Finite-time dynamic output feedback stabilization of linear time-varying systems by a piecewise constant method

Sufficient conditions for stability of linear time-varying systems

This work is concerned with the assignment of a desired PD-eigenstructure for linear time-varying systems. Despite its well-known limitations, gain scheduling control appeared to be a focus of the research efforts. Scheduling of frozen-time, frozen-state controllers for fast time-varying dynamics is known to be mathematically fallacious, and practically hazardous. Therefore, recent research efforts are being directed towards applying time-varying controllers. In this paper we: 1) introduce a differential algebraic eigenvalue theory for linear time-varying systems; and 2) propose a PD-eigenstructure assignment scheme via a differential Sylvester equation and a command generator tracker (CGT) for linear time-varying systems. The PD-eigenstructure assignment is utilized as a regulator. A feedforward gain for tracking control is computed by using the command generator tracker. The whole design procedures of the proposed PD-eigenstructure assignment scheme are systematic in nature. The scheme could be used to determine the stability of linear time-varying systems easily as well as to provide a new horizon of designing controllers for the linear time-varying systems. A missile flight control application is presented to validate the proposed schemes.

This technical note deals with asymptotic stabilization of general nonlinear discrete-time systems by dynamic output feedback control. First, a sufficient condition for the dynamic output feedback stabilization with the closed-loop domain of attraction (DOA) estimation is given. For a given Lyapunov function of the plant state and the controller state, the negative-definite domain (NDD) in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space is proposed, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z_+,u,z$</tex-math></inline-formula>, and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y$</tex-math></inline-formula> denote the next controller state, the current control input, the current controller state and the current plant output, respectively. All points <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+;u;z;y)$</tex-math></inline-formula> in the NDD can make the time difference of the Lyapunov function negative-definite for the controller state <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> and all plant states in a set of plant states, which is determined by the controller state <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula> and the plant output <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y$</tex-math></inline-formula>. The NDD in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space could be viewed as an unstructured dynamic output feedback stabilization controller set. And any invariant subset of the projection of the NDD in the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z_+$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$u$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$y)$</tex-math></inline-formula>-space onto the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(z$</tex-math></inline-formula>-<inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x)$</tex-math></inline-formula>-space is an estimate of the DOA for the closed-loop, where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$x$</tex-math></inline-formula> denotes the current plant state. Second, based on the NDD, the controller design problem is converted to a set approximation problem in Euclidean space. Using numerical set approximation methods, dealing with strong nonlinearities is no longer the bottleneck. An estimation method for the unstructured dynamic output feedback stabilization controller set is proposed based on interval analysis, which is a kind of guaranteed numerical method for approximating sets.

This paper is concerned with analyzing a novel model transformation of discrete-time Takagi-Sugeno (T-S) fuzzy systems with time-varying delays and applying it to dynamic output feedback (DOF) controller design. A new auxiliary model is proposed by employing a new approximation for time-varying delay state, and then delay partitioning method is used to analyze the scaled small gain of this auxiliary model. A sufficient condition on discrete-time T-S fuzzy systems with time-varying delays, which guarantees the corresponding closed-loop system to be asymptotically stable and has an induced l 2 disturbance attenuation performance, is derived by employing the scaled small gain theorem. Then the solvability condition for the induced l 2 DOF control is also established, by which the DOF controller can be found by solving a set of linear matrix inequalities (LMIs).

This paper investigates the problem of parameter-dependent state feedback control of continuous-time systems in the context of finite time stability. The controller is designed in order to guarantee that the closed-loop system is finite time stable. The system is considered time varying with the parameters modeled within a unit simplex. The design conditions obtained by means of Lyapunov functions are expressed as linear matrix inequalities. The finite time stability is assessed by using homogeneous polynomially parameter-dependent state feedback gains with arbitrary degree g. LMI relaxations are proposed based on Pólya's theorem. A controller is obtained by the solution of a factibility problem and the effect of the relaxation procedures analyzed by an optimization problem. Numerical examples are provided.

This paper considers the L <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> stability of systems with a convolution operator in the forward loop and a time-varying gain in the feedback loop. A frequency-domain condition involving the frequency response of the forward loop and the Fourier coefficients of the time-varying gain of the feedback loop is presented. The paper also contains a somewhat more involved time-domain condition in terms of the open-loop weighting pattern.

New results on dynamic output state feedback stabilization of some class of time-varying nonlinear Caputo derivative systems

This paper considers the design problem of dynamic output feedback control for fractional-order (FO) Takagi-Sugeno fuzzy systems with uncertain parameters. Via the application of FO system stability theory, robust control method and the linear matrix inequality techniques, sufficient conditions are given for the stability analysis and the existence of the dynamic fuzzy output feedback controllers. The dynamic output feedback controller designed here guarantees the robust stability of the closed-loop system. Finally, a fractional order chaotic system simulation example is given to illustrate the effectiveness of the design approach here.

This paper considers the problem of adaptive reliable guaranteed cost controller design for linear time-delay systems against actuator faults via dynamic output feedback. Based on indirect adaptive method, a new reliable dynamic output feedback controller is established. The controller gains are affinely dependent on the estimations of fault parameters, which are updating automatically according to the proposed adaptive laws. New delay-dependent sufficient conditions for the existence of the reliable guaranteed cost dynamic output feedback controller are derived in terms of linear matrix inequalities, which have less conservativeness than the corresponding reliable dynamic output feedback controller with fixed gains. An example is presented to show the effectiveness of the proposed method.

This paper considers the problem of simultaneously stabilizing a finite collection of linear time-varying systems via a stable linear time-varying controller. The class of controllers proposed in this paper are linear, time varying and infinite dimensional. However, the computer implementation of these controllers would not be computationally more demanding than the computer implementation of a finite-dimensional linear time-varying controller.

This paper addresses the dynamic output feedback stabilization problem for linear time-invariant systems where the process disturbance preview is available to the controller via communication networks. A lower bound of data rates of communication channels, above which there exists a feedback control policy to stabilize the unstable plant with unbounded disturbance, is presented. Furthermore, the problem of bandwidth allocation in the communication channel is analyzed based on the system dynamics. Simulation results show the validity of the proposed scheme.

In terminal guidance phase, only partial information concerning the interception geometry is available. To intercept a target in specified direction under this condition, an impact angle constrained guidance law is proposed based on the dynamic output feedback. The guidance law doesn't use the rate of line-of-sight angle, and can guarantee convergence speed of the closed-loop system. Utilizing the parametric algebraic Riccati equation, a dynamic output feedback controller is designed. The controller contains a feedback law and an observer is obtained. Through integrated design of the parameters of the feedback law and the observer, the closed-loop system can be transformed to a new system, whose system matrix is a Jordan matrix. Using Lyapunov approach, desired convergence speed of the new system can be obtained by tuning its poles. And the same convergence speed of the original closed-loop system can also be achieved. Based on the dynamic output feedback controller, an impact angle constrained guidance law is derived. The guidance law is demonstrated by numerical simulation. Convergence speed can be guaranteed by the guidance law, and high guidance precision is achieved.

On stability of linear time-varying systems

Stabilization and robustness analysis for time-varying systems with time-varying delays using a sequential subpredictors approach

The authors previously (Proc. 1990 IEEE International Symp. on Circuits and Systems, p.1393, 1990) presented an exact eigenvalue theory for the characterization of stability for linear time-varying systems. In this paper they extend these results to include precise characterizations and criteria for the quality of stability for linear, time-varying systems. These results provide analysts and control designers with a variety of mathematical tools and procedures for ensuring that the transient response of a linear, time-varying system is well behaved in the sense of settling-time, overshoot, etc. An example is given to illustrate the results.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>