Robust static output feedback infinite horizon RMPC for linear uncertain systems

This paper deals with the problem of static output feedback (SOF) control design for a class of diffusion partial differential equation (PDE) and ordinary differential equation (ODE) cascades, where the ODE model is used to describe the dynamics of the multi-input and multi-output (MIMO) plant and the diffusion PDE model is employed to represent the dynamics of actuators. The objective of this paper is to develop a simple as well as effective SOF controller via the Lyapunov's direct method such that the resulting closed-loop system is globally exponentially stable. By constructing a quadratic Lyapunov function, the sufficient condition on the globally exponential stability of the closed-loop cascaded system is presented in terms of linear matrix inequality (LMI). Then, an LMI-based design method of the SOF controller is developed on the basis of the obtained stability analysis result. Finally, two numerical examples are provided to illustrate the effectiveness of the proposed design method.

This paper will research the problem of static output feedback (SOF) stabilization of state-constrained switched linear systems via an improved average dwell time method (ADT). Firstly, an improved ADT method is adopted to establish sufficient conditions for SOF of the state-constrained switched linear systems in the form of matrix inequality. It has been shown that this method is less conservative than traditional ADT, which in view of different decay rates of a Lyapunov function related to an active subsystem on the basis of whether the saturations occur or not. Then, a new iterative algorithm is designed to solve the matrix inequality and a SOF controller can be added. In the iterative linear matrix inequality (ILMI) algorithm, it is important not only to overcome the typical bilinear matrix inequality (BMI) problem of SOF, but also to solve the non-convex problem caused by state constraints. Finally, the availability and the applicability of the proposed method is shown by the application of a boost converter.

ABSTRACTIn this work, synthesis of robust distributed model predictive control (MPC) is presented for a class of linear systems subject to structured time-varying uncertainties. By decomposing a global system into smaller dimensional subsystems, a set of distributed MPC controllers, instead of a centralised controller, are designed. To ensure the robust stability of the closed-loop system with respect to model uncertainties, distributed state feedback laws are obtained by solving a min–max optimisation problem. The design of robust distributed MPC is then transformed into solving a minimisation optimisation problem with linear matrix inequality constraints. An iterative online algorithm with adjustable maximum iteration is proposed to coordinate the distributed controllers to achieve a global performance. The simulation results show the effectiveness of the proposed robust distributed MPC algorithm.

this paper investigates the problem of robust gain-scheduled static output feedback (SOF) controller for a class of uncertain linear parameter varying systems (LPVSs). An uncertain LPVS is described by an interconnection of a nominal LPVS which is solely dependent on the measurable parameters, and a block structured uncertainty. For designing a H∞ gain-scheduled SOF controller as well as performing the stability analysis, the proposed method uses integral quadratic constraints (IQCs) to derive the linear matrix inequalities (LMIs) without any constraints on nominal system matrices. The performance and effectiveness of the proposed method are demonstrated based on an example.

This paper considers the problem of static output feedback (SOF) control for discrete-time switched linear systems under arbitrary switching. By the aid of switched quadratic Lyapunov functions combined with Finsler’s lemma, new sufficient conditions for SOF stabilization are obtained which guarantee a γ-performance of the closed-loop switched systems subject to input disturbances. The proposed method can work successfully in situations where the existing ones fail. Three examples are given to illustrate its effectiveness.

This paper studies the robust static output feedback (SOF) problem considering pole placement constraints for linear systems with polytopic uncertainty as well as linear parameter varying (LPV) systems. New linear matrix inequality (LMI) approaches are proposed for the SOF controller design while the pole placement, H2, and H∞ constraints are guaranteed. In addition, the gain-scheduled SOF controller will be designed for LPV systems if system parameters are measured. The proposed methods can be applied to general linear systems without imposing any constraints on system matrices. The performance and effectiveness of the proposed methods are shown using two examples.

Constant-air-volume (CAV) air-conditioning systems consist mainly of two local processes: an air-handling process and a room temperature process. A robust model predictive control (RMPC) strategy was developed for CAV air-conditioning systems, which adopted two uncertain first-order plus time-delay models to describe the dynamics of the local processes and used a linear matrix inequality (LMI)-based optimisation technique to optimise the control law. This paper develops a new control design, which reformulates the prediction models by shifting the uncertainties of the first model into the second one, and then uses the reformulated prediction models in the RMPC strategy. This paper will show that compared with the original design, the new control design can enhance the feasibility of the optimisation of control law, reduce the computational burden of the optimisation and also remove the requirement of a sensor for supply air temperature in the original design. Practical applications: The new design method is a further development of a RMPC strategy presented in Xu et al.13 It inherits the benefits of the original control design for practical application, i.e. uncertainties and constraints can be dealt with simultaneously in the design and the robustness of the controlled system can be enhanced. The new design improves the optimisation feasibility, reduces the computational complexity and does not need to measure the supply air temperature. When the new design is adopted to replace the traditional PI control, there is no necessity to change the existent input structure of the PI control. Hence, the new design can be realised in practice easier than the original design.

Considering the Linear Parameter Varying (LPV) systems of discrete time, in this paper an approach for the synthesis of robust Proporcional+Integral Observers (PIO) is presented. From LPV systems characterized with polytopical uncertainties, the method of design is based on considering a dynamics extended of the typical PIO, in order to transform the design of the matrices of the dynamics of the observer, as a design of the gain of Static Output Feedback (SOF) of a problem of robust control. Under these conditions and from the norms H2=H1 described as Linear Matrix Inequalities (LMI), the criteria to obtain the gain in the SOF problem are established; taking into account performance indices in H2 and H1, under the presence of uncertainties and disturbances.

The problem of fuzzy static output feedback H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> control is investigated for a class of nonlinear systems with time-varying delay. Specifically, the considered system is approximated by a Takagi-Sugeno (T-S) fuzzy model. By using linear matrix inequality (LMI) methods, we design a fuzzy static output feedback H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> controller, which guarantees asymptotic stability and a prescribed H <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</inf> index for the resulting closed-loop system. In contrast to the existing results, the gains of controller are obtained by solving a set of LMIs. The proposed method is easy to implement and can be applied to linear systems. A numerical example is provided to illustrate the effectiveness of the developed method.

In this article, a robust model predictive control strategy based on sum‐of‐squares (SOS‐RMPC) is proposed for hypersonic vehicle with state‐dependent input constraints and uncertain parameters. Based on feedback linearization model of hypersonic vehicle, the polytopic linear parameter varying error model with bounded disturbance is established for reference trajectory tracking. The real limits of the actual control inputs are transformed into the constraints of the virtual inputs equivalently with the state‐dependent nonlinear functions. Further, the multivariate linear fitting is used to approximate the state‐dependent input constraints into polynomials. A weighted composite virtual control is formulated as the combination of unconstrained control and auxiliary control. SOS technique is introduced to cast the polynomial constraints into the convex matrix SOS conditions via linear matrix inequality. The virtual control law can be obtained by solving a SOS‐RMPC convex optimization with infinite horizon. The invariant set is designed for the undisturbed error states and the norm‐bounding theory is applied to ensure that the predictive error states with disturbance can remain in the same invariant set. The real control law is obtained by inverse control. The simulation verifies the performance of the designed controller.

Any linear control system, including those with complex structure and dynamic compensation can systematically be formulated as a static output feedback problem where the control system parameters are contained in the output feedback matrix. For this reason, many numerical techniques have been developed specifically to solve the 7/2 optimal ouput feedback problem posed in static output feedback form. Current numerical techniques for computing the output feedback gain matrix recast the problem as a coupled set of Lyapunov equations. Various numerical techniques effectively solve the Lyapunov equations differently. Here, we propose a new numerical method for solving the output feedback gain matrix by direct solution of the cost function using Fade approximation. Using this technique, analytic expressions for cost function derivatives can be written compactly, allowing for efficient solution using second order optimiztion methods. Using a flight control system example, the new method is compared to existing optimal output feedback algorithms.

ABSTRACTThis paper investigates the robust control of discrete‐time switched systems in the presence of parametric uncertainty and time‐varying delays. To accommodate a more general model, nonlinear Lipschitz functions are incorporated. Additionally, the system is modeled with constraints on both control inputs and outputs. To achieve the control objectives, static and dynamic output feedback controllers are designed using the robust model predictive control (RMPC) framework. The underlying optimization problem is formulated and solved, leveraging Lyapunov‐Krasovskii functionals (LKF) and switched Lyapunov functions (SLF). The asymptotic stability of the closed‐loop system under arbitrary switching signals is ensured by satisfying sufficient conditions expressed as linear matrix inequalities (LMIs). A key advantage of the proposed method lies in its low conservatism. Moreover, the time delay and state variables are assumed unknown, and they ensure an acceptable transient response of the closed‐loop system. To demonstrate the effectiveness of the designed controllers, simulation results are provided and compared against prior related works, highlighting the improvements achieved.

This paper is concerned with the problem of ℓ2-ℓ∞ filtering for discrete-time Takagi-Sugeno (T-S) fuzzy systems with stochastic perturbation. Firstly, a basis-dependent existence condition of desirable ℓ2-ℓ∞ filters is proposed. Then by means of the convex linearisation technique, the derived condition is transformed into some strict linear matrix inequality (LMI) constraints, by which both full- and reduced-order filters can be designed. Moreover, for the reduced-order ℓ2-ℓ∞ filter design, a novel analysis and design method with the projection lemma is also provided. Finally, a numerical example is provided to illustrate the feasibility and effectiveness of the proposed full- and reduced-order ℓ2-ℓ∞ filter design methods.

A Nonsmooth, Nonconvex Optimization Approach to Robust Stabilization by Static Output Feedback and Low-Order Controllers

Fixed-Order H∞ Control Design via a Partially Augmented Lagrangian Method