Robust control system analysis and synthesis

Robust controller synthesis attracts reviving research interest, driven by\nthe rise of learning-based systems where uncertainty and perturbation are\nubiquitous. Facing an uncertain situation, a robustly stabilizing controller\nshould maintain stability while operating under a perturbed system deviating\nfrom its nominal specification. There have been numerous results for robust\ncontroller synthesis in multiple forms and with various goals, including\nmu-synthesis, robust primal-dual Youla, robust input-output, and robust system\nlevel parameterizations. However, their connections with one another are not\nclear, and we lack a general approach to robust controller analysis and\nsynthesis.\n To serve this purpose, we derive robust stability conditions for general\nsystems and formulate the general robust controller synthesis problem. The\nconditions hinge on the realization-stability lemma, a recent analysis tool\nthat unifies existing controller synthesis methods. Not only can the conditions\ninfer a wide range of existing robust results, but they also lead to easier\nderivations of new ones. Together, we demonstrate the effectiveness of the\nconditions and provide a unified approach to robust controller analysis and\nsynthesis.\n

This paper investigates the robust stability analysis and control synthesis of discrete-time switched systems with norm-bounded uncertainty under arbitrary switching by linear matrix inequality (LMI) approach together with switched Lyapunov function method. First, the equivalence between the existence of a switched Lyapunov function and two different matrix inequalities is established. These two matrix inequality conditions can be further proven to be equivalent to two LMI-based conditions. Then, some LMI-based conditions for stabilization via switched static state feedback as well as switched static output feedback are presented. In the case of state feedback, the conditions are necessary and sufficient, whereas for output feedback, the conditions are only sufficient as they depend on particular state-space representation used to describe the switched systems.

Linear matrix inequalities are a powerful tool in the field of robust control for analysis and synthesis of control systems. This work presents new formulations based on linear matrix inequalities for ∞ robust control synthesis by state-feedback for discrete time-invariant linear systems. The formulations include an additional matrix variable, which allows the Lyapunov function to be parameter-dependent, and two scalar parameters that must be chosen by the designer. It is proposed to apply evolutionary optimization to determine the optimal values of these two parameters. There are three possibilities for determining the two parameters, setting the first, setting the second, or optimizing both. The proposed formulations include existing formulations in the literature for specific values of the two scalar parameters. Case studies are presented to evaluate the proposed formulations.

Design and Analysis of Robust Control Systems in PARADISE

We consider several robust control analysis and design problems. As has become well known over the last few years, most of these problems are NP hard. We show that if instead of worst-case guaranteed conclusions, one is willing to draw conclusions with a high degree of confidence, then the computational complexity decreases dramatically.

This paper addresses the problems of robust stability analysis and L1-gain controller synthesis for uncertain impulsive positive systems. First, by employing the idea of impulse interval partitioning, an impulse-time-dependent discretized copositive Lyapunov function is proposed to analyze the robust stability and L1-gain performance of the considered system without control inputs, and several stability conditions and L1-gain criteria are respectively derived. Subsequently, a sufficient condition is formulated for the existence of state-feedback controllers with which not only the positivity and robust uniform asymptotic stability of the resulting closed-loop system are guaranteed, but also a prescribed L1-gain performance is satisfied simultaneously. Furthermore, to make the controller synthesis problem numerically tractable, we propose an iterative convex optimization algorithm to compute the desired controller parameters. Finally, three numerical examples and two realistic examples are presented to show the effectiveness and applicability of the proposed methodology.

This paper studies the dissipative control problems for the non‐linear descriptor system via a Takagi–Sugeno descriptor model with time delay and uncertainties. The considered systems are not necessarily regular or impulse free. Using the free‐weighting‐matrix approach combined with the developed linear matrix inequality techniques, the authors proposed an improved delay‐dependent stability criterion to guarantee the system to be admissible. Based on this criterion, the solvable conditions for the existence of dissipative controller are derived, which guarantees that the closed‐loop system is not only admissible but also strictly dissipative for all admissible uncertainties. The proposed method is more suitable for the robust stability analysis and control synthesis. Moreover, different control processes can be achieved for the targeted systems in the same design process. Therefore it could potentially reduce cost and time in the controller design for an actual physical system. The authors perform simulations to validate the obtained method for practical examples.

Modeling Approaches and Robust Stability Conditions for Networked Controlled Systems with Uncertain Delays

This paper presents a robust control synthesis method for linear time-invariant (LTI) differential systems with parametric uncertainty. The uncertain dynamics is described by a parametric kernel representation. We use parametric quadratic differential forms (QDF) to represent the dissipativity properties (both storage function and supply rate) of the uncertain system. The control design utilizes sum-of-squares (SOS) programming to search for a QDF supply rate for the controller such that the supply rate of the closed-loop system satisfies robust stability and performance conditions. Such a supply rate is then applied for control synthesis based J-factorization.

Automated Synthesis of Fixed Structure QFT Controller using Interval Constraint Satisfaction Techniques

Discrete-Time LPV Controller Synthesis Using Dilated LMIs with Application to an Arm-Driven Inverted Pendulum

Robust PID controller synthesis for plants with unknown or slowly varying parameters is still an open topic. A novel method is presented allowing to exactly determine all stabilizing controller gains for systems with interval and affine parametric uncertainties. This leads to far less conservative stability areas, compared to existing methods which are based on over-bounding by interval polynomials. As a consequence, iterative searches containing robust synthesis and analysis are omitted, which greatly reduces the time and effort to find potential controller parameters as robust stability is already guaranteed. Additionally, due to the inherent decoupling at singular frequencies, no frequency grinding is needed. The proposed method is based on a combination of a modified Parameter Space Approach with results from robust test set theory.

The μ/Km-synthesis formulation for robust controller synthesis with frequency dependent scalings/multipliers is shown to be equivalent to a bilinear matrix inequality (BMI) feasibility problem. Formulating the non-convex μ/Km-synthesis problem as a BMI problem allows local joint controller and multiplier synthesis using finite dimensional LMI-like optimization methods and also the application of global optimization techniques. The BMI approach improves on the standard D, G(SINGLE BOND)K-iteration, M(SINGLE BOND)K-iteration approaches to μ/Km-synthesis, and additionally allows controller order and structure constraints.

On the structure of generalized plant convexifying static [formula omitted] control problems

In this paper, the vehicle's dynamical behaviour is first described through a bifurcation analysis to give some insights for its robust control synthesis. Then a novel adaptive fractional-order sliding mode controller (AFOSMC) is designed to effectively control the uncertain supercavitating vehicle against payload changes, nonlinear planing force and external disturbances. The control scheme utilises the fractional calculus and the adaptive law for the sliding mode control (SMC) to provide active control actions. The fractional calculus can guarantee more flexibility and more freedom for tuning active control synthesis than the integer-order counterpart. The adaptation law has been presented to directly handle the payload change effects. The dynamic performance of the proposed controller is verified through extensive simulation results, which demonstrate faster responses than existing integer order controllers. Furthermore, the fractional order controller shows its effectiveness in dealing with external disturbance, when eliminating most of the uncertainty effect.