Parameter-dependent State Feedback Controllers Research Articles

Stabilization of polytopic discrete-time varying systems with rate and magnitude saturating actuators and bounded disturbances

This study tackles the critical challenge of designing parameter-dependent controllers for the wide class of discrete-time polytopic systems under rate and magnitude saturating actuators. Such a class finds widespread use in diverse applications spanning from medicine to industrial engineering. Our central contribution lies in developing novel convex parameter-dependent state feedback controller synthesis conditions that ensure regional and input-to-state stability, effectively mitigating the adverse effects of rate and magnitude-saturating actuators. Our approach considers the ℓ2/ℓ∞ gain between disturbance input and controlled output, offering performance specifications and retaining validity for specific initial conditions and amplitude-limited input disturbances. Moreover, our methodology is highly adaptable and suitable for a broad spectrum of systems, including LPV/quasi-LPV and T–S fuzzy systems. We leverage the position-type feedback model with speed limitation (PMSL) and the generalized sector condition to derive our controller synthesis method, resulting in a parallel-distributed-compensation strategy under standard assumptions, ensuring practicality and applicability to diverse system requirements. To highlight the effectiveness of our approach, we present numerical examples for comparative evaluation concerning the existing literature. Furthermore, we validate our methodology through real-time experiments conducted on a nonlinear coupled tank system, providing concrete evidence of its efficacy and feasibility for real-world implementation.

Fixed-structure parameter-dependent state feedback controller: A scaled autonomous vehicle path-tracking application

This paper introduces theoretical conditions for the computation of a novel type of state-feedback controller that makes use of Linear Parameter Varying approaches for synthesis. The novelty of this new State-Feedback controller lies on the fact that the controller has a fixed structure with constants matrix gains. However, the controller gains are affine on a parameter dependent basis function, which allows the controller to self-schedule according to real-time changes of the varying parameters. This type of controller is conceived with a focus on implementation, as in contrast with most other LPV approaches. The implementation of this Parameter-Dependent State-Feedback controller does not require any online interpolation or matrix inverse operations, independently of the number of varying parameters. The performance of the proposed approach is validated in a Scaled Autonomous Vehicle for steering control in a path tracking application.

A new design method to global asymptotic stabilization of strict-feedforward nonlinear systems with state and input delays

<abstract><p>This paper studies the global asymptotic stabilization problem of strict-feedforward nonlinear systems with state and input delays. We will first transform the considered system into an equivalent system by constructing the novel parameter-dependent state feedback controller and introducing the appropriate coordinate transformation. After that, the global asymptotic stability of the closed system is proved by giving the proper Lyapunov-Krasovskii functional and using the stability criterion of time-delay system.</p></abstract>

A new design method to global asymptotic stabilization of strict-feedforward stochastic nonlinear time delay systems

The stabilization problems of stochastic nonlinear systems remain challenging because many new obstacles will be encountered and need to be cautiously handled. This paper discusses the global asymptotic stabilization problem of stochastic nonlinear systems which possess strict-feedforward structure and multiple time delays. In the first place, we are trying to transform the considered system into an equivalent one by constructing the novel parameter-dependent state feedback controller and the suitable coordinate transformation. And then, through using the stochastic time delay system stability criterion and introducing the proper Lyapunov–Krasovskii functional, the global asymptotic stability of stochastic closed-loop system is proved. Finally, simulation results show the effectiveness of the parameter-dependent state feedback controller.

Memoryless parameter-dependent control strategy of stochastic strict-feedback time delay systems

For a class of stochastic strict-feedback nonlinear systems subject to different time delay states, this paper mainly concerns the problem of global asymptotic stabilization. Two new control strategies that the memoryless parameter-dependent state feedback control and the memoryless parameter-dependent output feedback control are taken into consideration, respectively. By skillfully constructing the Lyapunov-Krasovskii (L-K) functional, taking the proper determined parameter and employing the stochastic nonlinear time delay system (SNTDS) stability theory, the global asymptotic stability of the stochastic closed-loop system can be achieved. The proposed output feedback control scheme is finally utilized for the control design of the one-link manipulator system and two-stage chemical reactor system, which can verify the availability of the control approach.

A Direct Parameter-Error Codesign Approach of Discrete-Time Saturated LPV Systems

Considering a discrete-time linear parameter-varying systems with input constraints, this article deals with the codesign of a parameter-dependent state-feedback controller and a new parameter-dependent event-triggering mechanism. Two independent event-triggering policies are introduced at the sensor node to economize the limited network resources. They indicate whether the current state or the current scheduling parameters should be transmitted to the controller or not. In this sense, the controller scheduling parameters can differ from those of the system, yielding a certain degree of robustness concerning parameter deviations. Sufficient conditions, given in terms of linear matrix inequalities, ensure the regional asymptotic stability of the closed loop. To formulated them, we use the Lyapunov theory along with the generalized sector condition. Moreover, we formulate a convex optimization procedure indirectly reducing the transmission activity over the network. The proposed methodology effectiveness is attested through numerical examples and comparisons with related works from the literature.

Parameter-dependent controller design of stochastic strict feedforward nonlinear systems

In this paper, the exponentially stable problem is discussed for a class of stochastic strict feedforward nonlinear systems. Firstly, by employing the proper coordinate transformation and the novel parameter-dependent controller, the initial stochastic strict feedforward nonlinear system is converted to an equivalent system. After that, the implementable parameter-dependent controller, which is adopted to handle the nonlinearities, can be achieved by reasonably selecting the designed parameter. Finally, by means of stochastic Lyapunov stability theory, it is rigorously verified that the proposed parameter-dependent state-feedback controller and parameter-dependent output-feedback controller can guarantee that the stochastic closed-loop system is the fourth moment exponentially stable. Simulation results demonstrate the efficiency of the proposed parameter-dependent controller.

Gain-scheduled ℋ∞ controller synthesis for LPV systems subject to multiplicative noise

This paper proposes LMI conditions to design parameter-dependent (i.e. gain-scheduled) state-feedback controllers that ensure closed-loop stability with guaranteed ℋ∞ performance for both continuous and discrete-time LPV systems with state multiplicative noise. The state-space matrices and the multiplicative noise matrix are considered polytopic and independent. The time-varying parameters can be considered time-invariant, arbitrarily fast or with bounded rates of variation. The advantages of the proposed technique are illustrated by numerical examples borrowed from the literature.

Static output-feedback robust gain-scheduling control with guaranteed [formula omitted] performance

This paper develops synthesis conditions for Static Output-Feedback (SOF) Gain-Scheduling (GS) control with guaranteed upper bound of H2 performance for continuous-time Linear Parameter Varying (LPV) systems, where measurements of scheduling parameters are affected by uncertainties or measurement noise. The control problem is solved through an iterative two-stage design procedure. In the first stage, parameter-dependent state-feedback controller is obtained to minimize upper-bound of the H2 performance. Then, this controller is used as input to the second stage to synthesize Robust Gain-Scheduling (RGS) static output-feedback gain with minimal H2 performance. In both stages, the synthesis conditions are given in terms of Parametrized Linear Matrix Inequalities (PLMIs). Robust SOF (parameter-independent) controller can be synthesized as special case of the developed synthesis conditions. Two examples have been presented to illustrate the benefit of the proposed approach. One is an academic numerical example from literature and the other one is a realistic LPV model of Electric Variable Valve Timing (EVVT) actuator for automotive engines that utilizes engine speed and vehicle battery voltage as time-varying scheduling parameters.

State Feedback Controller Design for Affine Parameter-Dependent LPV Systems

Affine quadratic stability (AQS) is usually devoted to stability analysis of the linear parameter-varying (LPV) system. In this paper, the main contribution is to put forward state feedback control methods for the affine parameter-dependent LPV system based on AQS, guaranteed cost control and LPV linear quadratic regulator (LQR) included for good control performance, and test their conservatism. The parameter-dependent Lyapunov function from AQS could be transformed to linear matrix inequality (LMI) problems for tractability, and the convexity would ensure avoidance of solving infinite LMIs. In this way, parameter-dependent state feedback controllers could be gotten, and comparison between them is applied to analyze the admitted conservatism numerically by simulation results.

Improved Results on Asymptotic Stabilization for Stochastic Nonlinear Time-Delay Systems With Application to a Chemical Reactor System

The global asymptotic stabilization problem is investigated for a class of stochastic nonlinear time-varying delay systems under the weaker condition on nonlinear functions. The new parameter-dependent state and output feedback controllers are, respectively, proposed. Based on the stochastic time-delay system stability criterion, by tactfully introducing a suitable Lyapunov-Krasovskii functional, the globally asymptotically stable in probability of the closed-loop system is guaranteed by rigorous proof. As a practical application, the stochastic model of a two-stage chemical reactor system is established by reasonably introducing the Gaussian white noise. The developed approach is applied to the control design for this practical system. The simulation results demonstrate the efficiency of the proposed design approach.

An Offline Formulation of MPC for LPV Systems Using Linear Matrix Inequalities

An offline model predictive control (MPC) algorithm for linear parameter varying (LPV) systems is presented. The main contribution is to develop an offline MPC algorithm for LPV systems that can deal with both time-varying scheduling parameter and persistent disturbance. The norm-bounding technique is used to derive an offline MPC algorithm based on the parameter-dependent state feedback control law and the parameter-dependent Lyapunov functions. The online computational time is reduced by solving offline the linear matrix inequality (LMI) optimization problems to find the sequences of explicit state feedback control laws. At each sampling instant, a parameter-dependent state feedback control law is computed by linear interpolation between the precomputed state feedback control laws. The algorithm is illustrated with two examples. The results show that robust stability can be ensured in the presence of both time-varying scheduling parameter and persistent disturbance.

T–S model-based nonlinear moving-horizon [formula omitted] control and applications

Using a T–S fuzzy model, we propose a formulation for a moving horizon H∞ control problem for nonlinear systems. Our main contribution is combination of T–S fuzzy models and a moving horizon H∞ control scheme by Chen and Scherer to address the disturbance attenuation of nonlinear constrained system. A sufficient condition for the disturbance amplitude is given to guarantee the feasibility of the optimization problem at each time. A parameter-dependent state feedback control law is adopted, and the corresponding optimization problem is reduced to a convex optimization problem involving linear matrix inequalities (LMIs). The H∞ attenuation index for the nonlinear moving horizon H∞ control problem is adapted online to satisfy time-domain constraints. Finally, the effectiveness of the proposed scheme is successfully demonstrated by control of a continuous stirred tank reactor and active queue management for routers.

LMI Relaxations for Reduced-Order Robust ${\cal H}_{\infty}$ Control of Continuous-Time Uncertain Linear Systems

This technical note is concerned with the problem of reduced order robust <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> dynamic output feedback control design for uncertain continuous-time linear systems. The uncertain time-invariant parameters belong to a polytopic domain and affect all the system matrices. The search for a reduced-order controller is converted in a problem of static output feedback control design for an augmented system. To solve the problem, a two-stage linear matrix inequality (LMI) procedure is proposed. At the first step, a stabilizing state feedback scheduled controller with polynomial or rational dependence on the parameters is determined. This parameter-dependent state feedback controller is used at the second stage, which synthesizes the robust (parameter-independent) output feedback <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> dynamic controller. A homogeneous polynomially parameter-dependent Lyapunov function of arbitrary degree is used to assess closed-loop stability with a prescribed <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">H</i> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> attenuation level. As illustrated by numerical examples, the proposed method provides better results than other LMI based conditions from the literature.

Robust ℋ︁2 static output feedback design starting from a parameter-dependent state feedback controller for time-invariant discrete-time polytopic systems

This paper investigates the problem of computing robust ℋ︁2 static output feedback controllers for discrete-time uncertain linear systems with time-invariant parameters lying in polytopic domains. A two stages design procedure based on linear matrix inequalities is proposed as the main contribution. First, a parameter-dependent state feedback controller is synthesized and the resulting gains are used as an input condition for the second stage, which designs the desired robust static output feedback controller with an ℋ︁2 guaranteed cost. The conditions are based on parameter-dependent Lyapunov functions and, differently from most of existing approaches, can also cope with uncertainties in the output control matrix. Numerical examples, including a mass–spring system, illustrate the advantages of the proposed procedure when compared with other methods available in the literature. Copyright © 2009 John Wiley & Sons, Ltd.

Parameter dependent state-feedback control of LPV time delay systems with time varying delays using a projection approach

This paper is concerned with the stabilization of LPV time delay systems with time varying delays by parameter dependent state-feedback. First a stability test with H∞ performance is given through a parameter dependent LMI. This stability test is derived from a parameter dependent Lyapunov-Krasovskii functional combined with the Jensen's inequality. From this result we derive a state-feedback existence lemma expressed through a nonlinear matrix inequality (NMI). Using a result of the paper we are able to turn this (NMI) into a bilinear matrix inequality (BMI) involving a ‘slack’ variable. This BMI formulation is shown to be more flexible than the initial NMI formulation and is more adequate to be solved using algorithm such as ‘D-K iteration'. The controller construction is provided by two different ways. We finally discuss on the relaxation method and we show the efficiency of our method through several examples.

Stabilization of 2-D Linear Parameter-Varying Systems using Parameter-Dependent Lyapunov Function: An LMI Approach

This paper proposes a parameter-dependent state-feedback controller for the 2-D discrete linear parameter-varying (LPV) system with the Fornasini-Machesini (FM) first model. To find the stabilizing conditions of the system, we first transform the closed-loop system to a Roesser-type model, and then derive the conditions to linear matrix inequalities (LMIs) using a parameter-dependent Lyapunov function (PDLF) and a relaxation technique. The simulation results show that the designed controller is valid and the system asymptotically converges to the origin.

Linear matrix inequality characterisation for ℋ∞ and ℋ2 guaranteed cost gain-scheduling quadratic stabilisation of linear time-varying polytopic systems

Necessary and sufficient linear matrix inequality (LMI) conditions are provided to compute parameter-dependent state feedback control gains that ensure closed-loop quadratic stability for linear systems affected by arbitrarily fast time-varying parameters inside a polytope. The proposed conditions, based on an extension of Pólya&apos;s theorem and on the systematic construction of homogeneous polynomial solutions for parameter-dependent LMIs, are written as a sequence of progressively less and less conservative LMI conditions. Necessity is attained as the level of relaxation increases, providing a parameter-dependent state feedback gain that quadratically stabilises the system whenever such a gain exists. Moreover, parameter-dependent gains of arbitrary degree assuring quadratic stability with ℋ∞ and ℋ2 guaranteed costs are also provided. The convergence to the minimum values of the attainable ℋ∞ and ℋ2 guaranteed costs under closed-loop quadratic stability occurs as the degree of the polynomially parameter-dependent gain increases. Numerical results illustrate the efficiency of the proposed conditions.

Exponential stability and stabilization of uncertain linear time-varying systems using parameter dependent Lyapunov function

In this paper, the problem of exponential stability and stabilization for a class of uncertain linear time-varying systems is considered. The system matrix belongs to a polytope and the time-varying parameter as well as its time derivative are bounded. Based on a time-varying version of Lyapunov stability theorem, new sufficient conditions for the exponential stability and stabilization via parameter dependent state feedback controllers (i.e., a gain scheduling controllers) are given. Using parameter dependent Lyapunov function, the conditions are formulated in terms of two linear matrix inequalities without introducing extra useless decision variables and hence are simply verified. The results are illustrated by numerical examples.

LMI RELAXATIONS FOR H∞ CONTROL OF TIME-VARYING POLYTOPIC SYSTEMS BY MEANS OF PARAMETER-DEPENDENT QUADRATICALLY STABILIZING GAINS

This paper provides necessary and sufficient conditions in terms of LMI relaxations to compute H∞ parameter-dependent state feedback control gains that ensure closed-loop quadratic stability for linear systems affected by arbitrarily fast time-varying parameters inside a polytope. The proposed conditions, based on an extension of Pólya's Theorem and on the systematic construction of homogeneous polynomial solutions for parameter-dependent linear matrix inequalities, are written as a sequence of progressively less and less conservative linear matrix inequality conditions. Necessity is attained as the level of relaxation increases, providing an H∞ parameter-dependent state feedback gain that quadratically stabilizes the system whenever such a gain exists. An extension to compute decentralized control gains is also presented and numerical results illustrate the efficiency of the proposed conditions.