Quadratic Stabilization Problem Research Articles

Quadratic Stabilization of Linear Uncertain Positive Discrete-Time Systems

The paper provides extended methods for control linear positive discrete-time systems that are subject to parameter uncertainties, reflecting structural system parameter constraints and positive system properties when solving the problem of system quadratic stability. By using an extension of the Lyapunov approach, system quadratic stability is presented to become apparent in pre-existing positivity constraints in the design of feedback control. The approach prefers constraints representation in the form of linear matrix inequalities, reflects the diagonal stabilization principle in order to apply to positive systems the idea of matrix parameter positivity, applies observer-based linear state control to assert closed-loop system quadratic stability and projects design conditions, allowing minimization of an undesirable impact on matching parameter uncertainties. The method is utilised in numerical examples to illustrate the technique when applying the above strategy.

Quantized Feedback Control Based on Spherical Polar Coordinate Quantizer

Different from uniform quantizer and logarithmic quantizer, this article proposes a novel quantizer for the design problems of linear systems with quantized feedback. This quantizer is based on spherical polar coordinates and has a desired relation between the quantized vector and the corresponding quantization error. Utilizing the proposed quantizer with infinite data rate, the quadratic stabilization problems of linear systems via state feedback under quantization can be converted, respectively, to the same problems of the corresponding robust control systems with sector bound uncertainties. From a practical point of view, we are more concerned with quantized feedback control with finite data rate, so a quantizer with finite data rate is proposed for the abovementioned problems. Different from the quantizer with infinite data rate, a time-varying bounded quantization region needs to be determined to contain the quantized vector as the system evolves. Under this circumstance, the vector needed to be quantized cannot be quantized directly, so we give a quantization method to solve this problem.

Observer Design for Positive Uncertain Discrete-time Lipschitz Systems

Abstract For positive uncertain discrete-time Lipschitz systems this paper proposes a way to reflect matched uncertainties, structural system parameter constraints, positiveness and Lipschitz continuity in solving the problem of the state observer quadratic stability. The design conditions are proposed in the set of linear matrix inequalities to guarantee the observer strict positiveness, system parameter constraint representation and estimation error bounding in terms of achieved quadratic stability and nonnegative feedback gain matrix. It follows from the results obtained that the impact of nonnegative system matrix structures can be reflected in uncertainty matching problems. A numerical example is included to assess the feasibility of the technique and its applicability.

Control Design for Linear Uncertain Positive Discrete-time Systems

Abstract For linear uncertain positive discrete-time systems this paper proposes an extended way to reflect structural system parameter constraints and positiveness in solving the problem of the system quadratic stability. The new design conditions are proposed, exploiting a set of system parameter constraint representation in the form of linear matrix inequalities to guaranty the closed-loop strictly positiveness, while a Lyapunov principle is focused to guaranty the system quadratic stability. Moreover, only time-invariant strictly positive control law gain is used in the feedback loop. Closely connected to the obtained results is the fact that the impact of nonnegative system matrix structures can be successfully implemented. A numerical example is included to assess the feasibility of the technique and its applicability.

Quantized stabilization of stochastic systems with multiplicative noise under Markovian switching

A problem of quantized state feedback quadratic mean-square stabilization of discrete-time stochastic processes under Markovian switching and multiplicative noise is considered. A static quantizer is used in the feedback channel and the jump Markovian switching is modeled by a discrete-time Markov chain. The control input is simultaneously applied to both the rate vector and the diffusion term. It is shown that the coarsest quantization density that permits quadratic mean-square stabilization of this system is achieved with the use of a logarithmic quantizer, and the coarsest quantization density is determined by an algebraic Riccati equation, which is also the solution to a special linear stochastic Markovian switching control system. Also, sufficient conditions for exponential mean-square stabilization of such systems are also explored. An example is given to demonstrate the obtained results.

Reliability of disturbance rejection control based on the geometrical disturbance decoupling

AbstractIn this note, a reliable disturbance decoupled control system based on the geometrical technique is proposed. For this purpose, first, the conditions for nominal disturbance detection and rejection is introduced by using invariant subspaces. Then, the fragility issue of the controller/observer is addressed with appropriate H∞ constraints in the form of some linear matrix inequalities (LMIs). In contrast to conventional pole placement method suitable for systems by neglecting sources of uncertainty, in the presented method, the disturbance rejection problem reduces to a quadratic stability problem. It is observed that the proposed method has a better attenuation performance than classical observer‐based non‐fragile controller. © 2016 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

A stabilization method for steady motions with zero roots in the closed system

Based on previous results, we consider stabilization problems for both non-asymptotic stability and asymptotic stability with respect to all variables for equilibrium positions and stationary motions of mechanical systems with redundant coordinates. The linear stabilizing control is defined by the solution of a linear---quadratic stabilization problem for an allocated linear controllable subsystem of as small dimension as possible. We find sufficient conditions under which a complete nonlinear system closed by this control is ensured asymptotic stability despite the presence of at least as many zero roots of the characteristic equation as the number of geometric relations. We prove a theorem on the stabilization of the control equilibrium applied only with respect to redundant coordinates and constructed from the estimate of the phase state vector obtained by a measurement of as small dimension as possible.

The Model and Quadratic Stability Problem of Buck Converter in DCM

Quadratic stability is an important performance for control systems. At first, the model of Buck Converter in DCM is built based on the theories of hybrid systems and switched linear systems primarily. Then quadratic stability of SLS and hybrid feedback switching rule are introduced. The problem of Buck Converter’s quadratic stability is researched afterwards. In the end, the simulation analysis and verification are provided. Both experimental verification and theoretical analysis results indicate that the output of Buck Converter in DCM has an excellent performance via quadratic stability control and switching rules.

A sliding sector approach to quantized feedback variable structure control

This paper is concerned with the quantized feedback quadratic stabilization problem for linear time-invariant systems. Sliding sector based quantized state feedback variable structure control schemes are established. The main benefit of the sliding sector technique is that it can avoid chattering caused by the utilization of variable structure control strategy. With the proposed discrete on-line adjustment of the quantization parameter, it is shown that the proposed sliding sector based sliding mode controllers can tackle state quantization and guarantee quadratic stability of the closed-loop system. Simulation results are given to verify the effectiveness of the proposed method.

First Order Perturbation Bounds of the Discrete-Time LMI-Based H∞ Quadratic Stability Problem for Descriptor Systems

Abstract In this paper we propose an approach to obtain first order perturbation bounds for the discrete-time Linear Matrix Inequalities (LMI) based H∞ quadratic stability problem for descriptor systems. Applying the considered approach we are able to compute tight first order perturbation bounds for the LMIs’ solutions to the H∞ quadratic stability problem for discrete-time descriptor systems. In the paper we present an approach to compute the estimates of the individual condition numbers of the considered LMIs. To illustrate the performance and applicability of the results obtained we present a numerical example

Disturbance Decoupling via Dynamic Output Feedback for Switched Linear Systems

AbstractIn this paper two disturbance decoupling problems without stability and with quadratic stability via dynamic output feedback for switched linear systems are investigated. Firstly, simultaneous (C, A, B)-pair which incorporates both the concept of simultaneous (A, B)-invariant subspace and its dual simultaneous (C, A)-invariant subspace for a family of linear systems are investigated. Secondly, disturbance decoupling problem without stability via dynamic output feedback for switched linear system is formulated and it's solvabililty conditions are presented by using the concept of simultaneous (C, A, B)-pair. Further, the quadratic stability problem for disturbance decoupled switched system is also investigated.

Input and Output Quantized Feedback Linear Systems

Although there has been a lot of research on analysis and synthesis of quantized feedback control systems, most results are developed for the case of a single quantizer (either measurement quantization or control signal quantization). In this technical note, we investigate the case of feedback control systems subject to both input and output quantization. This is motivated by the fact that it is common in remotely controlled systems that measurement and control signals are shared over a single digital network. More specifically, we consider a single-input single-output linear system with memoryless logarithmic quantizers. We firstly show that the output feedback quadratic stabilization problem in this setting can be addressed with no conservatism by means of a sector bound approach. Secondly, we provide a sufficient condition for quadratic stabilization via the solution of a scaled H <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">¿</sub> control problem. Finally, we analyze a problem of bandwidth allocation in the communication channel for finite-level input and output quantizers.

On the quadratic stability of descriptor systems with uncertainties in the derivative matrix

This article considers the quadratic stability problem of uncertain descriptor systems. The underlying system may be subjected to linear fractional uncertainties both in the derivative matrix and in the state matrix. Furthermore, the rank of the derivative matrix may change with respect to the uncertainties. By using an S-procedure and sum-of-squares technique, a sufficient condition on the quadratic stability problem of such type of uncertain descriptor systems is presented. An illustrative example is explored to sustain the findings of this article.

Quadratic stabilization of switched nonlinear systems

In this paper, the problem of quadratic stabilization of multi-input multi-output switched nonlinear systems under an arbitrary switching law is investigated. When switched nonlinear systems have uniform normal form and the zero dynamics of uniform normal form is asymptotically stable under an arbitrary switching law, state feedbacks are designed and a common quadratic Lyapunov function of all the closed-loop subsystems is constructed to realize quadratic stabilizability of the class of switched nonlinear systems under an arbitrary switching law. The results of this paper are also applied to switched linear systems.

Composite Systems with Uncertain Couplings of Fixed Structure: Scaled Riccati Equations and the Problem of Quadratic Stability

We consider large scale systems consisting of a finite number of separate uncertain subsystems which interact via uncertain couplings of arbitrarily prescribed structure. The couplings are viewed as structured perturbations of the block-diagonal system representing the collection of the separate nominal subsystems (the “nominal system”). We define spectral value sets and stability radii for these time-invariant structured perturbations and derive formulas for their computation. Scaled Riccati equations are introduced to obtain explicit formulas for the stability radii with respect to time-varying and possibly nonlinear perturbations of the given structure. From these we derive necessary and sufficient conditions under which the stability radii with respect to time-invariant and time-varying perturbations are equal. These results are obtained by constructing joint quadratic Liapunov functions of optimal robustness. With their help we prove necessary and sufficient conditions for quadratic stability and sufficient conditions for the validity of a generalized Aizerman conjecture.

GENERALIZED QUADRATIC STABILIZATION FOR DISCRETE-TIME SINGULAR SYSTEMS WITH TIME-DELAY AND NONLINEAR PERTURBATION

This paper discusses a generalized quadratic stabilization problem for a class of discrete-time singular systems with time-delay and nonlinear perturbation (DSSDP), which the satisfies Lipschitz condition. By means of the S-procedure approach, necessary and sufficient conditions are presented via a matrix inequality such that the control system is generalized quadratically stabilizable. An explicit expression of the static state feedback controllers is obtained via some free choices of parameters. It is shown in this paper that generalized quadratic stability also implies exponential stability for linear discrete-time singular systems or more generally, DSSDP. In addition, this new approach for discrete singular systems (DSS) is developed in order to cast the problem as a convex optimization involving linear matrix inequalities (LMIs), such that the controller can stabilize the overall system. This approach provides generalized quadratic stabilization for uncertain DSS and also extends the existing robust stabilization results for non-singular discrete systems with perturbation. The approach is illustrated here by means of numerical examples.

New LMI-Based Conditions for Quadratic Stabilization of LPV Systems

This paper is concerned with quadratic stabilization problem of linear parameter varying (LPV) systems, where arbitrary time-varying dependent parameters are belonging to a polytope. It provides improved linear matrix inequality- (LMI-) based conditions to compute a gain-scheduling state-feedback gain that makes closed-loop system quadratically stable. The proposed conditions, based on the philosophy of Pólya's theorem, are written as a sequence of progressively less and less conservative LMI. More importantly, by adding an additional decision variable, at each step, these new conditions provide less conservative or at least the same results than previous methods in the literature.

Stability and stabilization of block-cascading switched linear systems

The main purpose of this paper is to investigate the problem of quadratic stability and stabilization in switched linear systems using reducible Lie algebra. First, we investigate the structure of all real invariant subspaces for a given linear system. The result is then used to provide a comparable cascading form for switching models. Using the common cascading form, a common quadratic Lyapunov function is (QLFs) is explored by finding common QLFs of diagonal blocks. In addition, a cascading Quaker Lemma is proved. Combining it with stability results, the problem of feedback stabilization for a class of switched linear systems is solved.

Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation

This note considers the generalized quadratic stability problem for continuous-time singular system with nonlinear perturbation. The perturbation is a function of time and system state and satisfies a Lipschitz constraint. In this work, a sufficient condition for the existence and uniqueness of solution to the singular system is firstly presented. Then by using S-procedure and matrix inequality approach, a necessary and sufficient condition is presented in terms of linear matrix inequality, under which the maximal perturbation bound is obtained to guarantee the generalized quadratic stability of the system. That is, the system remains exponential stable and the nominal system is regular and impulse free. Furthermore, robust stability for nonsingular systems with perturbation can be obtained as a special case. Finally, the effectiveness of the developed approach for both singular and nonsingular systems is illustrated by numerical examples.

New results of robust quadratically stabilizing control for uncertain linear time-delay systems

The problem of robust quadratic stabilization for a class of uncertain linear dynamic systems with time-delays in both state and control is discussed in this paper. The uncertain time-delay systems under consideration are described by state differential equations with unknown-but-bounded uncertain parameters that are confined into convex polyhedrons. A new necessary and sufficient condition for the existence of quadratically stabilizing control law and the corresponding synthesis method are derived, the results are given in terms of a set of linear matrix inequalities (LMIs). Finally, numerical examples are presented to demonstrate the effectiveness of the obtained results.