Controllable Canonical Form Research Articles

Robust controller design for fuzzy parametric uncertain systems: An optimal control approach

A new approach of designing a robust controller for fuzzy parametric uncertain systems is proposed. A linear time invariant (LTI) system with fuzzy coefficients is called as fuzzy parametric uncertain system (FPUS). The proposed method envisages conversion of the FPUS into an uncertain (interval) state space controllable canonical form system in terms of its alpha cut. Further, the problem of designing a robust controller is translated into an optimal control problem minimizing a cost function. For matched uncertainty, it is shown that the optimal control problem is a linear quadratic regulator (LQR) problem, which can be solved to obtain a robust controller for FPUS. The numerical examples and simulation results show the effectiveness of the proposed method in terms of robustness of the controller.

A transfer function approach to the realisation problem of nonlinear systems

This article studies the nonlinear realisation problem, i.e. the problem of finding the state equations of a nonlinear system from the transfer function representation being easily computable from the higher order input–output differential equation. The realisation in both observer and controller canonical forms is studied. The results demonstrate a clear connection with those from linear theory. In the solution the concept of adjoint polynomials, adjoint transfer function and right factorisation of the transfer function play a key role. Finally, the results are applied for system linearisation up to input–output injection used in the observer design.

Q-Difference linear control systems

We propose q-difference versions of some basic results of complete controllability and observability of linear systems such as criterions of controllability and observability, controllability and observability canonical forms, the q-duality theorem, the interconnection between these concepts and that of polynomials without common zeros. As an appendix, is given a simple but computationally meaningful Maple Procedure for the controllability criterion for time constant systems.

Canonical Forms of Generic Piecewise Linear Continuous Systems

This technical note reports a simple, but effective strategy to derive a single coordinate transformation of a generic <formula formulatype="inline" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex Notation="TeX">$n$</tex></formula> -dimensional piecewise-linear continuous vector field that puts both of its subsystems into the same observer or control canonical form. The only hypothesis required is that both subsystems are either observable or controllable. To derive the result, the structure of the system is used in order to prove that it suffices to consider the same transformation for both subsystems. The theoretical results are illustrated by means of the Colpitts oscillator as a practical case of study.

Model reference robust adaptive control for a class of uncertain switched linear systems

SUMMARYIn this paper, we proposed a model reference robust adaptive control approach for a class of uncertain switched linear systems, in which subsystems of the switched linear system are in control canonical form. The control architecture is composed of a switched reference system (SRS) and a switched adaptive controller (SAC). The SRS specifies the desired dynamics of the uncertain switched linear system, while the SAC makes the uncertain switched linear system dynamics track the SRS dynamics. By multiple Lyapunov functions method, we prove that the closed‐loop switched system is uniformly bounded under arbitrary switching laws, provided that a linear matrix inequality (LMI)‐based sufficient condition is satisfied. We apply the proposed approach to a typical servo‐hydraulic positioning system. The simulation results show that the proposed approach is fairly insensitive to disturbances, uncertainties and non‐smoothly varying dynamics, and performs better than a proportional‐derivative controller or a minimal controller synthesis controller. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Minimal Realizations of Linear Systems: The “Shortest Basis” Approach

Given a discrete-time linear system C, a shortest basis for C is a set of linearly independent generators for C with the least possible lengths. A basis B is a shortest basis if and only if it has the predictable span property (i.e., has the predictable delay and degree properties, and is non-catastrophic), or alternatively if and only if it has the subsystem basis property (for any interval J, the generators in B whose span is in J is a basis for the subsystem CJ). The dimensions of the minimal state spaces and minimal transition spaces of C are simply the numbers of generators in a shortest basis B that are active at any given state or symbol time, respectively. A minimal linear realization for C in controller canonical form follows directly from a shortest basis for C, and a minimal linear realization for C in observer canonical form follows directly from a shortest basis for the orthogonal system C⊥. This approach seems conceptually simpler than that of classical minimal realization theory.

Active radial basis function sliding mode controller for unified chaotic system with disturbance and uncertainties

An adaptive active radial basis function (RBF) sliding mode controller is designed to control a unified chaotic system with parametric uncertainties under external disturbance. The controlled system is divided into a controllable subsystem and a free subsystem. Based on the controllable canonical form of controllable sub-systems state errors at the target points, a sliding surface is defined as the only input to the RBF controller. The weight of the controller is tuned on-line based on the sliding mode reaching law. The simulation results show that this method is applicable and effective, and the robustness to parametric uncertainties and external disturbance is provided. And the chattering of conventional sliding controls doesn’t occur.

Discrete-time minimal control synthesis adaptive algorithm

This article proposes a discrete-time Minimal Control Synthesis (MCS) algorithm for a class of single-input single-output discrete-time systems written in controllable canonical form. As it happens with the continuous-time MCS strategy, the algorithm arises from the family of hyperstability-based discrete-time model reference adaptive controllers introduced in (Landau, Y. (1979), Adaptive Control: The Model Reference Approach, New York: Marcel Dekker, Inc.) and is able to ensure tracking of the states of a given reference model with minimal knowledge about the plant. The control design shows robustness to parameter uncertainties, slow parameter variation and matched disturbances. Furthermore, it is proved that the proposed discrete-time MCS algorithm can be used to control discretised continuous-time plants with the same performance features. Contrary to previous discrete-time implementations of the continuous-time MCS algorithm, here a formal proof of asymptotic stability is given for generic n-dimensional plants in controllable canonical form. The theoretical approach is validated by means of simulation results.

Robust and Retunable State Realizations of Transfer Functions with known Poles and Zeros

The standard controllable and observable canonical realizations have A-matrices whose eigenvalue locations are highly sensitive to small perturbations to the nonzero entries of the A-matrix that correspond to the coefficients of the characteristic polynomial. On the other hand, the eigenvalue locations of tridiagonal A-matrices are relatively insensitive to similar perturbations. Given a set of known poles and zeros from which the transfer function of a SISO is computed, this paper details an approach to state space realizations with tridiagonal A-matrices using a genetic algorithm and the equivalence of the realization with the controllable canonical form. In addition to the insensitivity of the tridiagonal realization to perturbations, an additional genetic algorithm is used to reconfigure input and output matrices when nonzero gains in the B- and C-matrices are perturbed to zero, i.e., disconnections (shorts/opens) which account for over 70% of field failures. The reconfigured matrices achieve an equivalent realization, thereby maintaining the same input–output behavior. Reconfigurable systems are critical to maintaining exact input–output behavior (automatically) without being able to repair the disconnection.

Nonlinear model-based control of pneumatic artificial muscle servo systems

This paper proposes a control methodology of the pneumatic artificial muscle (PAM) actuated servo systems. The four major processes in such systems, including the flow dynamics, pressure dynamics, force dynamics, and load dynamics, are studied to develop a full nonlinear model that encompasses all the major nonlinearities. The developed single-input-single-output model takes the valve command as the input and calculates the third-order derivative of the load position as the output. Since this model is expressed in the control canonical form, nonlinear control approaches can be easily applied. Based on this model, a sliding model controller is developed to obtain robust control in the existence of model uncertainties and disturbances. The controller is implemented on an experimental system, and the effectiveness of the proposed approach demonstrated by step response and sinusoidal tracking at different frequencies.

T-S 퍼지모델을 이용한 이산 시간 비선형계통의 상태 궤환 선형화

In this paper, a novel feedback linearization is proposed for discrete-time nonlinear systems described by discrete-time T-S fuzzy models. The local linear models of a T-S fuzzy model are transformed to a controllable canonical form respectively, and their T-S fuzzy combination results in a feedback linearizable Tagaki-Sugeno fuzzy model. Based on this model, a nonlinear state feedback linearizing input is determined. Nonlinear state transformation is inferred from the linear state transformations for the controllable canonical forms. The proposed method of this paper is more intuitive and easier to understand mathematically compared to the well-known feedback linearization technique which requires a profound mathematical background. The feedback linearizable condition of this paper is also weakened compared to the conventional feedback linearization. This means that larger class of nonlinear systems is linearizable compared to the case of classical linearization.

A MacWilliams Identity for Convolutional Codes: The General Case

A MacWilliams identity for convolutional codes will be established. It makes use of the weight adjacency matrices of the code and its dual, based on state space realizations (the controller canonical form) of the codes in question. The MacWilliams identity applies to various notions of duality appearing in the literature on convolutional coding theory.

A Systematic Method for Gain Selection of Robust PID Control for Nonlinear Plants of Second-Order Controller Canonical Form

A systematic method to select gains of a discrete proportional-integral-derivative (PID) controller is presented. The PID controller with the gains obtained by the proposed method can robustly control nonlinear multiple-input-multiple-output (MIMO) plants in a second-order controller canonical form, such as robot dynamics. This method has been made possible by the finding that the discrete PID control is equivalent to the discrete form of time-delay control (TDC), a robust control method for nonlinear plants with uncertainty. By using this equivalence relationships are obtained between PID gains and parameters of TDC, which enable a systematic method for the select PID gains. In addition, based on the systematic method, a simple and effective method is proposed to tune PID gains applicable to nonlinear plants with inaccurate models. This method incorporates a set of independent tuning parameters that is far less than those for conventional methods for PID gain selection. The usefulness of the proposed methods is verified through the ease and simplicity of determining PID gains for six degrees-of-freedom (DOF) programmable universal machine for assembly (PUMA)-type robot manipulator; the effectiveness of these PID gains is confirmed by the adequate and robust performance through experimentation on the robot.

A new theorem on higher order derivatives of Lyapunov functions

The Lyapunov stability analysis method for nonlinear dynamic systems requires a non positive first derivative of the Lyapunov functions along the system’s trajectories. In this paper, a new method is developed to relax this requirement. A new sufficient condition is developed for the stability analysis of nonlinear systems, introducing some inequalities for higher order derivatives of the Lyapunov function. The differential inequalities can be considered as a new controllable canonical form linear time invariant system with negative inputs. The stability analysis of a given nonlinear system is then reduced to check if the characteristic equation for the new linear time invariant system is Hurwitz. Some examples are presented to establish the approach.

Strong feedback cyclization for systems over rings

For nonnecessarily reachable systems over a commutative ring R, we define a strong form of feedback cyclization ( FC). With this natural generalization of the FC property we obtain a feedback reduced form for systems over strong FC rings (i.e. rings for which every system verifies the FC property). In the particular case of reachable single input systems, this gives the usual control canonical form of Bumby et al. [Remarks on the pole-shifting problem over rings, J. Pure Appl. Algebra 20 (1981) 113–127]. Also it is proved that a commutative ring with 1 in its stable range has the strong FC property if and only if it has the UCU property, which is the natural parallel form of the result given by Brewer et al. [Pole assignability in polynomial rings, power series rings and Prüfer domains, J. Algebra 106 (1987) 265–286] for the reachable case. Many classes of rings which were known to be FC rings are in fact strong FC rings, but there are FC rings which are not strong FC rings.

Global optimization for the -norm model reduction problem

A branch and bound algorithm is proposed to solve the -norm model reduction problem for continuous and discrete-time linear systems, with convergence to the global optimum in a finite time. The lower and upper bounds in the optimization procedure are described by linear matrix inequalities (LMI). Also proposed are two methods with which to reduce the convergence time of the branch and bound algorithm: the first one uses the Hankel singular values as a sufficient condition to stop the algorithm, providing to the method a fast convergence to the global optimum. The second one assumes that the reduced model is in the controllable or observable canonical form. The -norm of the error between the original model and the reduced model is considered. Examples illustrate the application of the proposed method.

A new recurrent neurofuzzy network for identification of dynamic systems

In this paper a new structure of a recurrent neurofuzzy network is proposed. The network is based on two interconnected Fuzzy Inference Systems (FISs), one recurrent and another static, that intend to model the behavior of an unknown dynamic system from input–output data. In the proposed structure each rule involves a linear system in a controllable canonical form in order to reduce the online computational load and facilitate the online checking of the stability of the resulted network. The training for the recurrent FIS is made by a gradient-based Real-Time Recurrent Learning Algorithm (RTRLA), while the training for the static FIS is based on a simple gradient method. The initial parameter conditions prior to training are obtained by extracting information from a static FIS trained with delayed input–output signals. To demonstrate the effectiveness of the proposed structure, two nonlinear systems are identified.

Balanced parameterization of multichannel blind deconvolutive systems: A continuous time realization

A key problem in multichannel blind deconvolution (MDB) is how to properly model the mixer so that the demixer can be properly modelled correspondingly. This problem naturally triggers one question, viz. how to determine some key model orders, e.g. the number of states, if the problem is analyzed using a state space model. In this paper, to answer this question, we will apply a balanced parameterization approach to a MBD problem with the mixer being modelled as a continuous time linear time invariant (LTI) system. Besides allowing the determination of the number of states required in the demixer, compared with the controller canonical form representation or the observer canonical form representation of the LTI continuous time system, our approach has also the advantages of numerical robustness. The proposed method is validated through practical examples using speech signals and electroencephalographic (EEG) data.

자이로의 불규칙 혼합잡음을 고려한 보조항법시스템 칼만 필터 설계

Using the equivalent ARMA model representation of the mixed random errors, we propose Klaman filter design methods for aided INS(Inertial Navigation System) which contains the gyroscope mixed random errors. At first step, considering the characteristic of indirect feedback Kalman filter used in the aided INS, we perform the time difference of equivalent ARMA model. Next, according to the order of the time differenced ARMA model, we achieve the state space conversion of that by two methods. If the order of AR part is greater than MA part, we use controllable or observable canonical form. Otherwise, we establish the state apace equation via the method that several step ahead predicts are included in the state variable, where we can derive high and low order models depending on the variable which is compensated from gyroscope output. At final step, we include the state space equation of gyroscope mixed random errors into aided INS Kalman filter model. Through the simulation, we show that both the high and low order filter models proposed give less navigation errors compared to the conventional filter which assume the mixed random errors as white noise.

和分型／差分型状態方程式表現に基づくロバスト制御系の解析／設計条件の関連性について

This paper considers robust control analysis and synthesis problems via the expression of our previously proposed summational type state equation. The summational type state equation is a mathematical expression to solve two essential problems, i.e., one of them is a physical problem of discontinuity in mathematical expressions e.g. the controllable canonical form for different orders of the existing state equation, and the other one is an engineering problem of disunification in which continuous-time and discrete-time systems are not described with consistency. First, this paper introduces the summational type state equation. Next, a robust stability analysis condition and a feasible condition for the scaled H∞ control synthesis problem are derived from the summational type state equation. Furthermore, the relation between analysis/synthesis conditions based on summational and difference type state equations is clarified. Finally, the effectiveness of the summational type state equation is shown by numerical examples on a sensitivity minimization problem. From these results, this paper points out that the summational type state equation is one of possible important mathematical expressions in control theory.