Optimal Decoupled Control of Two-Input Two-Output Systems

The paper presents some approaches for the design of a diagonal proportional-integral-derivative (PID) controller for a two-input two-output (TITO) system. First, a new autotuning procedure is given for a TITO system and from the results of this a design method is developed based on moving the critical point to a desired position on the compensated characteristic loci. Second, an optimisation method using integral performance criteria is presented. Examples are given to illustrate both design methods.

Current mode circuits have attracted the attention of the design engineers due to their unique characteristics that are not found in their voltage mode counterparts. This article presents a series expansion method based design of current mode circuits for the control of a two input two output (TITO) system. The unique contribution of this work is the realization of the TITO controller using the second generation current conveyor circuits. The controller was practically implemented using AD844 integrated circuits. Simulation and the experimental results are presented for a pragmatic case study to demonstrate the efficacy of the proposed design methodology.

The design of decentralized proportional integral (PI) controller based on biggest log modulus tuning (BLT) tuning method is extended to design centralized PI controllers for two-input–two-output (TITO) systems. The multivariable PI control structure proposed by Tanttu and Lieslehto (1991) is considered. The single detuning parameter (F) is selected in order to get the biggest log modulus as 4 dB. Decentralized PI controllers are also designed using the IMC to get the biggest log modulus as 4 dB. The centralized PI controllers give improved main responses and decreased interactions. This improvement is shown particularly for TITO systems having the relative gain element, λij < 1. The robustness of these controllers is evaluated by the inverse maximum singular value versus frequency plot for both the input and output multiplicative uncertainties. Simulation results are given for two TITO examples. The present controllers give better performances over that of the original Tanttu and Lieslehto (TL) method. The present decentralized PI controllers give improved performances over the decentralized PI controllers proposed by Xiong and Cai (2006).

This study describes the design and implementation of an algebraic approach for a dynamic optimal decoupling controller in a multi-variable liquid-level system. This method aligns with model-matching criteria and the frequency-matching technique. The prime objective of the proposed approach is to create a closed-loop feedback system using a PID controller that aligns with a user-defined linear system model in terms of both system dynamics and static behavioral response. To illustrate robust stability, the study incorporates multiplicative input and output uncertainty. Simulation results verify the efficacy of the proposed decentralized controller, presenting its effectiveness in achieving both set-point accuracy and disturbance attenuation. Furthermore, the study employs disk margin analysis to ascertain safe ranges for gain and phase margin.

In the control of industrial robots, both safety and reference tracking performance are required. For safe human-robot interaction, robots should exhibit low mechanical (or controlled) impedance so that they react to the interaction forces in a compliant manner. On the other hand, the reference tracking requires for the robots to reject exogenous disturbances, which results in an increased impedance. In order to achieve these two conflicting objectives, a frequency-shaped impedance control (FSIC) method is proposed in this paper. The proposed method utilizes the two different functionalities of the disturbance observer (DOB): a disturbance estimation function as an observer and a disturbance rejection function as a feedback controller. Namely, the DOB is utilized as an observer at the frequencies where the robots interact with humans, while it is used as a feedback controller (i.e., disturbance rejection controller) at the frequencies where the reference tracking is required. The proposed approach is realized by shaping a filter of the DOB in the frequency domain so that the impedance is manipulated to achieve both the compliant interaction and reference tracking. The compromised reference tracking performance in the frequency range, where the impedance is set low, can also be supplemented by feedforward control. A typical feedback controller and a feedforward controller are designed in addition to the DOB-controlled system as the whole control system to enhance reference tracking performance and the betterment of stability robustness. The proposed method is verified by experimental results in this paper.

Bellman’s discrete dynamic programming is one of the most general approaches to solve optimal control problems. For discrete-time dynamical systems, it is, at least theoretically, capable to determine globally optimal control laws. In most practical cases, both state and control variables are subject to constraints. Due to the necessity for gridding of the range of both state and control variables in numerical implementations of dynamic programming, the computational effort grows exponentially with increasing system dimensions. This fact is well known as the curse of dimensionality. Furthermore, gridding of intervals representing uncertain system parameters is inevitable, if dynamic programming is used for the design of optimal controllers for systems with uncertainties. In this contribution, an interval arithmetic procedure for the design of optimal and robust controllers is presented. This procedure relies on the basic concepts of dynamic programming. Sophisticated techniques for the exclusion of non-optimal control strategies significantly reduce the computational burden. Since interval techniques can be applied to both continuous-time and discrete-time dynamical systems, the interval arithmetic optimization approach presented in this chapter is applicable to both cases. In addition, the inclusion of effects of uncertain parameters in the underlying optimality criteria is demonstrated. For that purpose, interval arithmetic routines for analysis and design of optimal and robust controllers have been developed. Details about computationally efficient implementations of interval arithmetic optimization procedures and numerical results for a mechanical positioning system with statedependent switchings between different dynamical models for viscous and Coulomb friction are summarized.

This paper considers the problem of dynamic output feedback H ∞ controller design for linear systems with input and output imprecision. In the controlled systems, the measurement and control signal transmissions from the plant to the controller and the controller to the plant, respectively, are assumed to be imprecise, and norm-bounded uncertainties are utilized to model the phenomenon of the imprecise transmissions. Attention is focused on the design of an H ∞ controller such that the closed-loop system is asymptotically stable and has a prescribed H ∞ performance with respect to the imprecision effects. A linear fractional transformation approach is proposed to obtain LMI-based design conditions for the H ∞ controller. Finally, the usefulness of the developed approach will be demonstrated by a numerical example.

The gap metric has been employed to formulate weighting functions for multi-model control of nonlinear systems. Compared to traditional weighting method, there is only one tuning parameter, so complicated tuning task is avoided. Besides, the gap metric based weighting methods extend the merits of traditional ones: they can also be computed offline and stored in lookup tables. Comparison among three kinds of gap metric based weighting method has been made in this work. A Two-input-two -output (TITO) system is studied. Simulations demonstrate that each of the gap metric based weighting method has its typical advantage. And they are all superior to traditional weighting methods.

The main objective of this research work is the design of decentralized PID controller with fractional order filter for strongly connected multivariable distillation column system. Wherein, the major contribution of this paper is the proposition of compensators for the decoupling process, in addition to extending the application of the proposed approach from Single-Input Single-Output (SISO) systems into Multi-Input Multi-Output (MIMO) systems. More precisely, the proposed idea is based on eliminating the strongly interactive terms between the inputs and outputs of the multivariable distillation column systems so that, the separated multi-loop parts can be treated as a set of SISO subsystems, which have an equivalent dynamic of the whole original system. Thereafter, a PID controller with a fractional filter has been introduced to the decoupled subsystems in decentralized scheme in order to improve the control system robustness. Two multivariable systems have been carefully considered in this paper; the first is a theoretical Two-Input Two-Output (TITO) system while the second is the Wood and Berry distillation column. The obtained results have confirmed the superiority of the proposed approach over the two degrees of freedom technique, which is based on Internal Model Control method. Indeed, the proposed approach has reduced the dynamic error, the settling time, and the rise time. Furthermore, the designed decentralized PID controller with fractional filter has demonstrated rejection to structured perturbation from one loop to another.

In this paper, we present a global optimal and stable fuzzy controller design method for both continuous- and discrete-time fuzzy systems under both finite and infinite horizons. First, a sufficient condition is proposed which indicates that the global optimal effect can be achieved by the fuzzily combined local optimal controllers. Based on this sufficient condition, we derive a local concept approach to designing the optimal fuzzy controller by applying traditional linear optimal control theory. The stability of the entire closed-loop continuous fuzzy system can be ensured by the designed optimal fuzzy controller. The optimal feedback continuous fuzzy system can not only be guaranteed to be exponentially stable, but also be stabilized to any desired degree. Also, the total energy of system output is absolutely finite. Moreover, the resultant feedback continuous fuzzy system possesses an infinite gain margin; that is, its stability is guaranteed no matter how large the feedback gain becomes. Two examples are given to illustrate the proposed optimal fuzzy controller design approach and to demonstrate the proved stability properties.

In this study, the problem of finding an optimal controller for nonlinear systems with one input and a reference tracking signal is approached. With the problem's formulation, any desired signal can be tracked instantly with a closed-loop controller without the need for integral terms. Presentation lies at the heart of optimal control. This study, however, does not consider the integral term, allowing tracking and stability to occur naturally. It has a broad scope with a wide range of applications, namely when dealing with affine nonlinear systems, which provide geometric control unification with asymptotic stability in some cases. A common scenario that comes from optimal control involves the minimization of integral cost functionals. Issues like asymptotic stability or even tracking to the desired reference signal have always been the main limitations. In this study, the main theorem allows the solution of optimal control problems with no-integral terms, in other words tracking problems with input/state constraints, providing closed-loop controllers. A DC motor with a pendulum in upright position is an example of an application for which singular optimal control is tested in this study. The results confirm both asymptotic stability and optimal tracking with an accuracy of 95%. The main contributions of this study include an optimal closed-loop controller with no mixed initial/final conditions, input/state constraints, asymptotic stability guarantee, a strong connection with geometric tools and finally the possibility to generalize to systems with multiple inputs. As a conclusion, general nonlinear control systems can be included in the optimal control methodology presented in this study including input/state constraints. Due to the lack of integral terms, the problem can be solved in closed form by using an optimal closed-loop controller.

Observer-based dynamic optimal recoil controller design for deepwater drilling riser systems

This paper investigates fuzzy adaptive output feedback optimal control problem for a class of strict-feedback nonlinear systems. With the help of fuzzy logic systems approximating the unknown nonlinear functions and cost function, the unmeasured states are estimated by designing fuzzy adaptive state observer. Combining state observer with backstepping design technique, a feedforward controller is designed. Based on the designed feedforward control strategy, the controlled nonlinear system can be converted to an equivalence nonlinear system in affine-form. Finally, a fuzzy adaptive optimal controller with parameters adaptive laws is developed. The whole control scheme consists of a feedforward controller and a feedback optimal controller. It is shown that the proposed output feedback optimal control approach can guarantee that all signals in the closed-loop system are bounded, and the system output can track the reference signal. In addition, the proposed control approach can guarantee cost function is the smallest. Simulation results are given to demonstrate the effectiveness of the proposed control approach.

SUMMARY The development of control strategies for controllable passive dampers — ‘semiactive’ damping devices — is complicated by the nonlinear and dissipative nature of the devices and the nonlinear nature of the closed-loop system with any form of parametric feedback control. Control design for nonlinear systems is often achieved by designing a control for a linearized model since strategies for linear systems are straightforward. One such approach is clipped-optimal control in which the desired damper forces are determined from an optimal controller (e.g., linear quadratic regulator, linear quadratic Gaussian, H2 and H∞), which is designed assuming that the damping devices are fully linear actuators that can exert any forces (dissipative or nondissipative), and a secondary bang–bang controller commands the controllable damper to exert forces as close as possible to the desired forces. However, designs using any linearized model generally result in suboptimal (and sometimes very poor) performance because the linear actuator assumption differs from the actual implementation with a dissipative damping device. Thus, one must generally resort to a large-scale parameter study in which the nonlinear system is simulated many times to determine control strategies that are actually optimal for the nonlinear controlled closed-loop system. This paper demonstrates how an approach developed by the authors, which can reduce the model of a system with local modifications to a lower order set of Volterra integral equations in the modification forces, can significantly decrease the computational burden of a complex control design study for controllable dampers. The approach is first adapted to use a state feedback control with a bang–bang on–off secondary controller. Then, the structural and damping device models are detailed. The optimal semiactive control design is demonstrated via both a parameter study and an optimization algorithm. Numerical examples demonstrate that the approach is effective and, compared with a conventional simulation approach, can reduce computation time by two orders of magnitude. Copyright © 2014 John Wiley & Sons, Ltd.

This paper describes a formal method of designing optimal feedback controllers using the method of dynamic programming. Adopting a quadratic performance criterion, a multi-stage control law is easily programmed for a digital computer thus providing convergent feedback matrices. A visual display system is employed which allows the designer to simulate control of the process for a variety of conditions, enabling him to select the performance function which gives the desired system response.