Guaranteed Asymptotic Stability for Some Linear Systems With Bounded Uncertainties

1979 ◽  
Vol 101 (3) ◽  
pp. 212-216 ◽  
Author(s):  
G. Leitmann
Keyword(s):  
Asymptotic Stability ◽  
State Feedback ◽  
Linear Feedback ◽  
Linear Matrix ◽  
Compact Sets ◽  
Nonlinear Controller ◽  
Linear Dynamical Systems ◽  
Linear Matrix Equation ◽  
Time Invariant ◽  
Bounded Uncertainties

We consider a class of linear dynamical systems in which the system and input matrices, as well as the input, are uncertain. The nominal system is time-invariant, while the uncertainties are assumed to be measurable functions of time whose values may range in given compact sets. Utilizing solely the knowledge of the sets from which uncertain quantities take their values, we derive a state feedback controller that guarantees global uniform asymptotic (Lyapunov) stability of the zero state in the presence of admissible uncertainties. The controller is nonlinear, namely componentwise switching; however, its construction requires only the solution of a linear matrix equation. Unlike linear feedback, this nonlinear controller assures asymptotic stability for any admissible realization of the system; this is illustrated by means of a simple example.

Hinf Control of Uncertain Takagi-Sugeno Fuzzy Descriptor Systems

2013 ◽  
Vol 694-697 ◽  
pp. 2110-2115
Author(s):  
Bao Ping Ma ◽  
Ming Chen
Keyword(s):  
State Feedback ◽  
Descriptor Systems ◽  
Descriptor System ◽  
Linear Matrix ◽  
Matrix Inequalities ◽  
Control Laws ◽  
Invariant Norm ◽  
Norm Bound ◽  
Takagi Sugeno ◽  
Time Invariant

This paper focuses on the problem of Hinf control for uncertain Takagi-Sugeno fuzzy descriptor system with time-invariant norm-bound uncertainty. Sufficient condition for robust Hinf control with state feedback is derived. It is shown that the control laws can be obtained by solving a set of linear matrix inequalities (LMIs) which is numerically tractable with commercially available software. Numerical example is given to demonstrate the advantage of the proposed method.

scholarly journals Resilient set-membership state estimation for nonlinear complex networks with time-delay and incomplete measurements

2021 ◽  
Vol 336 ◽  
pp. 08017
Author(s):  
Ning Yang ◽  
Dongyan Chen ◽  
Jun Hu
Keyword(s):  
Complex Networks ◽  
State Estimation ◽  
Taylor Series Expansion ◽  
Interval Matrix ◽  
Linear Matrix ◽  
Convex Optimization Problem ◽  
Set Membership ◽  
Linearization Errors ◽  
Time Invariant ◽  
Bounded Uncertainties

Taking the incomplete measurements and the weighted try-once-discard (WTOD) protocol into account, this paper develops a novel resilient set-membership state estimation (RSMSE) method for time-varying nonlinear complex networks with time-invariant delay. A classic interval matrix technique is utilized to describe incomplete measurements. The Taylor series expansion is applied to dispose the nonlinearities, where the high-order terms of the linearization errors are described by norm-bounded uncertainties. To mitigate the communication burden, the WTOD protocol is introduced, where only one node can send updated data through a shared communication network at each certain transmission step. Using the recursive linear matrix inequalities (RLMIs), a series of ellipsoidal sets including the state vector can be determined. The desirable estimator gain and a smallest possible estimation ellipsoid can be calculated via solving the convex optimization problem. Lastly, we use an illustrative example to show the feasibility of the introduced RSMSE technique.

scholarly journals Robust Position Control of a Two-Sided 1-DoF Impacting Mechanical Oscillator Subject to an External Persistent Disturbance by Means of a State-Feedback Controller

Complexity ◽  
2019 ◽  
Vol 2019 ◽  
pp. 1-14 ◽  
Author(s):  
Firas Turki ◽  
Hassène Gritli ◽  
Safya Belghith
Keyword(s):  
State Feedback ◽  
Position Control ◽  
External Disturbance ◽  
Feedback Controller ◽  
Linear Matrix ◽  
Stability Conditions ◽  
Mechanical Oscillator ◽  
Impact Oscillator ◽  
State Feedback Controller ◽  
The Impact

This paper proposes a state-feedback controller using the linear matrix inequality (LMI) approach for the robust position control of a 1-DoF, periodically forced, impact mechanical oscillator subject to asymmetric two-sided rigid end-stops. The periodic forcing input is considered as a persistent external disturbance. The motion of the impacting oscillator is modeled by an impulsive hybrid dynamics. Thus, the control problem of the impact oscillator is recast as a problem of the robust control of such disturbed impulsive hybrid system. To synthesize stability conditions, we introduce the S-procedure and the Finsler lemmas by only considering the region within which the state evolves. We show that the stability conditions are first expressed in terms of bilinear matrix inequalities (BMIs). Using some technical lemmas, we convert these BMIs into LMIs. Finally, some numerical results and simulations are given. We show the effectiveness of the designed state-feedback controller in the robust stabilization of the position of the impact mechanical oscillator under the disturbance.

State Feedback Robust H∞ Control for Generalized Discrete System and Simulation

2013 ◽  
Vol 467 ◽  
pp. 621-626
Author(s):  
Chen Fang ◽  
Jiang Hong Shi ◽  
Kun Yu Li ◽  
Zheng Wang
Keyword(s):  
Discrete System ◽  
State Feedback ◽  
Closed Loop ◽  
The State ◽  
Linear Matrix ◽  
Sufficient Condition ◽  
Matrix Inequalities ◽  
Parameter Uncertainties ◽  
Robust Controller ◽  
Closed Loop Systems

For a class of uncertain generalized discrete linear system with norm-bounded parameter uncertainties, the state feedback robust control problem is studied. One sufficient condition for the solvability of the problem and the state feedback robust controller are obtained in terms of linear matrix inequalities. The designed controller guarantees that the closed-loop systems is regular, causal, stable and satisfies a prescribed norm bounded constraint for all admissible uncertain parameters under some conditions. The result of the normal discrete system can be regarded as a particular form of our conclusion. A simulation example is given to demonstrate the effectiveness of the proposed method.

A Unified Approach to H∞ Control of Singular Markovian Jump Systems

2013 ◽  
Vol 135 (6) ◽  
Author(s):  
Guoliang Wang ◽  
Hongyi Li
Keyword(s):  
Transition Probability ◽  
Markovian Jump Systems ◽  
Linear Matrix ◽  
Markovian Jump ◽  
New Approach ◽  
Control Approach ◽  
Singular Markovian Jump Systems ◽  
Jump Systems ◽  
Bounded Uncertainties ◽  
The Given

This paper considers the H∞ control problem for a class of singular Markovian jump systems (SMJSs), where the jumping signal is not always available. The main contribution of this paper introduces a new approach to a mode-independent (MI) H∞ controller by exploiting the nonfragile method. Based on the given method, a unified control approach establishing a direct connection between mode-dependent (MD) and mode-independent controllers is presented, where both existence conditions are given in terms of linear matrix inequalities. Moreover, another three cases of transition probability rate matrix (TRPM) with elementwise bounded uncertainties, being partially unknown and to be designed are analyzed, respectively. Numerical examples are used to demonstrate the effectiveness of the proposed methods.

Robust H2/H∞Control Strategy for Linear Markovian Jump Systems

2014 ◽  
Vol 525 ◽  
pp. 646-652
Author(s):  
Min Bian ◽  
Qing Yun Guo
Keyword(s):  
Control Strategy ◽  
State Feedback ◽  
Control Strategies ◽  
State Feedback Control ◽  
Linear Matrix ◽  
Bounded Real Lemma ◽  
Finite State ◽  
Linear State ◽  
Feasible Solutions ◽  
Jump Systems

The robust H2/<em>H</em>∞ control strategy for a class of linear continuous-time uncertain systems with randomly jumping parameters is investigated. The transition of the jumping parameters is decided by a finite-state Markov process. The uncertainties are supposed to be norm-bounded. It is desired to design a linear state feedback control strategies such that the closed-loop system satisfies H performance and minimizes the H2 norm of the system. A sufficient condition is first established on the existence of the robust H2/<em>H</em>∞controller bases on the bounded real lemma. Then the corresponding state-feedback law is given in terms of a set of linear matrix inequalities (LMIs). It is showed that this condition is equivalent to the feasible solutions problem of LMI. Furthermore, the control strategy design problem is converted into a convex optimization problem subject to LMI constraints, which can be easily solved by standard numerical software.

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