Uncertain Multivariable Systems Research Articles

Global adaptive stabilization for a class of uncertain multivariable systems with nonlinear parametrization

Global adaptive stabilization for a class of uncertain multivariable systems with nonlinear parametrization

Comparing multivariable uncertain model structures for data‐driven robust control: Visualization and application to a continuously variable transmission

AbstractThe selection of uncertainty structures is an important aspect of system identification for robust control. The aim of this paper is to provide insight into uncertain multivariable systems for robust control. A unified method for visualizing model sets is developed by generating Bode plots of multivariable uncertain systems, both in magnitude and phase. In addition, these model sets are compared from the viewpoint of the control objective, allowing a quantitative analysis as well. An experimental case study on an automotive transmission application demonstrates these connections and confirms the importance of the developed framework for control applications. In addition, the experimental results provide new insights into the shape of associated model sets by using the presented visualization procedure. Both the theoretical and experimental results confirm that a recently developed robust‐control‐relevant uncertainty structure outperforms general dual‐Youla‐Kučera uncertainty, which in turn outperforms traditional uncertainty structures, including additive uncertainty.

IMC‐PID tuning method based on maximum sensitivity for uncertain multivariable systems

SummaryA controller design method based on effective open‐loop transfer function (EOTF) and internal model control (IMC) is proposed for square multivariable systems with uncertain parameters. In this paper, two equivalent substitutions are used to deal with the multivariable system with uncertain parameters. In the first equivalent substitution, the EOTF method is used to solve the coupling problem of the system. In order to prevent the interval explosion problem of parameters, the affine algorithm is adopted in the decoupling process to determine the value region of system parameters. Then, the multivariable system with uncertain parameters is transformed into a set of single‐variable system with uncertain parameters. The second equivalent substitution is based on Kharitonov's theorem to process every single variable system with uncertain parameters and replace it by a set of boundary models. Meanwhile, in order to ensure that the tuning controller can not only make the system stable but also meet a certain degree of robustness, based on the idea of IMC, the maximum sensitivity is introduced to design the PID controller. Finally, numerical simulation results are given to verify the effectiveness of the proposed method.

Optimized reduced order ARX modelling and robust control for multivariable uncertain systems

This paper deals with modelling and robust control of linear multivariable uncertain systems based on the new linear multivariable ARX (Auto-Regressive with eXogenous input)-Laguerre model and the loop-shaping design procedure with the relative gain array theory. In fact, to guarantee significant parameter number reduction of the multivariable ARX-Laguerre model compared with the classic multivariable ARX model, we propose the optimization of the Laguerre poles of the model using the genetic algorithm where the Fourier coefficients are identified using the recursive least square method. Later, the optimized multivariable ARX-Laguerre model is exploited to propose a robust control algorithm for linear multivariable uncertain systems. Indeed, we propose to combine the loop-shaping design procedure approach and the relative gain array theory to develop a simplified final robust controller guaranteeing reference tracking and robust stability against parametric uncertainties. This proposition aims for the simplification of the control’s scheme by adjusting the weighting matrix and the final robust controller calculated from the loop-shaping design procedure for a simpler and more experimentally applicable control scheme. The identification of Fourier coefficients and Laguerre poles as well as the robust multivariable control algorithm are experimentally validated on a laboratory coupled two-tank system showing good results in terms of modelling and attending desired control performances.

Robust nonlinear model predictive sliding mode control algorithm for saturated uncertain multivariable mechanical systems

In this paper, robust stabilization and predictive tracking control for a class of nonlinear uncertain multivariable systems is presented. The control scheme is built by incorporating nonlinear model predictive control in discrete sliding mode control to obtain optimal results while satisfying hard constraints and closed-loop robustness in the presence of external disturbance (matched or unmatched) and parametric uncertainty. Additionally, the control input domain limitation is involved in construction of the model predictive control scheme by employing sliding functions admissible bounds. The problem of cost minimization is solved through optimal selection of terminal cost gain and allocation of system dynamics with respect to sliding surfaces while robust stabilization and feasibility are ensured through the prediction horizon.

Multivariable binary adaptive control using higher-order sliding modes applied to inertially stabilized platforms

This paper presents an extension of the Binary Model Reference Adaptive Control (BMRAC) for uncertain multivariable (square) systems with non-uniform arbitrary relative degrees using only output feedback and its application to inertially stabilized platforms using a two degree of freedom gimbal as actuator. The BMRAC is a robust adaptive strategy with good transient performance, thus useful for uncertain systems, and the multivariable framework is suitable to deal with mechanical unbalances. Using a newly proposed differentiator with dynamic gains based on higher-order sliding mode, the proposed controller achieves global and exact tracking. To illustrate the effective of the proposed solution, simulations are presented using real-word data obtained from an instrumented vehicle in an irregular ground.

Local input‐to‐state stability analysis on compact set for disturbance observer‐based uncertain multivariable nonlinear control systems

AbstractThis article proposes a local input‐to‐state stability (LISS) analysis as a practical stability analysis approach. In existing LISS analysis approaches, a region including a state trajectory is not specified, and the decrement of a Lyapunov function must be guaranteed in the entire domain of the system state. This reduces the advantage of LISS by limiting the state region, and limits the class of systems for which stability can be guaranteed. To expand the class of systems for which stability can be guaranteed, the LISS Lyapunov function on a compact set (LISS LF CS) is proposed to guarantee LISS. By using the LISS LF CS, the region containing the state trajectory was revealed and the LISS condition was relaxed. In addition, the convergence rate and steady state error bound were determined for systems with LISS LF CS. The input‐to‐state stability (ISS) and LISS were analyzed for a multivariable disturbance observer‐based control system using the ISS and LISS LFs, respectively. The advantages of the proposed LISS analysis are demonstrated by comparing the ISS and LISS conditions. The analytical results were verified by conducting numerical simulations.

Output Feedback Tracking Sliding Mode Control for Systems with State- and Input-Dependent Disturbances

In this paper, output feedback tracking sliding mode control was considered for uncertain multivariable linear systems. The uncertainties included external disturbance, the system state, and control input. A new property of the loop transfer recovery (LTR) observer was first established: the state estimation error of the LTR observer can be made arbitrarily small with respect to state- and input-dependent system uncertainties. Observer-based output feedback tracking sliding mode control using the LTR observer is presented. The proposed sliding mode control approach can maintain the boundedness of the system state and drive the system outputs arbitrarily close to the desired reference outputs; the degree of closeness was determined by a design parameter in the LTR observer. In the proposed approach, the most general and simple observer-based output feedback control formulation was used to achieve global tracking. Simulations with a two-degree-of-freedom (DOF) robotic manipulator application illustrated the claimed properties, and a peaking and chattering reduction technique was demonstrated to protect the actuator.

Distributed adaptive stabilization

In this paper we consider distributed adaptive stabilization for uncertain multivariable linear systems with a time-varying diagonal matrix gain. We show that uncertain multivariable linear systems are stabilizable by diagonal matrix high gains if the system matrix is an H-matrix with positive diagonal entries. Based on matrix measure and stability theory for diagonally dominant systems, we consider two classes of uncertain linear systems, and derive a threshold condition to ensure their exponential stability by a monotonically increasing diagonal gain matrix. When each individual gain function in the matrix gain is updated by state-dependent functions using only local state information, the boundedness and convergence of both system states and adaptive matrix gains are guaranteed. We apply the adaptive distributed stabilization approach to adaptive synchronization control for large-scale complex networks consisting of nonlinear node dynamics and time-varying coupling weights. A unified framework for adaptive synchronization is proposed that includes several general design approaches for adaptive coupling weights to guarantee network synchronization.

Recursive filtering adaptive neural fault-tolerant control for uncertain multivariable nonlinear systems

This paper synthesizes a recursive filtering adaptive neural fault-tolerant controller for uncertain multivariable nonlinear systems. The proposed control scheme adopts an estimation/cancellation strategy to deal with uncertainties and/or disturbances. The nonlinear uncertainties are approximated by a Gaussian radial basis function (GRBF)-based neural network incorporated with a piecewise constant adaptive law, where the adaptive law will generate adaptive parameters by solving the error dynamics between the real system and the state predictor with the neglection of unknowns, and recursive least squares (RLS) is applied to distribute the total uncertainty estimates into matched and mismatched components. The cooperation of GRBF learning method and piecewise constant adaptive law relaxes the stringent constraint on the hardware CPU speed and achieves fast adaptation. The filtering control law delivers a satisfactory performance with guaranteed robustness. Two numerical examples are provided to illustrate the effectiveness of the proposed control architecture via comparisons.

Filtered adaptive constrained sampled‐data control for uncertain multivariable nonlinear systems

AbstractA filtered adaptive constrained sampled‐data controller for uncertain multivariable nonlinear systems in the presence of various constraints is synthesized in this paper. A piecewise constant adaptive law drives that estimation error dynamics to zero at each sampling time instant yields adaptive parameters. The filtered control scheme consists of two components. Based on an estimation/cancellation strategy, a disturbance rejection control law is designed to compensate the nonlinear uncertainties within the bandwidth of low‐pass filters, whereas a constraint violation avoidance control law is designed to solve an online constrained optimization problem. Although a reduced sampling time helps to minimize the estimation error caused by the neglect of unknowns, the resulting aggressive signals put more restrictions on the control law. Greater sacrifice of tracking performance is required to satisfy the constraints. The constraints violation avoidance control law is in favor of a larger sampling time. Sufficient conditions are given to guarantee the stability of the closed‐loop system with the sampled‐data controller, where the input/output signals are held constant over the sampling period. Numerical examples are provided to validate the theoretical results, comparisons between the constrained sampled‐data controller and unconstrained adaptive controller with the implementation of different sampling times are carried out.

Filtered adaptive constrained sampled-data output feedback control for uncertain multivariable nonlinear systems

This paper synthesises a constrained sampled-data output feedback controller for uncertain multivariable nonlinear systems in the presence of various constraints, using the filtered adaptive control architecture. Nonlinear uncertainties are estimated online by solving the error dynamics between the output predictor and the real system with the neglection of unknowns, the yielded adaptive parameters are piecewise constant. In the proposed control scheme, the nonlinear uncertainties are compensated for within the bandwidth of low-pass filters, while the trade-off between tracking and constraints violation avoidance is formulated as a numerical constrained optimisation problem which is solved periodically. Priority is given to constraints violation avoidance at the cost of deteriorated tracking performance. System transformations are performed to find the sufficient conditions that can guarantee the stability of the closed-loop system with the sampled-data controller, where both the states and control inputs are held constant over the sampling period.

An improved optimal integral sliding mode control for uncertain robotic manipulators with reduced tracking error, chattering, and energy consumption

This paper proposes an improved optimal integral sliding mode control (IOISMC) for a robust and optimal tracking control of robotic manipulators under the presence of coupling effects, parameter variations, and external disturbances. By representing the system dynamics in the error coordinate and introducing an additional optimal integral term in the nominal control part of IOISMC, the effects of unmeasured forces, reaching phase issue, chattering issue, and position offset associated with conventional optimal integral sliding mode control are significantly addressed. Hence, the proposed approach has several notable merits, namely fast recovery response, low chattering, high tracking performance, high energy efficiency, low computational complexity, and robustness against uncertainties. The idea has been proven in two ways: a strong theoretical framework using Lyapunov stability theory and extensive simulation verification. The superiority of the proposed method has been confirmed through significant reductions of chattering and energy consumption without significant performance loss. Furthermore, the proposed method is capable of sustaining high tracking performance despite the presence of the aforementioned uncertainties and eliminates the reaching phase. It is envisaged that the proposed method can be useful in designing a robust and optimal tracking control for all types of uncertain multivariate systems without compromising on the computational cost.

Graphical PID tuning method for uncertain fractional-order multivariable systems

In this paper, a graphical tuning method for controllers parameters based on the open-loop fractional transfer function (FO-EOTF) method is proposed for fractional-order parameter uncertain multivariable system. The FO-EOTF method is proposed to transform the parameter uncertain fractional-order multivariable system into a set of independent parameter uncertain fractional-order univariate systems and determine the parameters regions of the univariate systems. The gain phase margin tester is used to further guarantee the robust performance of the controlled system. Finally, simulation result from the numerical simulation is presented to demonstrate the effectiveness of this method.

Non-sequential QFT Design Methodology for Disturbance Rejection Problem in Uncertain Multivariable Systems

A new design approach for quantitative feedback synthesis (QFT) to the disturbance rejection problem in an uncertain multi-input multi-output (MIMO) linear time-invariant system is proposed. The proposed method is centered on converting the problem on uncertain system into a nominal system and the uncertainties are transferred as a output disturbances whose range is known. In particular, we derive the controller bounds for each of the equivalent nominal single-input single-output (SISO) problems. The proposed non-sequential approach is attractive as it brings simplicity and less conservative than an existing independent SISO method. The effectiveness of the proposed approach is illustrated via challenging MIMO systems with both parametric and non-parametric uncertainty. The proposed solution is less conservative than the existing design which leads to reduced feedback cost.

Adaptive unit vector control of multivariable systems using monitoring functions

SummaryAn adaptive sliding‐mode unit vector control approach based on monitoring functions to deal with disturbances of unknown bounds is proposed. An uncertain multivariable linear system is considered with a quite general class of nonsmooth disturbances. Global stabilization/tracking is demonstrated using either state or output feedback. The proposed adaptation method makes the control gain less conservative, becoming large enough when the disturbance grows and becoming smaller when it decreases, leading to reduced chattering effects. In contrast to previous methods, the new switching scheme is able to guarantee a prespecified transient time, maximum overshoot, and steady‐state error for multivariable uncertain plants. The proposed technique is applied to the trajectory tracking control of a surface vessel subjected to ocean currents, wind, and waves. Simulations are presented to show the performance of the new adaptation scheme in this adverse scenario of possibly growing, temporarily large, or vanishing exogenous disturbances.

Sliding mode predictive control of linear uncertain systems with delays

In this paper, a continuous control strategy for robust stabilization of a class of uncertain multivariable linear systems with delays in both the state and control variables is proposed. A predictor is designed to compensate the delay effect in the control input, and then an integral sliding mode control technique along with super-twisting algorithm is applied to compensate partially the effect of the perturbation term. Finally, a nominal delay-free part of the control input is designed to stabilize the sliding mode dynamics. The proposed control scheme is extended to the class of systems modeled in Regular form. For this class of perturbed systems with delay in the state, a transformation to the systems with the delay-free state is proposed. The stability conditions of the closed-loop uncertain system are derived, and the results obtained in this work are compared against previous works. To show the effectiveness of the proposed method, simulation results are presented.

Robust Predictor-Based Control of Uncertain Linear Systems with Delay via Integral Sliding Mode Control

In this paper, a continuous control strategy for robust stabilization of a class of uncertain multivariable linear systems with delays in both the state and control variables is proposed. A predictor is designed to compensate the delay effect in control input, and then an integral sliding mode control technique along with super-twisting algorithm is applied to compensate partially the effect of the perturbation term. Finally, a nominal delay-free part of the control input is designed to stabilize the sliding mode dynamics. The method is extended to the class of systems modeled in Regular form, it is proposed a transformation for the systems with delay in state then results in a system delay-free in state. Conditions for the stability of the closed-loop uncertain system are derived. Simulation results are presented and compared against previous works.

Fixed-Order Decoupling of Interval Plant Families

A method for the reduction of interactions in linear time invariant (LTI) multivariable uncertain systems is proposed. An H∞-norm metric is proposed for the assessment of interactions in interval uncertain multiple-input multiple-output (MIMO) plants. Based on this, a procedure for the design of fixed-order dynamic decoupling precompensators for MIMO plants with interval uncertainty is outlined which can be solved using efficient solvers such as cvx. The proposed methodology is used to develop a low-order robust multivariable controller for voltage and frequency control of an islanded distributed generation (DG) unit.

From adaptive control to variable structure systems – seeking harmony

From adaptive control to variable structure systems – seeking harmony