Controllable Canonical Form Research Articles

Immersion and Invariance Adaptive Control through Polynomial Adaptation

Abstract In conventional immersion and invariance (I&I) adaptive control design, control parameter adaptation is typically linear with respect to the parameter-error-induced perturbation, resulting in quadratic-rate dissipation of energy associated with the off-the-manifold variable. Departing from such a convention, this article contributes a novel strategy - polynomial adaptation. As the name suggests, control parameter adaptation in this approach takes the form of a general polynomial in relation to the perturbation. Accordingly, this new design induces polynomial-rate energy dissipation, which is faster than the quadratic one in the conventional scheme, thereby enhancing closed-loop control performance. The theoretical underpinnings of the new approach are demonstrated through the design of an I&I adaptive tracking control law for a general nth-order, single-input-single-output, parametrically uncertain, nonlinear system in the controllable canonical form. Additionally, a numerical study of the proposed method on the second-order forced Duffing oscillator showcases its improved transient performance in comparison to a baseline controller developed with the standard I&I adaptive control technique.

Bond graph realizations of state space linear time-invariant multivariable descriptions

Given a state space description of a linear time-invariant multivariable system, a similarity transformation is applied to get a controllability canonical form. This transformation assures a Bond Graph (BG) realization and avoids the use of active bonds. Active or signal bonds do not preserve the continuity of power flows and consequently, energy conservation properties are lost. The state matrix of the system in transformed coordinates contains an antisymmetric matrix associated with the storage field and the rest is associated with the dissipative field. Thus, in general, the dissipative field is implemented with a multiport element, and in order to give physical meaning, the storage field is implemented with one-port elements. The resulting BG model does not have zero-order causal paths, that is, algebraic loops between the dissipative elements. The results are applied to the BG realization of a linear robust multivariable controller that is designed for a two mass system modelled in BG.

Approximation-Based Approach to Adaptive Control of Linear Time-Varying Systems

An adaptive state-feedback control system is proposed for a class of linear timevarying systems represented in the controller canonical form. The adaptation problem is reduced to the one of Taylor series-based first approximations of the ideal controller parameters. The exponential convergence of identification and tracking errors of such an approximation to an arbitrarily small and adjustable neighbourhood of the equilibrium point is ensured if the condition of the regressor persistent excitation with a sufficiently small time period is satisfied. The obtained theoretical results are validated via numerical experiments.

Prescribed‐time global stabilization for MIMO nonlinear systems with a linear time‐varying control mechanism

AbstractIn this article, the prescribed‐time global stabilization problem is taken into consideration when it comes to multi‐input‐multi‐output (MIMO) systems with unknown mismatched coupling nonlinear items. It is assumed that the unknown nonlinearities satisfy the linear growth conditions. First, a class of MIMO coupling nonlinear systems with controllable canonical form are considered. Then, by wielding a time‐varying Lyapunov‐like function and properties of parametric Lyapunov equations, a high‐gain linear time‐varying state feedback control law is developed to attenuate the impact of unknown nonlinear terms. Meanwhile, all state signals and control inputs are globally bounded and the target system becomes stable within the prescribed‐time. Finally, the simulation results on an example system with 3 inputs and 11 state variables verify that the developed control mechanism accomplishes the goal of global stabilization in the pre‐defined time.

On the explicit Hermitian solutions of the continuous‐time algebraic Riccati matrix equation for controllable systems

AbstractThis paper proposes explicit solutions for the algebraic Riccati matrix equation. For single‐input systems in controllable canonical form, the explicit Hermitian solutions of the non‐homogeneous Riccati equation are obtained using the entries of the system matrix, the closed‐loop system matrix, and the weighting matrix. The unknown entries of the closed‐loop system matrix are solved by scalar quadratic equations. For a homogeneous Riccati equation with a zero weighting matrix, the explicit solutions are proposed analytically in terms of the system eigenvalues. The advantages of the explicit solutions are threefold: first, if the system is controllable, the solution is directly given and the invariant subspaces of the Hamiltonian matrix are not required; second, if the system is near singularity, the explicit solution has higher numerical precision compared with the solution computed by numerical algorithms; third, for a real system in the controllable canonical form, the non‐negativity can be analysed for the explicit almost stabilizing solution.

STRUCTURE-PRESERVING MODEL ORDER REDUCTION OF RANDOM PARAMETRIC LINEAR SYSTEMS VIA REGRESSION

We investigate model order reduction (MOR) of random parametric linear systems via the regression method. By sampling the random parameters contained in the coefficient matrices of the systems, the iterative rational Krylov algorithm (IRKA) is used to generate sample reduced models corresponding to the sample data.We assemble the resulting reduced models by interpolating the coefficient matrices of reduced sample models with the regression technique, where the generalized polynomial chaos (gPC) is adopted to characterize the random dependence coming from the original systems. Noting the invariance of the transfer function with respect to restricted equivalence transformations, the regression method is conducted based on the controllable canonical form of reduced sample models in such a way to improve the accuracy of reduced models greatly.We also provide a posteriori error bound for the projection reduction method in the stochastic setting. We showcase the efficiency of the proposed approach by two large-scale systems along with random parameters: a synthetic model and a mass-spring-damper system.

An Investigation of Stability, Controllability and Observability of a Three Degree of Freedom Translational Mechanical System using State Space Approach

State-space modelling approach which is essentially time-domain developed in the late 1960s is a new approach to the analysis and design of complex control systems. Several researches have carried out modelling and analysis of mechanical systems with a number of degrees of freedom of movement using state space approach among which is the work of Sivak and Darina (2012) who modelled a system with two degrees of freedom. In this paper, therefore, we provide an extension of the work of Sivak and Darina to model and analyse a three degree of freedom translational mechanical system using state-space approach. The system was first presented in equivalent free body diagrams, then Newton's second law of motion was used to derive its equations of motion. The State-space formulation in the controllable canonical form obtained from the time-domain differential equations is adopted and the Laplace transform method was used in the analysis to determine the poles (natural frequencies) of the system using two numerical examples in MATLAB software. The software was also used to determine the controllability and observability matrices of the system. The stability, controllability and observability of the system were then discussed from the poles, controllability matrix and observability matrix respectively. Tables of numerical values and line graphs were used to present the results obtained from the analysis. Finally, in these results, the system was found to be stable, controllable and observable suggesting that a state feedback ( pole placement ) control design for the system is possible.

Control of distributed-parameter systems using normal forms: an introduction

Abstract This paper gives an overview of the control of distributed-parameter systems using normal forms. Considering linear controllable PDE-ODE systems of hyperbolic type, two methods derive tracking controllers by mapping the system into a form that is advantageous for the control design, analogous to the finite-dimensional case. A flatness-based controller makes use of the hyperbolic controller canonical form that follows from a parametrization of the system’s solutions. A backstepping design exploits the strict-feedback form of the system to recursively stabilize and transform the subsystems.

A Modular Modeling Method of Mistuned Blisk Based on State-Space Theory with Fitting Aerodynamic Excitation

In order to improve the solution efficiency of vibration characteristics for the mistuned blisk, a modular method combined with state-space method based on lumped-parametric model is developed for the first attempt, which provides a general modeling framework which can be employed to conveniently construct the lumped-parametric models of bladed disk systems with multiple mistuning and variable blade number. It is noteworthy that a pressure fit in the axial-span-pressure space is performed to obtain the continuous equations of aerodynamic forces according to the aerodynamic loads acting on the blade surface. The function of traveling wake excitation is established by considering the rotational velocity. Subsequently, 1N-degree-of-freedom (1N-DOF) and 2N-DOF lumped-parametirc models are respectively used to construct the mistuned blisk. A modular solution method of multiple mistuned lumped-parametirc model is derived from the controllable canonical form and minimum realisation, which reveals that the minimum dimension of the state space is two times of its DOF. The correctness of the present method is varified by several comparisions. Vibration localization is found in the transient response and steady-state response of the blisk. The effects of the mistuning on the blades and the disk are discussed, respectively.

Innovative non-asymptotic and robust estimation method using auxiliary modulating dynamical systems

In this paper, we aim to introduce an innovative non-asymptotic and robust estimation method based on the observable canonical form by means of a set of auxiliary modulating dynamical systems. The latter auxiliary systems are given by the controllable canonical form with zero initial condition. The proposed method can be applied to many kinds of linear and nonlinear systems. In this paper, it is applied to estimate the states and the output’s derivatives for linear singular systems with multiple inputs and multiple outputs in noisy environment. First, the considered singular system is transformed into a form similar to the Brunovsky’s observable canonical form with the injection of the inputs’ and outputs’ derivatives. Second, algebraic integral formulas are obtained both for the state variables and the outputs’ derivatives. After giving solutions of the required auxiliary systems, error analysis in discrete noisy case is addressed, where a provided noise error bound can be used to select design parameters. In the end, numerical simulations are given to illustrate the performance of the proposed method.

Certainty equivalence-based robust sliding mode control strategy and its application to uncertain PMSG-WECS.

This work focuses on maximum power extraction via certainty equivalence-based robust sliding mode control protocols for an uncertain Permanent Magnet Synchronous Generator-based Wind Energy Conversion System (PMSG-WECS). The considered system is subjected to both structured and unstructured disturbances, which may occur through the input channel. Initially, the PMSG-WECS system is transformed into a Bronwsky form, i.e., controllable canonical form, which is composed of both internal and visible dynamics. The internal dynamics are proved stable, i.e., the system is in the minimum phase. However, the control of visible dynamics, to track the desired trajectory, is the main concern. To carry out this task, the certainty equivalence-based control strategies, i.e., conventional sliding mode control, terminal sliding mode control and integral sliding mode control are designed. Consequently, a chattering phenomenon is suppressed by the employment of equivalent estimated disturbances, which also enhance the robustness of the proposed control strategies. Eventually, a comprehensive stability analysis of the proposed control techniques is presented. All the theoretical claims are verified via computer simulations, which are performed in MATLAB/Simulink.

Flatness Analysis for the Sampled-data Model of a Single Mast Stacker Crane

We show that the Euler-discretization of the nonlinear continuous-time model of a single mast stacker crane is flat. The construction of the flat output is based on a transformation of a subsystem into the linear time-variant discrete-time controller canonical form. Based on the derived flat output, which is also a function of backward-shifts of the system variables, we discuss the planning of trajectories to achieve a transition between two rest positions and compute the corresponding discrete-time feedforward control.

Optimal sliding mode preview repetitive control for continuous-time nonlinear systems

In this paper, the problem of sliding mode preview repetitive control (SMPRC) for continuous-time nonlinear systems is proposed. First, an augmented delay system is constructed, in which the output of the modified repetitive controller is taken as a part of the augmented state vector. The augmented delay system is then transformed into the controllable canonical form by using state and linear transformations. To introduce preview compensation, a virtual optimal preview control (PC) law is designed for the uncontrollable subsystem. Next, a sliding mode control (SMC) law is cooperated with the virtual optimal PC law to ensure the robustness of the overall system in the presence of uncertainties and external disturbances. Stability of the proposed controller is also proved. Finally, the numerical simulation results demonstrate the effectiveness of the proposed method. Abbreviations: LQR: linear quadratic regulator; MRC: modified repetitive control; PC:preview control; PRC: preview repetitive control; RC: repetitive control; SFC: statefeedback control; SMC: sliding mode control; SMPRC: sliding mode preview repetitivecontrol.

Variable Structure Control of Second Order Systems With Nonlinear Uncertain Discontinuous Input Functions Using Approximate Inverses

We consider second order nonlinear systems in control canonical form, where the functional relation between the control variable and the system acceleration is nonlinear, uncertain and discontinuous. We approximate the nonlinear uncertain and discontinuous function with an invertible one, and use variable structure control theory to suppress the error of the approximate inverse. We rewrite the finite time reaching law or the terminal reaching law into the form of inequalities and use these inequalities to design a control law that drives the system state to the sliding surface in finite time, and one that ensures the convergence of the system speed and position errors to a narrow, adjustable width band around zero. The stability of the control system is also analyzed and proven. We demonstrate how to ensure the sufficiency of the available control effort, by the proper design of control parameters. The method is then validated on a numerical model of a nonlinear system with nonlinear, uncertain and discontinuous input function.

PID with a Switching Action Controller for Nonlinear Systems of Second-order Controller Canonical Form

The desired performance of nonlinear systems by using only proportional-integral-derivative (PID) controllers is difficult to obtain because of its simple structure. To improve the control performance of nonlinear systems, we proposed a PID with a switching action (PIDSA) controller that only adds the switching action of a sliding mode control (SMC) while maintaining the structure of the PID. The PIDSA controller has the simplicity of PID and the robustness of SMC. In addition, a double integral sliding surface (DISS) was designed to match the characteristics of the PID with the switching action. The stability of the overall system was proved for nonlinear systems by performing the Lyapunov stability analysis. Finally, experiments were conducted on a hydraulic manipulator, which is a highly nonlinear system. The results show that the proposed controller exhibits excellent performance and that DISS enhances the performance compared with using an integral sliding surface.

Jerk forms dynamics of a Chua’s family and their new unified circuit implementation

A scheme to implement the Jerk form of the Chua system family using a controllable canonical form applied in linear systems is proposed. The main thought is that the nonlinear function with a single independent variable input can be superposed by a multiple linear function, which is regarded as the input of the system, and its single variable is regarded as the output of the system. Its state space could be reconstructed into the controllable canonical form. The output after transformation is fed back to the nonlinear function as its input independent variable, thus the controllable canonical form is transformed into the Jerk form of the Chua’s chaotic system. All Jerk forms of three-order Chua system, and part Jerk forms of four-order Chua system are presented. Analysis of the Lyapunov exponent spectrum, eigenvalues of the original three-order, four-order Chua systems and their Jerk forms, and the same values demonstrate the equivalent of the systems for both structures. According to the complexity and National Institute of Standards and Technology (NIST) test results, the pseudo-random performance of the chaotic sequence brought by the Jerk forms of the Chua system is better. The simplest unified circuit block diagram for the Jerk forms of the Chua’s family is also given. By changing their resistance parameters, the bifurcation diagrams show that Jerk forms’ systems are entering chaos, these circuits implement results of 2–6 scroll attractors. Experimental observations are provided for confirmation.

Automated synthesis of a local model network based nonlinear model predictive controller applied to the engine air path

The efficient and emission reducing control of the air path of an internal combustion engine is a challenging task due to its nonlinear and multivariate nature. By applying the well-known local model network approach to describe the nonlinear process in terms of linear operating point approximations, a fast and efficient model generation through data-driven system identification can be achieved. In this paper it is demonstrated how a nonlinear multivariate model predictive controller can be synthesised from the identified model directly by exploiting its representation in flatness coordinates. For the proposed controller, a compactly formulated quadratic programme results. Because of the uniform representation of all local models in controllability canonical form, a state observer is rendered unnecessary. Additionally, input and output constraints can be taken into account in the optimisation directly. The effectiveness of the control scheme is demonstrated successfully in jointly controlling the exhaust manifold pressure and the engine out NOx concentration for a heavy-duty engine on the testbed.

Inverse and Direct Solutions of the Discrete Time Lyapunov Equation With System Matrix in Companion Form

The discrete time Lyapunov equation is used in many applications and there is interest in its inverse and direct solutions. New methods are proposed to obtain solutions for cases where the system matrix is in controllable canonical form. The approach is based on the relationship between the discrete Lyapunov equation and the entries of one of the stability tables presented by Jury. It is shown that the inverse solution, which is based on this stability table, can be obtained using LDLt decomposition. Also the direct solution of the discrete Lyapunov equation can be obtained directly from the entries of this stability table. The proposed algorithms are illustrated by numerical examples.

Pinning Control of Higher Order Nonlinear Network Systems

In this letter, we study the problem of controlling via pinning the motion of nonlinear network systems of any order whose dynamics are in controllable canonical form. Different from existing works that either focus on spontaneous synchronization, assume linear dynamics or rely on dynamics cancellation, here we provide a constructive method to prove pinning controllability towards the desired trajectory selected by the pinner. We introduce an algorithmic procedure that associates to any connected topology a suitable Lyapunov function for the network system. The approach is demonstrated on an illustrative example.

Robust Intelligent Control of SISO Nonlinear Systems Using Switching Mechanism.

In this article, a robust adaptive learning control strategy for uncertain single-input-single-output systems in strict-feedback form and controllability canonical form (CCF) is studied. For the strict-feedback system, the dynamic surface control is introduced while for the controllability canonical system, sliding-mode control is further constructed. The finite-time design is introduced for fast convergence. Under the switching mechanism, the intelligent design and the robust technique work together to obtain robust tracking performance. Once the states run out of the domain of intelligent control, the robust item will pull the states back while inside the neural working domain, the composite learning is developed to achieve higher approximation precision by building the prediction error for the weight update. The closed-loop system stability is analyzed via the Lyapunov approach. Especially for the CCF, the finite-time convergence is achieved while the system signals are globally uniformly ultimately bounded. Simulation studies on the general nonlinear systems and the flight dynamics show that the new design scheme obtains better tracking performance with higher precision and stronger robustness.