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

Agulhari, Cristiano Marcos; Oliveira, Ricardo C. L. F.; Peres, Pedro L. D.

doi:10.1109/tac.2011.2174693

Cited by 107 publications

(95 citation statements)

References 23 publications

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“…Results given in Table I indicate that the proposed method in this paper and [18] lead to the best results among the others. Example 2: Consider the following continuous-time polytopic system with two vertices [19]: The objective here is to design a static output feedback H ∞ controller with linearly parameter-dependent Lyapunov matrices.…”

Section: Simulation Resultsmentioning

confidence: 78%

“…The minimum and maximum values of a can be determined by a bisection algorithm. The results of Theorem 1 are compared with ones of [18], [19], [30], [31] in Table I. The proposed methods in Theorem 1 and [18] are both initialized with the same parameter-dependent state feedback K sf (λ).…”

Section: Simulation Resultsmentioning

confidence: 99%

“…In this subsection, we show that the proposed approach in [18] relies on the concept of SPRness of a transfer function where A-matrix is fixed by a parameter-dependent gain.…”

Section: B Relation Between the Proposed Stabilizing Static Outputmentioning

confidence: 99%

“…If there are a Lyapunov matrix P (λ) and matrices F (λ), V (λ), X, and L such that (21) holds [18]. Then, K = X −1 L is a robust static output feedback controller which stabilizes the augmented continuous-time system given in (5).…”

Section: B Relation Between the Proposed Stabilizing Static Outputmentioning

confidence: 99%

“…Furthermore, it is theoretically and numerically demonstrated that the proposed approach allows fixedorder stabilizing (H ∞ ) controller synthesis which potentially use less decision variables than some existing approaches, e.g. [18].…”

Section: Introductionmentioning

confidence: 99%

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Fixed-order control of LTI systems subject to polytopic uncertainty via the concept of Strictly Positive Realness

Sadabadi

Karimi

2015

2015 American Control Conference (ACC)

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Abstract-This paper deals with the problem of fixed-order controller design of LTI continuous-time and discrete-time polytopic systems via homogeneous polynomially parameterdependent Lyapunov matrices. The proposed method is based on the concept of Strictly Positive Realness (SPRness) of a transfer function depending on a parameter-dependent gain. To convert the problem to a set of LMI conditions, the parameterdependent gain is determined a priori by means of a parameterdependent state feedback controller. Simulation results and a comparison with recent existing methods demonstrate the effectiveness of the proposed approach.

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Section: Simulation Resultsmentioning

confidence: 78%

Section: Simulation Resultsmentioning

confidence: 99%

“…In this subsection, we show that the proposed approach in [18] relies on the concept of SPRness of a transfer function where A-matrix is fixed by a parameter-dependent gain.…”

Section: B Relation Between the Proposed Stabilizing Static Outputmentioning

confidence: 99%

Section: B Relation Between the Proposed Stabilizing Static Outputmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Fixed-order control of LTI systems subject to polytopic uncertainty via the concept of Strictly Positive Realness

Sadabadi

Karimi

2015

2015 American Control Conference (ACC)

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A convex optimization approach to adaptive stabilization of discrete‐time LTI systems with polytopic uncertainties

Lee

Joo

Tak

2014

Adaptive Control & Signal

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This paper suggests a simple convex optimization approach to state-feedback adaptive stabilization problem for a class of discrete-time LTI systems subject to polytopic uncertainties. The proposed method relies on estimating the uncertain parameters by solving an online optimization at each time step, such as a linear or quadratic programming, and then, on tuning the control law with that information, which can be conceptually viewed as a kind of gain-scheduling or indirect adaptive control. Specifically, an admissible domain of stabilizing state-feedback gain matrices is designed offline by means of linear matrix inequality problems, and based on the online estimation of the uncertain parameters, the state-feedback gain matrix is calculated over the set of stabilizing feedback gains. The proposed stabilization algorithm guarantees the asymptotic stability of the overall closed-loop control system. An example is given to show the effectiveness of the proposed approach.A CONVEX OPTIMIZATION APPROACH TO ADAPTIVE STABILIZATION 1117 deal of research has been dedicated to developing more general types of gain-scheduling controllers, for instance, path-dependent control laws [20] and polynomially parameter-dependent control laws [21,22].However, this is no longer the case for uncertain LTI systems in the sense that the uncertain parameters are literally assumed not to be measurable. For this reason, the development of less conservative design methods for the static state-feedback has become a fundamental and challenging problem. In the late 1990s, the so-called extended Schur complement and slack variable approaches were developed by the pioneering work [8,9], which played a significant role in the subsequent development of the LMI-based robust analysis and control approaches. More recently, an important advance on robust stabilization of discrete-time LTI systems was accomplished in [23,24], where the so-called periodically time-varying state-feedback controller was proposed and turned out to be effective in reducing the conservatism in the static state-feedback approaches.On the other hand, adaptive control technology [25-29] has a long history of interest in the control community [30][31][32][33] and has attracted considerable attention to date. One of the advantages of the adaptive control approach is the ability to adjust the controller so as to cope with complex, uncertain, and highly nonlinear nature of the plant especially in the absence of complete information on the system. The adaptive control strategies may be broadly classified into the following two categories: direct [29] and indirect [27,28] adaptive controls. The indirect adaptive control is a feedback control scheme in which the plant dynamics is estimated first, and the feedback controller is calculated by exploiting the estimation [34,35], while in the direct case, the ideal control law is approximated without the estimation of the plant.The philosophy of adaptive control and the gain-scheduling control leads us to the key motivation for the main re...

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New iterative linear matrix inequality based procedure for H₂ and H_∞ state feedback control of continuous‐time polytopic systems

Jaimoukha

2020

Intl J Robust & Nonlinear

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This article provides new linear matrix inequality (LMI) sufficient conditions for a generalized robust state feedback control synthesis problem for linear continuous-time polytopic systems. This generalized problem includes the robust stability, H 2-norm, and H ∞-norm problems as special cases. Using a novel general separation result, which separates the state feedback gain from the Lyapunov matrix but with the state feedback gain synthesized from the slack variable, then allows the formulation of LMI sufficient conditions for the generalized problem. Compared to existing parameterized LMI based conditions, where auxiliary scalar parameters are introduced in order to include the quadratic stability conditions (ie, assuming a constant Lyapunov matrix) as a special case, the proposed new conditions are true LMIs and contain as a particular case the optimal quadratic stability solution. Utilizing any initial solution derived by the quadratic or some existing methods as a starting solution, we propose an algorithm based on an iterative procedure, which is recursively feasible in each update, to compute a sequence of nonincreasing upper bounds for the H 2-norm and H ∞-norm. In addition, if no feasible initial solution can be found for some uncertain systems using any existing methods, another algorithm is presented that offers the possibility of obtaining a robust stabilizing gain. Numerical examples from the literature demonstrate that our algorithms can provide less conservative results than existing methods, and they can also find feasible solutions where all other methods fail. K E Y W O R D S H 2 and H ∞ norms, LMI relaxations, polytopic uncertainties, robust state feedback control, slack variables, uncertain linear systems 1 INTRODUCTION The design of robust H 2-norm and H ∞-norm controllers for linear time-invariant systems with polytopic uncertainties has received considerable attention in the past decades. Many studies devoted to investigating robust stability and This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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

Cited by 107 publications

References 23 publications

Fixed-order control of LTI systems subject to polytopic uncertainty via the concept of Strictly Positive Realness

Fixed-order control of LTI systems subject to polytopic uncertainty via the concept of Strictly Positive Realness

A convex optimization approach to adaptive stabilization of discrete‐time LTI systems with polytopic uncertainties

New iterative linear matrix inequality based procedure for H₂ and H_∞ state feedback control of continuous‐time polytopic systems

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

Cited by 107 publications

References 23 publications

Fixed-order control of LTI systems subject to polytopic uncertainty via the concept of Strictly Positive Realness

Fixed-order control of LTI systems subject to polytopic uncertainty via the concept of Strictly Positive Realness

A convex optimization approach to adaptive stabilization of discrete‐time LTI systems with polytopic uncertainties

New iterative linear matrix inequality based procedure for H2 and H∞ state feedback control of continuous‐time polytopic systems

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New iterative linear matrix inequality based procedure for H₂ and H_∞ state feedback control of continuous‐time polytopic systems