Low‐order H∞ controller design on the frequency domain by partial optimization

In this paper, an optimization method of low‐order multivariable controllers for H_∞ control is proposed. Starting from a low‐order stabilizing controller, our method gives a sequence of controllers for which the H_∞ norm performance index is monotonically non‐increasing by tuning the numerator coefficient matrices of the low‐order controller. This controller class includes multivariable PID controllers. The proposed method is a descent method where the feasible direction is calculated by solving a linear matrix inequality that represents a sufficient condition for the H_∞ criterion for each frequency. Usefulness is shown by two numerical examples. Copyright © 2009 John Wiley & Sons, Ltd.     

Robust Controller Design And Pid Tuning For Multivariable Processes

In this paper, a robust controller design method is first formulated to deal with both performance and robust stability specifications for multivariable processes. The optimum problem is then dealt with using a loop‐shaping H_∞ approach, which gives a sub‐optimal solution. Then a PID approximation method is proposed to reduce a high‐order controller. The whole procedure involves selecting several parameters and the computation is simple, so it serves as a PID tuning method for multivariable processes. Examples show that the method is easy to use and the resulting PID settings have good time‐domain performance and robustness.     

Non-fragile multivariable PID controller design via system augmentation

In this paper, the issue of designing non-fragile H_∞ multivariable proportional-integral-derivative (PID) controllers with derivative filters is investigated. In order to obtain the controller gains, the original system is associated with an extended system such that the PID controller design can be formulated as a static output-feedback control problem. By taking the system augmentation approach, the conditions with slack matrices for solving the non-fragile H_∞ multivariable PID controller gains are established. Based on the results, linear matrix inequality -based iterative algorithms are provided to compute the controller gains. Simulations are conducted to verify the effectiveness of the proposed approaches.     

STATE‐SPACE DIGITAL PID CONTROLLER DESIGN FOR MULTIVARIABLE ANALOG SYSTEMS WITH MULTIPLE TIME DELAYS

This paper presents a discrete‐time state‐space methodology for optimal design of digital PID controllers for multivariable analog systems with multiple time delays. The multiple time‐delayed multivariable analog systems are formulated in a state‐space generic form so that the exact discrete‐time state‐space model can be constructed. Then, the optimal digital PID controller is designed via a state‐feedback and state‐feedforward LQR approach. The developed PID controller can be applied to a general time‐delayed multivariable analog system represented by a semi‐proper or strictly proper transfer function matrix. Illustrative examples are given to compare the performance of the proposed approach with alternative techniques.     

A genetic-multivariable fractional order PID control to multi-input multi-output processes

A multivariable fractional order PID controller is designed and to get suitable coefficients for the controller, a genetic algorithm with a new topology to generate a new population is proposed. The three parts of the genetic algorithm such as reproduction, mutation, and crossover are employed and some variations in the methods are fulfilled so that a better performance is gained. The genetic algorithm is applied to design FOPID controllers for a multivariable process and the results are compared with the responses of a H_∞ based multivariable FOPID controller. The simulation responses show that in all cases, the genetic-multivariable FOPID controller has suitable performance, and the output of the system has a smaller error. Also, in the proposed method, variations in one output have a smaller effect on another output which is shown the ability of the proposed method to overcome the interaction in the multivariable processes.     

Multivariable PID control by decoupling

This paper presents a new methodology to design multivariable proportional-integral-derivative (PID) controllers based on decoupling control. The method is presented for general n × n processes. In the design procedure, an ideal decoupling control with integral action is designed to minimise interactions. It depends on the desired open-loop processes that are specified according to realisability conditions and desired closed-loop performance specifications. These realisability conditions are stated and three common cases to define the open-loop processes are studied and proposed. Then, controller elements are approximated to PID structure. From a practical point of view, the wind-up problem is also considered and a new anti-wind-up scheme for multivariable PID controller is proposed. Comparisons with other works demonstrate the effectiveness of the methodology through the use of several simulation examples and an experimental lab process.     

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.     

Plant zero structure and further order reduction of a singular H∞ controller

A new class of reduced‐order controllers is obtained for the H_∞ problem. The reduced‐order controller does not compromise the performance attained by the full‐order controller. Algorithms for deriving reduced‐order H_∞ controllers are presented in both continuous and discrete time. The reduction in order is related to unstable transmission zeros of the subsystem from disturbance inputs to measurement outputs. In the case where the subsystem has no infinite zeros, the resulting order of the H_∞ controller is lower than that of the existing reduced‐order H_∞ controller designs which are based on reduced‐order observer design. Furthermore, the mechanism of the controller order reduction is analysed on the basis of the two‐Riccati equation approach. The structure of the reduced‐order H_∞ controller is investigated. Copyright © 2002 John Wiley & Sons, Ltd.     

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.     

On the design of multivariable PID controllers via LMI approach

In this paper, we study the design problem of multivariable PID controllers which guarantee the stability of the closed loop systems, H₂ or H_∞ performance specifications, or maximum output control requirement, respectively. Algorithms based on iterative linear matrix inequality technique are developed to find the feedback gains of PID controllers corresponding to the above mentioned four cases. A numerical example on the design of PID controllers for aircraft is provided to illustrate the effectiveness of the proposed method.     

A Stable Output Feedback Position Control With Integral Action For Robot Manipulators

In this paper, the problem of robust regulation of robot manipulators using only position measurements is addressed. The main idea of the control design methodology is to use an observer to estimate simultaneously the velocity and the modeling error signal induced by model/system mismatches. The controller is obtained by replacing the velocity and the modeling error in an inverse dynamics feedback by their estimates, which leads to a certainty equivalence controller. The resulting controller has a PID‐type structure which, under least prior knowledge, reduces to the PI₂D regulator studied in [20]. Moreover, the controller is endowed with a natural antireset windup (ARW) scheme to cope with control torque saturations. Regarding the closed‐loop behavior, it is proven that the region of attraction can be arbitrarily enlarged with high observer gains only, thus we prove semiglobal asymptotic stability. Our result supersedes previous works in the direction of performance estimates; specifically, it is also proven that the performance induced by a saturated inverse dynamics controller can be recovered by our PID‐type controller. In this sense, our work reveals some connections between PID‐type and inverse dynamics controllers.     

Robust ℋ︁∞ PID control for multivariable networked control systems with disturbance/noise attenuation

In this paper, we study the design problem of PID controllers for networked control systems (NCSs) with polyhedral uncertainties. The load disturbance and measurement noise are both taken into account in the modeling to better reflect the practical scenario. By using a novel technique, the design problem of PID controllers is converted into a design problem of output feedback controllers. Our goal of this paper is two‐fold: (1) To design the robust PID tracking controllers for practical models; (2) To develop the robust ??_∞ PID control such that load and reference disturbances can be attenuated with a prescribed level. Sufficient conditions are derived by employing advanced techniques for achieving delay dependence. The proposed controller can be readily designed based on iterative suboptimal algorithms. Finally, four examples are presented to show the effectiveness of the proposed methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Multivariable PID control with set-point weighting via BMI optimisation

The paper focuses on the design of multivariable PID controllers with set-point weighting. The advantage of this PID structure is that the responses of the system to disturbances and to changes in the set-point can be adjusted separately. The proposed design methods rely on the transformation of the tuning of the controller gains into a static output feedback (SOF) problem. Hence, multivariable PID controllers can be designed by solving an optimisation problem with bilinear matrix inequalities (BMIs). The paper addresses the design of both time-invariant and gain-scheduled robust controllers. All of the tuning methods discussed through the paper are based on a PID structure with filtered derivative term, thus guaranteeing the well-posedness of the closed loop system.     

Robust non‐minimal order filter and smoother design for discrete‐time uncertain systems

This paper is concerned with the design of robust non‐minimal order H_∞ filters for uncertain discrete‐time linear systems. The uncertainty is assumed to be time‐invariant and to belong to a polytope. The novelty is that a convex filtering design procedure with Linear Matrix Inequality constraints is proposed to synthesize guaranteed‐cost filters with order greater than the order of the system. An H_∞‐norm bound for the transfer‐function from the system input to the filtering error is adopted as performance criterion. The non‐minimal order filters proposed generalize other existing filters with augmented structures from the literature and can provide better performance. An extension to the problem of robust smoothing is proposed as well. The procedure is illustrated by a numerical example. Copyright © 2016 John Wiley & Sons, Ltd.     

Robust Performance Controller Synthesis Based on Discrete‐Time Noncausal Linear Periodically Time‐Varying Scaling

This paper is concerned with the technique called discrete‐time noncausal linear periodically time‐varying (LPTV) scaling for robust stability analysis and synthesis. It is defined through the lifting treatment of discrete‐time systems, and naturally leads to a sort of noncausal operation of signals. In the robust stability analysis of linear time‐invariant (LTI) systems, it has been shown that even static noncausal LPTV scaling induces some frequency‐dependent scaling when it is interpreted in the context of lifting‐free treatment. This paper first discusses in detail different aspects of the effectiveness of noncausal LPTV scaling, with the aim of showing its effectiveness in controller synthesis. More precisely, we study the robust performance controller synthesis problem, where we allow the controllers to be LPTV. As in the LTI robust performance controller synthesis problem, we tackle our problem with an iterative method without guaranteed convergence to a globally optimal controller. Despite such a design procedure, the closed‐loop H_∞ performance is expected to improve as the period of the controller is increased, and we discuss how the frequency‐domain properties of noncausal LPTV scaling could contribute to such improvement. We demonstrate with a numerical example that an effective LPTV controller can be designed for a class of uncertainties for which the well‐known μ‐synthesis fails to derive even a robust stabilization controller.     

Robust PID controller tuning based on the constrained particle swarm optimization

This paper proposes a novel tuning strategy for robust proportional-integral-derivative (PID) controllers based on the augmented Lagrangian particle swarm optimization (ALPSO). First, the problem of PID controller tuning satisfying multiple H_∞ performance criteria is considered, which is known to suffer from computational intractability and conservatism when any existing method is adopted. In order to give some remedy to such a design problem without using any complicated manipulations, the ALPSO based robust gain tuning scheme for PID controllers is introduced. It does not need any conservative assumption unlike the conventional methods, and often enables us to find the desired PID gains just by solving the constrained optimization problem in a straightforward way. However, it is difficult to guarantee its effectiveness in a theoretical way, because PSO is essentially a stochastic approach. Therefore, it is evaluated by several simulation examples, which demonstrate that the proposed approach works well to obtain PID controller parameters satisfying the multiple H_∞ performance criteria.     

Multi-loop design of multi-scale controllers for multivariable processes

Based on the recently proposed (SISO) multi-scale control scheme, a new approach is introduced to design multi-loop controllers for multivariable processes. The basic feature of the multi-scale control scheme is to decompose a given plant into a sum of basic modes. To achieve good nominal control performance and performance robustness, a set of sub-controllers are designed based on the plant modes in such a way that they are mutually enhanced with each other so as to optimize the overall control objective. It is shown that the designed multi-scale controller is equivalent to a conventional PID controller augmented with a filter. The multi-scale control scheme offers a systematic approach to designing multi-loop PID controllers augmented with filters. Numerical studies show that the proposed multi-loop multi-scale controllers provide improved nominal performance and performance robustness over some well-established multi-loop PID controller schemes.     

A Design Of Multiloop Predictive Self‐Tuning Pid Controllers

In this paper, a new design scheme of multiloop predictive self‐tuning PID controllers is proposed for multivariable systems. The proposed scheme firstly uses a static pre‐compensator as an approximately decoupling device, in order to roughly reduced the interaction terms of the controlled object. The static matrix pre‐compensator is adjusted by an on‐line estimator. Furthermore, by regarding the approximately decoupled system as a series of single‐input single‐output subsystems, a single‐input single‐output PID controller is designed for each subsystem. The PID parameters are calculated on‐line based on the relationship between the PID control and the generalized predictive control laws. The proposed scheme is numerically evaluated on a simulation example.     

Multirate sampled‐data stabilization for a class of low‐order lower‐triangular nonlinear systems

This paper investigates the problem of sampled‐data controller design for a class of lower‐triangular systems in the p‐normal form (0<p<1). A multirate digital feedback control scheme is proposed to achieve the global strong stabilization of the sampled‐data closed‐loop system under some assumptions. In the design of the controller, the input‐Lyapunov matching strategy and multirate control approach are combined to obtain better stabilizing performance. Unlike the design method based on the approximate discrete‐time model, our controller is obtained from the exact discrete‐time equivalent model, which does not need to be computed completely. The approximate multirate digital controllers are proved to be effective in the practical implementation. It is shown that, compared with the emulated control scheme, our controller may provide faster decrease of Lyapunov function for each subsystem. This will lead to allow large sampling periods. An illustrative example is provided to verify the effectiveness of the proposed control scheme.