Low order integral-action controller synthesis

A simple controller synthesis method is developed for certain classes of linear, time-invariant, multi-input multi-output plants. The number of poles in each entry of these controllers depends on the number of right-half plane plant zeros, and is independent of the number of poles of the plant to be stabilized. Furthermore, these controllers have integral-action so that they achieve asymptotic tracking of step input references with zero steady-state error. The designed controller’s poles and zeros are all in the stable region with the exception of one pole at the origin for the integral-action design requirement. The freedom available in the design parameters may be used for additional performance objectives, although the only goal here is stabilization and tracking of constant references.     

Simultaneously stabilizing controller design for a class of MIMO systems

It is shown that a class of linear, time-invariant, multi-input multi-output plants can be simultaneously stabilized. This class of plants all have the same number of zeros at infinity, at zero, or both, but no other zeros in the unstable region. If they have zeros at zero or infinity, then their gain matrices at zero and infinity also satisfy a positive-definiteness condition. There is no restriction on the poles of the plants considered in this class. An explicit design procedure is proposed to achieve simultaneously stabilizing controllers. All simultaneously stabilizing controllers for this class of plants are also characterized in terms of a parameter matrix that satisfies a unimodularity condition.     

MIMO PID tuning via iterated LMI restriction

We formulate multi‐input multi‐output proportional integral derivative controller design as an optimization problem that involves nonconvex quadratic matrix inequalities. We propose a simple method that replaces the nonconvex matrix inequalities with a linear matrix inequality restriction, and iterates to convergence. This method can be interpreted as a matrix extension of the convex–concave procedure, or as a particular majorization–minimization method. Convergence to a local minimum can be guaranteed. While we do not know that the resulting controller is globally optimal, the method works well in practice, and provides a simple automated method for tuning multi‐input multi‐output proportional integral derivative controllers. The method is readily extended in many ways, for example, to the design of more complex, structured controllers. Copyright © 2015 John Wiley & Sons, Ltd.     

Simultaneous stabilization of multiplicatively perturbed plants

Necessary and sufficient conditions are obtained for simultaneous stabilizability of a given linear, time-invariant, multi-input multi-output nominal plant and a multiplicatively perturbed plant, where the multiplicative perturbation is stable. These conditions are derived from the general parity-interlacing-property applicable to any two arbitrary plants and are expressed explicitly in terms of the real-axis poles and zeros of the nominal plant and the perturbation. The class of perturbations for which simultaneous stabilization is achievable are characterized using these conditions. A special class of unknown diagonal perturbations is also considered and simultaneously stabilizing controllers are designed for the nominal plant under any such unknown perturbation.     

Reliable decentralised control of delayed MIMO plants

Reliable decentralised proportional–integral–derivative controller synthesis methods are presented for closed-loop stabilisation of linear time-invariant plants with two multi-input, multi-output (MIMO) channels subject to time delays. The finite-dimensional part of plants in the classes considered here are either stable or they have at most two poles in the unstable region. Closed-loop stability is maintained with only one of the two controllers when the other controller is turned off and taken out of service.     

纯量反馈系统稳定零极近似相消的积分约束

导出了单输入单输出反馈控制系统误差响应基于名义系统稳定的零、极点近似相消的时间域积分约束，此积分约束是任何反馈控制系统均应满足的．这一约束给出了单输入单输出反馈系统固有折中的新的观点．名义系统稳定的零、极点近似相消的存在导致反馈控制系统的调节时间延长或者误差响应的无穷范数变大．因此，在反馈控制系统设计中，尽量避免补偿器的零、极点与名义系统的极、零点近似相消(即使这些零、极点是稳定的)．     

Performance and robustness preservation in MIMO systems when applying SPR Substitutions†

We are concerned in this article with controlled linear time-invariant multi input multi output systems in continuous-time. We study here the preservation under strict positive real substitutions (of zero relative degree) of coprime factorisations as well as stabilising properties in parametrised controllers. We also tackle here both the preservation of well-posedness (of the tracking feedback control scheme) and the preservation of H _∞-optimality properties in some classes of closed-loop systems affected by multiplicative unstructured uncertainty, including the so-called suboptimal H _∞-control problem by stable controllers (i.e. preservation of both strong stabilisation and H _∞-boundedness).     

Decentralized blocking zeros and the decentralized strongstabilization problem

This paper is concerned with a new system theoretic concept, decentralized blocking zeros, and its applications in the design of decentralized controllers for linear time-invariant finite-dimensional systems. The concept of decentralized blocking zeros is a generalization of its centralized counterpart to multichannel systems under decentralized control. Decentralized blocking zeros are defined as the common blocking zeros of the main diagonal transfer matrices and various complementary transfer matrices of a given plant. As an application of this concept, we consider the decentralized strong stabilization problem (DSSP) where the objective is to stabilize a plant using a stable decentralized controller. It is shown that a parity interlacing property should be satisfied among the real unstable poles and real unstable decentralized blocking zeros of the plant for the DSSP to be solvable. That parity interlacing property is also sufficient for the solution of the DSSP for a large class of plants satisfying a certain connectivity condition. The DSSP is exploited in the solution of a special decentralized simultaneous stabilization problem, called the decentralized concurrent stabilization problem (DCSP). Various applications of the DCSP in the design of controllers for large-scale systems are also discussed     

Fundamental limitation on achievable decentralized performance

A commonly accepted fact is that the diagonal structure of the decentralized controller poses fundamental limitations on the achievable performance, but few quantitative results are available for measuring these limitations. This paper provides a lower bound on the achievable quality of disturbance rejection using a decentralized controller for stable discrete time linear systems with time delays, which do not contain any finite zeros on or outside the unit circle. The proposed result is useful for assessing when full multivariable controllers can provide significantly improved performance, as compared to decentralized controllers. The results are also extended to the case, where the individual subcontrollers are restricted to be PID controllers.     

A robustness study of a finite‐time/exponential tracking continuous control scheme for constrained‐input mechanical systems: Analysis and experiments

The closed‐loop analysis of a recently proposed continuous scheme for the finite‐time or exponential tracking control of constrained‐input mechanical systems is reformulated under the consideration of an input‐matching bounded perturbation term. This is motivated by the poor number of works devoted to support the so‐cited argument claiming that continuous finite‐time controllers are more robust than asymptotical (infinite‐time) ones under uncertainties and the limitations of their results. We achieve to analytically prove that, for a perturbation term with sufficiently small bound, the considered tracking continuous control scheme leads the closed‐loop error variable trajectories to get into an origin‐centered ball whose radius becomes smaller in the finite‐time convergence case, entailing smaller posttransient variations than in the exponential case. Moreover, this is shown to be achieved for any initial condition, avoiding to restrain any of the parameters involved in the control design, and under the suitable consideration of the nonautonomous nature of the closed loop. The study is further corroborated through experimental tests on a multi‐degree‐of‐freedom robotic manipulator, which do not only confirm the analytical result but also explore the scope or limitations of its conclusions under adverse perturbation conditions.     

Transient response comparison of feedback and feed‐forward compensation methods in systems with zero steady state error

Proportional–Integral–Derivative (PID) controllers with integral action are conventionally used as feedback controllers. They are used to obtain zero steady‐state error when the reference input or the disturbance are steps and where zero‐type systems (i.e. with no poles at the origin) are controlled. The controller meets its objective when the controlled system is non linear, but usually introduces undesirable changes in the dynamics that must be compensated by readjusting the proportional gain. In this paper, we compare PID with two alternative techniques based on the use of a feed‐forward system and a multiple feedback system, respectively. A detailed comparison of the transient response obtained with these methods is presented and validated with some simulation examples.     

Multi‐objective robust fuzzy fractional order proportional–integral–derivative controller design for nonlinear hydraulic turbine governing system using evolutionary computation techniques

The present paper proposes a novel multi‐objective robust fuzzy fractional order proportional–integral–derivative (PID) controller design for nonlinear hydraulic turbine governing system (HTGS) by using evolutionary computation techniques. The fuzzy fractional order PID (FOPID) controller takes closed loop error and its fractional derivative as inputs and performs fuzzy logic operations. Then, it produces the output through the fractional order integrator. The predominant advantages of the proposed controller are its capability to handle complex nonlinear processes like HTGS in heuristic manner, due to fuzzy incorporation and extending an additional flexibility in tuning the order of fractional derivative/integral terms to enhance the closed loop performance. The present work formulates the optimal tuning problem of fuzzy FOPID controller for HTGS as a multi‐objective one instead of a traditional single‐objective one towards satisfying the conflicting criteria such as less settling time and minimum damped oscillations simultaneously to ensure the improved dynamic performance of HTGS. The multi‐objective evolutionary computation techniques such as non‐dominated sorting genetic algorithm‐II (NSGA‐II) and modified NSGA‐II have been utilized to find the optimal input/output scaling factors of the proposed controller along with the order of fractional derivative/integral terms for HTGS system under no load and load turbulence conditions. The performance of the proposed fuzzy FOPID controller is compared with PID and FOPID controllers. The simulations have been conducted to test the tracking capability and robust performance of HTGS during dynamic set point changes for a wide range of operating conditions and model parameter variations, respectively. The proposed robust fuzzy FOPID controller has ensured better fitness value and better time domain specifications than the PID and FOPID controllers, during optimization towards satisfying the conflicting objectives such as less settling time and minimum damped oscillations simultaneously, due to its special inheritance of fuzzy and FOPID properties.     

PID Stabilization of MIMO Plants

Closed-loop stabilization using proportional-integral-derivative (PID) controllers is investigated for linear multiple-input-multiple-output (MIMO) plants. General necessary conditions for existence of PID-controllers are derived. Several plant classes that admit PID-con- trollers are explicitly described. Plants with only one or two unstable zeros at or "close" to the origin (alternatively, at or close to infinity) as well as plants with only one or two unstable poles which are at or close to origin are among these classes. Systematic PID-controller synthesis procedures are developed for these classes of plants.     

Designs and analyses of various fuzzy controllers with region‐wise linear PID subcontrollers

To reduce the complexity of PID‐like fuzzy controllers with three inputs, designs and analyses of various fuzzy controllers with region‐wise linear PID subcontrollers are presented in this paper. The proposed region‐wise linear PID subcontrollers are composed of one dimensional fuzzy mechanism. And the triangular‐type membership functions are adopted for the input variables of the fuzzy controllers. All the possible structures of fuzzy controllers with region‐wise linear PID subcontrollers are discussed. According to the number of one‐dimensional fuzzy mechanisms included in the structure of the fuzzy controllers, the fuzzy controllers are classified into three main categories. An algorithm is provided to construct effective fuzzy controllers with lowest complexity among all the possible structures. Also, the properties of various designs of fuzzy controllers with region‐wise linear PID subcontrollers are compared. The simulation results are included to demonstrate the performances of three basic types of proposed fuzzy controllers with the linear, nonlinear, and delayed plants. © 2001 John Wiley & Sons, Inc.     

Robust tracking and disturbance rejection of bounded rate signals for uncertain/non-linear systems

We consider the problem of designing controllers for non-linear/uncertain systems to achieve tracking of a reference output signal in the presence of a disturbance input signal. When the exogenous signal (combined reference and disturbance) is constant, we require that the tracking error eventually go to zero. When the exogenous signal has a bounded rate, we require that the tracking error be eventually bounded with a bound which only depends on the bound on the magnitude of the rate of the exogenous signal. We also require that the state and control input are bounded when the exogenous signal is bounded. We propose controllers which have a classical PI (proportional integral) structure using state feedback. For linear systems and specific classes of non-linear/uncertain systems we present conditions whose satisfaction guarantees the existence of controllers which achieve the desired behaviour. Satisfaction of these conditions also yields the controller gain matrices.     

Reliable decentralized integral-action controller design

Reliable stabilizing controller design with integral-action is considered for linear time-invariant, multi-input-multi-output decentralized systems with stable plants. Design methods are proposed to achieve reliable closed-loop stability with integral-action in each output channel for asymptotic tracking of step-input references applied at each input. The design approaches guarantee stability and integral-action in the active channels when all controllers are operational and when any of the controllers is set equal to zero due to failure     

Robustness with integrators

Two important practical criteria for most real control systems are that the controller has an integrator and that the closed-loop system has robust stability. In this paper we identify and characterize a grouping or classification of linear scalar discrete-time plants comprising three classes according to the best possible stability robustness margins against coprime factor uncertainty that can be obtained with integral controllers.     

An upper bound for the gain of stabilizing proportional controllers

A stable linear system controlled by a proportional controller is closed-loop stable provided the controller has sufficiently small gain. If the system has an unstable zero then any proportional controller with sufficiently large gain is destabilizing.In this note we give an upper bound for the gain of stabilizing proportional controllers of stable systems that have one or more unstable zeros.     

On the parametrization of solutions to the servomechanism problem

This paper addresses a servomechanism problem for linear multivariable plant/sensor systems and two-parameter controllers. The controllers are required to ensure asymptotic tracking of a class of reference signals and asymptotic rejection of two classes of distinct disturbances signals which affect simultaneously the plant input and the sensor output. The solutions of the servomechanism problem are explicitly parametrized by free, stable, rational and proper matrices under two sets of assumptions: one which is necessary and sufficient for the existence of solutions and another (stronger) set which avoids the Kronecker products and associated higher-dimension factorizations involved in the first case.     

Design of Adaptive Block Backstepping Controllers with Perturbations Estimation for Nonlinear State‐Delayed Systems in Semi‐Strict Feedback Form

This paper is an extended study of an existing block backstepping control scheme designed for a class of perturbed multi‐input systems with multiple time‐varying delays to solve regulation problems, where the time‐varying delays must be linear with state variables. A new control scheme is proposed in this research where all the unknown multiple time‐varying delay terms in the dynamic equations can be nonlinear state functions in non‐strict feedback form, and the upper bounds of the time‐delays as well as their derivatives need not to be known in advance. Another improvement is to further alleviate the problem of “explosion of complexity,” i.e., to reduce the number of time derivatives of virtual inputs that the designers have to compute in the design of controllers. This is done by utilizing an existent derivative estimation algorithm to estimate the perturbations in the designing of proposed controllers. Adaptive mechanisms are also embedded in the controllers so that the upper bounds of perturbations and perturbation estimation errors are not required to be known beforehand. The resultant controlled systems guarantee asymptotic stability in accordance with the Lyapunov stability theorem. Finally, a numerical example and a practical application are demonstrated to verify the merits and feasibility of the proposed control scheme.