Robust Control Synthesis for Linear Differential Systems with Parametric Uncertainty

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

Dissipative Dynamical Systems

This paper develops some dissipativity conditions for linear time-invariant (LTI) systems in the behavioural framework. The behaviour of a system is characterised by its persistently exciting trajectories. For the dissipativity conditions, both the supply rate and the storage function are represented using quadratic difference forms (QdFs) using past steps. We show that it is possible to define an LTI system of arbitrary length using trajectories with low order of excitation. The system can be defined in a similar way as an image representation and the dissipativity conditions can hence be derived using a similar logic. The conditions are presented in the form of linear matrix inequalities (LMIs).

Given a nominal model, an integrated uncertainty model identification and mu-synthesis algorithm is presented for linear time-invariant (LTI) systems. The goal is 1) to construct an uncertainty model set characterized by parameterized weighting functions of dynamic perturbations and disturbances in the general linear fractional transformation (LFT) form - customary representation in modern robust control and 2) to select from this set according to closed-loop control objectives. The motivation is the simultaneous model validation and model optimization with respect to closed-loop requirements. The algorithm works on sampled, bounded-energy experimental data on the frequency-domain and integrates nominal model invalidation, uncertainty model construction and control synthesis in order to achieve robust performance. Standard D-K iteration steps are combined with an optimization step on a group of selected data. The method is demonstrated on a simple numerical example, where the results are comparable with analytic computations

A robust dynamic output feedback controller synthesis algorithm considering H∞/H2performance and regional pole placement is addressed for a nonlinear system with parameter uncertainties and external disturbance. First, the formulation of a gain‐scheduled mixed‐objective robust dynamic output feedback controller for continuous‐time polytopic linear parameter varying (LPV) systems is presented. To reduce conservativeness, some auxiliary slack variables and parameter‐dependent Lyapunov functions are employed in addition to well‐established performance conditions. Then, sufficient conditions for the desired gain‐scheduled mixed‐objective robust dynamic output feedback controllers are cast into an efficiently tractable finite‐dimensional convex optimization problem in terms of linear matrix inequalities (LMIs). Finally, numerical simulation shows the validity of the proposed controller, which has good stability, strong robustness, satisfied disturbance attenuation ability, and smooth dynamic properties.

In this paper, a MIMO PI design procedure is proposed for linear time invariant (LTI) systems with multiple time delays. The controller tuning is established in two stages and guarantees performances for set‐point changes, disturbance variations, and parametric uncertainties. In the first stage, an iterative linear matrix inequality (ILMI) approach is extended to design PI controllers for systems with multiple time delays without performance guarantee, a priori. The second stage is devoted to improve the closed‐loop performances by minimizing sensitivity functions. Simulations results carried out on the unstable distillation column, the stable industrial scale polymerization (ISP) reactor, and the non‐minimum phase 4‐tank benchmark prove the efficiency of the proposed approach. A comparative analysis with the conventional internal model control (IMC) approach, a multiloop IMC‐PI approach, and a previous ILMI PID approach proves the superiority of the proposed approach compared to the related ones.

ROBUST STABILITY ANALYSIS OF UNCERTAIN INTERCONNECTION IN THE BEHAVIORAL FRAMEWORK

Frequency-domain stability conditions for asynchronously sampled decentralized LTI systems

This study investigates characteristic root equivalency relations between commensurate order and integer order Linear Time Invariant (LTI) systems. Author introduces some useful properties of a special class of commensurate order systems, which is called characteristic root equivalency class of LTI systems. These properties present potential to facilitate design and analysis efforts of this class of commensurate order systems. In this sense, straightforward stability checking procedures and design approaches for commensurate order root equivalent systems of the first and second order LTI systems are demonstrated. Findings of the study are validated by illustrative examples.

In this paper, the cooperative stabilization problem of linear time-invariant (LTI) systems is studied based on a class of distributed Luenberger observers. A given LTI system is estimated by a network of observers. In case of observability configuration or external attack, each observer has to possess different measurement output of the LTI system. Classical distributed control strategies cannot be directly used because they are assumed to obtain identical information. In view of this problem, a class of distributed observers are designed under an undirected graph, which can recover the state of the LTI system by limited local information. By using any single channel of the estimated state, a control input is proposed to stabilize the LTI system. By analysing the stability of an augmented system which involves the LTI system and the estimation error system, some necessary and sufficient conditions are derived for the existence of control gains that guarantee the stabilization of the LTI system. In addition, some algorithms are proposed to specify the control gains. Lastly, some simulation examples are given to verify the effectiveness of the theoretical result.

One of the key performance requirements for different control systems is non-overshooting step response, so that the controllable value should not overcome the reference value within a transient process. The problem of providing a non-overshooting step response was examined in this paper. Despite much scientific research being dedicated to the overshoot elimination problem, there are little to no results regarding parametric uncertainty for the discussed problem. Consideration of parametric uncertainty, particularly in the form of interval-given parameters, is essential, since in many physical processes, electronic devices and control systems parameter values can be obtained with acceptable error, and they can vary under different conditions. The main result of our research is the development of a proportional-integral-derivative (PID)-controller tuning approach for systems with interval-given parameters that provides a non-overshooting step response for such classes of control systems. This paper investigates analytical conditions and constraints for linear time invariant (LTI) systems in order to have no overshoot, enhances them with respect to parametric uncertainty, and formulates rules for tuning choices of parameters.

In this paper, the dynamic modelling of a new configuration spacecraft is investigated. The significance of dumbbell-shaped spacecraft to deep space exploration and the configuration of dumbbell-shaped spacecraft are introduced firstly. Then, the vibration problem of the dumbbell-shaped spacecraft of large-angle attitude maneuver is investigated, and a control program based on the combination of adaptive robust control (ARC) and component synthesis vibration suppression method-seven-section path planning (CSVS-SPP) is proposed. The large-angle attitude maneuver route of the spacecraft, which serves as the reference path, is planned using the CSVS-SPP approach, and the attitude controller is designed using the ARC. This program can effectively reduce the influence of external disturbance and parameter uncertainty on the system performance while completing attitude maneuver and suppress the vibration of the flexible beam during large-angle attitude maneuver. The numerical simulations show the superiority and effectiveness of the proposed ARC+CSVS-SPP.

This paper considers two topics in the area of robust control system analysis and synthesis. The first topic is concerned with the testing for zero inclusion of 2-D convex polytopes, which is essential in the robustness analysis of linear systems based on generalisations of Kharitonov's four-polynomial concept. Barmish (1989) has extended the Kharitonov approach to the more general case of polynomials subject to linearly dependent coefficient perturbations and more general zero location regions, but this is accompanied by various computational problems. These are analysed and two new more direct and computationally less intensive approaches, which produce much more accurate results, are presented. The second topic is concerned with a modification to existing dyadic and full-rank pole assignment design methods for MIMO uncertain systems having affine linear perturbation structures, in which a gain factor is introduced in order to ensure robustness stability and performance. An illustrative example is also included.

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.

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.