded actuator amplitude and rate non-linearities. Application of the proposed framework is demonstrated via a flight control example involving actuator amplitude and rate saturation constraints.     

Stabilization of nonlinear time-delay systems with input saturation via anti-windup fuzzy design

This investigation considers stability analysis and control design for nonlinear time-delay systems subject to input saturation. An anti-windup fuzzy control approach, based on fuzzy modeling of nonlinear systems, is developed to deal with the problems of stabilization of the closed-loop system and enlargement of the domain of attraction. To facilitate the designing work, the nonlinearity of saturation is first characterized by sector conditions, which provide a basis for analysis and synthesis of the anti-windup fuzzy control scheme. Then, the Lyapunov–Krasovskii delay-independent and delay-dependent functional approaches are applied to establish sufficient conditions that ensure convergence of all admissible initial states within the domain of attraction. These conditions are formulated as a convex optimization problem with constraints provided by a set of linear matrix inequalities. Finally, numeric examples are given to validate the proposed method.     

Control of systems with actuator saturation non-linearities: An LMI approach

In this paper, we develop a static, full-state feedback and a dynamic, output feedback control design framework for continuous-time, multivariable, linear, time-invariant systems subject to time-invariant, sector-bounded, input non-linearities. The proposed framework directly accounts for robust stability and robust performance over the class of input non-linearities. Specifically, the problem of feedback control design in the presence of time-invariant, sector-bounded, input non-linearities is embedded within a Lure-Postnikov Lyapunov function framework by constructing a set of linear-matrix-inequality conditions whose solution guarantees closed-loop asymptotic stability with guaranteed domains of attraction in the face of time-invariant, sector-bounded, actuator non-linearities. A detailed numerical algorithm is provided for solving the linear-matrix-inequality conditions arising in actuator saturation control. Three illustrative numerical examples are presented to demonstrate the effectiveness of the proposed approach.