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

Time-varying stability analysis of linear systems with linear matrix inequalities

Aerospace attitude control systems are often modeled as time-varying linear systems. In industry, these systems are analyzed with linear time-invariant (LTI) methods by treating the system as slowly varying. Stability analysis with parameter dependent Lyapunov functions and linear matrix inequalities (LMIs) enables the consideration of bounds on system parameters' rates of variation while accounting for time-varying behavior. The LMI criteria are adapted to predict robustness in time-varying systems. In a case study, stability envelopes are created for time-varying uncertain parameters in a spacecraft. The time-of-flight is divided into intervals and analyzed using typical trajectories of time-varying parameters. For the uncertain parameter combinations considered, LMI stability criteria deduce that the system is stable and possesses stability margins that meet or exceed requirements for the time intervals that can be approximated by linear system models

New Delay-Range-Dependent Robust Exponential Stability Criteria of Uncertain Impulsive Switched Linear Systems with Mixed Interval Nondifferentiable Time-Varying Delays and Nonlinear Perturbations

We investigate the problem of robust exponential stability analysis for uncertain impulsive switched linear systems with time-varying delays and nonlinear perturbations. The time delays are continuous functions belonging to the given interval delays, which mean that the lower and upper bounds for the time-varying delays are available, but the delay functions are not necessary to be differentiable. The uncertainties under consideration are nonlinear time-varying parameter uncertainties and norm-bounded uncertainties, respectively. Based on the combination of mixed model transformation, Halanay inequality, utilization of zero equations, decomposition technique of coefficient matrices, and a common Lyapunov functional, new delay-range-dependent robust exponential stability criteria are established for the systems in terms of linear matrix inequalities (LMIs). A numerical example is presented to illustrate the effectiveness of the proposed method

A new condition and equivalence results for robust stability analysis of rationally time-varying uncertain linear systems

Uncertain systems is a fundamental area of automatic control. This paper addresses robust stability of uncertain linear systems with rational dependence on unknown time-varying parameters constrained in a polytope. For this problem, a new sufficient condition based on the search for a common homogeneous polynomial Lyapunov function is proposed through a particular representation of parameter-dependent polynomials and LMIs. Relationships with existing conditions based on the same class of Lyapunov functions are hence investigated, showing that the proposed condition is either equivalent to or less conservative than existing ones. As a matter of fact, the proposed condition turns out to be also necessary for a class of systems. Some numerical examples illustrate the use of the proposed condition and its benefits. © 2011 IEEE.published_or_final_versionThe 2011 IEEE International Symposium on Computer-Aided Control System Design (CACSD), Denver, CO., 28-30 September 2011. In IEEE CACSD International Symposium Proceedings, 2011, p. 234-23

State-feedback control of LPV sampled-data systems

In this paper, we address the analysis and the state-feedback synthesis problems for linear parameter-varying (LPV) sampled-data control systems. We assume that the state-space data of the plant and the sampling interval depend on parameters that are measurable in real-time and vary in a compact set with bounded variation rates. We explore criteria such as the stability, the energy-to-energy gain (induced L2 norm) and the energy-to-peak gain (induced L2 -to- L∞ norm) of such sampled-data LPV systems using parameter-dependent Lyapunov functions. Based on these analysis results, the sampled-data state-feedback control synthesis problems are examined. Both analysis and synthesis conditions are formulated in terms of linear matrix inequalities that can be solved via efficient interior-point algorithms

Robust Guaranteed Cost Output-Feedback Gain-Scheduled Controller Design

In the paper a new robust guaranteed cost output-feedback gain-scheduled PID controller design technique is presented for affine linear parameter-varying systems under polytopic model uncertainty, with the assumption that the scheduled parameters are affected with absolute uncertainty. The proposed centralized or decentralized method is based on the Bellman-Lyapunov equation, guaranteed cost, and parameter-dependent quadratic stability. The robust stability and performance conditions are translated to an optimization problem subject to bilinear matrix inequalities, which can be solved or further linearized. As the main result, the suggested stability and performance conditions without any restrictions on the controller structure are convex functions of the scheduling and uncertainty parameters. Hence, there is no need for applying multi-convexity or other relaxation techniques and consequently the proposed solution delivers a less conservative design method. The viability of the novel design technique is demonstrated and evaluated through numerical examples

Parameter-Dependent Lyapunov Functions for Linear Systems With Constant Uncertainties

Robust stability of linear time-invariant systems with respect to structured uncertainties is considered. The small gain condition is sufficient to prove robust stability and scalings are typically used to reduce the conservatism of this condition. It is known that if the small gain condition is satisfied with constant scalings then there is a single quadratic Lyapunov function which proves robust stability with respect to all allowable time-varying perturbations. In this technical note we show that if the small gain condition is satisfied with frequency-varying scalings then an explicit parameter dependent Lyapunov function can be constructed to prove robust stability with respect to constant uncertainties. This Lyapunov function has a rational quadratic dependence on the uncertainties

Generalized robust gain-scheduled PID controller design for affine LPV systems with polytopic uncertainty

In the paper a generalized guaranteed cost output-feedback robust gain-scheduled PID controller synthesis is presented for affine linear parameter-varying systems under polytopic model uncertainty. The controller synthesis is generalized in a sense that it covers robust, robust gain-scheduled, and robust switched (with arbitrary switching algorithm) PID controller design. The proposed centralized/decentralized controller method is based on Bellman–Lyapunov equation, guaranteed cost, and parameter-dependent quadratic stability. The proposed sufficient robust stability and performance conditions are derived in the form of bilinear matrix inequalities (BMI) which can efficiently be solved or further linearized. As the main result, the suggested performance and stability conditions without any restriction on the controller structure are convex functions of the scheduling and uncertainty parameters. Hence, there is no need for applying multi-convexity or other relaxation techniques and consequently the proposed solution delivers a less conservative design method. The viability of the novel design technique is demonstrated and evaluated through numerical examples

Gain-scheduled H∞ control via parameter-dependent Lyapunov functions

Synthesising a gain-scheduled output feedback H∞ controller via parameter-dependent Lyapunov functions for linear parameter-varying (LPV) plant models involves solving an infinite number of linear matrix inequalities (LMIs). In practice, for affine LPV models, a finite number of LMIs can be achieved using convexifying techniques. This paper proposes an alternative approach to achieve a finite number of LMIs. By simple manipulations on the bounded real lemma inequality, a symmetric matrix polytope inequality can be formed. Hence, the LMIs need only to be evaluated at all vertices of such a symmetric matrix polytope. In addition, a construction technique of the intermediate controller variables is also proposed as an affine matrix-valued function in the polytopic coordinates of the scheduled parameters. Computational results on a numerical example using the approach were compared with those from a multi-convexity approach in order to demonstrate the impacts of the approach on parameter-dependent Lyapunov-based stability and performance analysis. Furthermore, numerical simulation results show the effectiveness of these proposed techniques