State feedback design for singularly perturbed system using unified approach

The problem of robust performance analysis for the delta operator formulated uncertain systems is considered. Sufficient conditions for D-stability robustness of the unified delta operator systems with structured uncertainty are presented in terms of matrix measure approach, which the desired closed-loop poles are constrained to locate within a prespecified D-region. The proposed results can unify related conclusions of continuous time and discrete time systems into the delta operator framework.

One of the most common methods of synthesis of the nonlinear control systems is the method of a feedback linearization (FL). The idea of this method consists in conversion of the original nonlinear system into a linear one by means of a state feedback and coordinate transformation. Then, the methods of control theory for the linear systems are used for the system design. If the original nonlinear system cannot be linearized exactly by the state feedback, the method of the approximate feedback linearization (AFL) is used. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system (not of the entire system). In this paper, the author proposes a method of an approximate feedback linearization control of the nonlinear singularly perturbed (SP) systems. The proposed method is based on a decomposition of the original SP system and construction of AFL control in the form of composite FL controls for the slow and fast subsystems. In general, a nonlinear SP system cannot be easily separated into slow and fast subsystems, because the conditions of Tikhonov theorem are not complied. In order to overcome this, the author proposes to perform the feedback linearization method at first for the system's part, which describes the fast state variables. Thus, a fast control is chosen, so that the conditions of Tikhonov theorem would be met. Then, using a standard singular perturbation technique, we obtain a slow subsystem. Further the problem of FL control for a slow subsystem is solved. The resulting AFL control is obtained in the form of a composite control. Application of the proposed approach is illustrated with two examples.

Aiming at high-speed signal sampling system with internal parameter perturbation and external disturbance, a novel delta operator saturated variable rate reaching law based on sigmoid function is presented, and soft sliding mode controller design is addressed for delta operator systems with uncertainties in the paper. The structural feature of uncertain delta operator system is concerned. Soft sliding mode controller design for continuous-time system and discrete-time system can be converted to the case of delta operator system. The stability of uncertain delta operator soft sliding mode control (SMC) system is guaranteed by Lyapunov stability theory. Soft sliding mode controller design for uncertain delta operator systems is achieved by the proposed reaching law under a certain matching condition. It is illustrated that delta operator soft SMC system by the proposed reaching law in a simulation example can not only reach the scope of the switching plane in limited time, but also reduce system chattering effectively. The system on stability and process steady is better than that of delta operator exponential reaching law and delta operator variable rate reaching law. At the same time, the system has a good completely robustness compared with delta operator statured reaching law.

This paper combines a Smith Predictor with an approach for graphically determining all Proportional-Integral-Derivative (PID) controllers in either the Continuous-Time (CT) or Discrete-Time (DT) domains that meet Gain Margin (GM) and Phase Margin (PM) specifications. Smith Predictors (SP) are often used when designing controllers for systems that exhibit "relatively" large delays that may cause the system's stability and/or performance to deteriorate. The PID controller gains, namely Proportional gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> ), Integral gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ) and Derivative gain (K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> ), will be determined graphically using only the frequency response of the system's components, i.e., plant with delay and SP structure. The inclusion of a SP along with a PID controller can significantly improve stability margins and performance when compared to relying solely on a PID controller. The improvement can be observed even if there is a mismatch between the actual process and its corresponding SP model. By using the delta operator, the procedure outlined applies to both CT and DT systems, hence, the unified approach designation. Additionally, this procedure will ensure meeting predesign requirements in terms of gain and phase margins. The stability boundaries of the PID controller gains will be determined graphically where within the boundaries, nominal stability is guaranteed and stability margins requirements are met.

In this paper the H/sub /spl infin//-optimal control and risk-sensitive control of linear singularly perturbed, discrete-time systems are considered. It is shown that the Riccati equation associated with the solution of the H/sub /spl infin//-optimal control problem, can be approximated by an outer series solution, and a boundary-layer correction series solution. As the singular perturbation parameter /spl epsiv/ tends to zero, it is shown that the full-order Riccati equation can be approximated by the zeroth-order outer-series solution. It is also recognized that the outer-series solution and the boundary-layer solution, are equivalent to the resulting asymptotic series expansion in powers of /spl epsiv/, of the Riccati equation associated with a certain risk-sensitive control problem. When a two-time-scale decomposition is considered, it is shown that an approximate composite control can be formed, as the sum of the slow and fast subsystems. Unlike the asymptotic expansion approach, it is shown that the original system can be approximated by the maximum of the performance associated with the slow and fast subsystems. Finally, it is shown that the slow and fast subsystems, are equivalent to certain slow and fast subsystems, respectively, resulting from applying a two-time-scale decomposition to a certain risk-sensitive control problem. >

In this paper, the regulation problem of a class of nonlinear singularly perturbed discrete-time systems is investigated. Using the theory of singular perturbations and time scales, the nonlinear system is decoupled into reduced-order slow and fast (boundary layer) subsystems. Then, a composite controller consisting of two sub-controllers for the slow and fast subsystems is developed using the discrete-time state-dependent Riccati equation (D-SDRE). It is proved that the equilibrium point of the original closed-loop system with a composite controller is locally asymptotically stable. Moreover, the region of attraction of the closed-loop system is estimated by using linear matrix inequality. One example is given to illustrate the effectiveness of the results obtained.

The object of this paper is to find the stability regions of an arbitrary order transfer function with time delay using a unified approach for continuous and discrete-time systems. The stability boundaries of the proportional-integral (PI) controller, proportional-derivative (PD) controller and proportional-integral-derivative (PID) controller are found in terms of the proportional gain ( K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">p</inf> ), integral gain ( K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> ) and derivative gain ( K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">d</inf> ). The delta operator is used to describe the controllers because it provides not only numerical properties superior to the discrete time shift operator, but also converges to the continuous time derivative operator as the sampling period approaches zero. A key advantage of this approach is that stability boundaries can be found when only the frequency response and not the parameters of the plant transfer function are known. A unified approach allows us to use the same procedure for finding the discrete time and continuous time stability regions. If the plant transfer function is known, the stability regions can be found analytically. Regions where phase and gain margin specifications are met can also be found.

This paper studies the guaranteed cost control for singularly perturbed uncertain systems. The system is decomposed into slow and fast sub-systems. The uncertainties in the fast subsystem are transferred to the composite control gain and are considered as the controller gain perturbations. The composite control is achieved based on the slow subsystem under controller gain perturbations and the fast subsystem

This article is devoted to the problem of composite control design for continuous nonlinear singularly perturbed (SP) system using approximate feedback linearization (AFL) method. The essence of AFL method lies in the feedback linearization only of a certain part of the original nonlinear system. According to AFL approach, we suggest to solve feedback linearization problems for continuous nonlinear SP system by reducing it to two feedback linearization problems for slow and fast subsystems separately. The resulting AFL control is constructed in the form of asymptotic composition (composite control). Standard procedure for the composite control design consists of the following steps: 1) system decomposition, 2) solution of control problem for fast subsystem, 3) solution of control problem for slow subsystem, 4) construction of the resulting control in the form of the composition of slow and fast controls. The main difficulty during system decomposition is associated with dynamics separation condition for nonlinear SP system. To overcome this, we propose to change the sequence of the design procedure: 1) solving the control problem for fast state variables part, 2) system decomposition, 3) solving the control problem for slow state variables part, 4) construction of the resulting composite control. By this way, fast feedback linearizing control is chosen so that the dynamics separation condition would be met and the fast subsystem would be stabilizable. The application of the proposed approach is illustrated through several examples.

The stability of an inverted pendulum is a textbook example of control. The easiest case is to put the pendulum on a cart and apply feedback force control on it to keep the upright position stable. This paper compares the no delay case (feed‐in‐time control: an anticipatory effect) and the delay differential equation approach. Then we study both continuous and discrete time systems. The main aim of the work is to investigate the behaviour of such systems at the stability boundaries by using numerical simulation. The principal points of interest are how continuous time systems differ from discrete time system at a bifurcation point and how time delay or an anticipatory feed‐in‐time control acts on its behaviour. Other exciting questions are how sampling delay can be taken into consideration and is bifurcation a robust phenomenon.

A delta operator based Discrete Sliding Mode Fuzzy Controller (DSMFC) for dc-dc buck Converter is described. The discrete systems by means of shift operator turns into mathematically disordered at an unexpected sampling rates. In order to face this difficulty, the delta operator is introduced. The delta operator can convert continuous-time system into discrete-time system and vice-versa. The main aim is to increase the rate of response, develop efficiency, get excellent Robustness and chattering phenomenon inhibition. This method is easy to implement and the overall system is efficient and cost-effective. During discretization, to circumvent any data damage, quick sampling has to be done which can be achieved with the help of delta operator because of its well defined mathematical properties even at rapid sampling.

We study time-scale separation and robust controller design for a class of singularly perturbed nonlinear systems under perfect state measurements. The system dynamics are taken to be jointly linear in the fast state variables, control and disturbance inputs, but nonlinear in the slow state variables. Since global timescale separation may not always be possible for nonlinear singularly perturbed systems, we restrict our attention here to some closed subset of the state space, on which a timescale separation holds for sufficiently small values of the singular perturbation parameter. We construct a slow controller and a composite controller based on the solutions of particular slow and fast games obtained using time-scale separation. For the class of systems for which the slow controller can be selected to be robust with respect to small regular structural perturbations on the slow subsystem, we show under some growth conditions that the composite controller can achieve any desired level of performance that is larger than the maximum of the performance levels for the slow and fast subsystems,. A slow controller, however, is not generally as robust as the composite controller; but, still under some conditions which are delineated in the paper, the fast dynamics can be totally ignored. The paper also presents a numerical example to illustrate the theoretical results.

The contributions of this paper are threefold. We first explore the differences in uncertainty representations and control designs between continuous time systems and discrete time systems, particularly the high frequency interpretation does not work for discrete time systems. It is shown that uncertainties appeared in sampled data systems may be highly structured, however they do not fall into any category which is currently available in the literature. Secondly, we define a measure on these uncertainties which confines a neighborhood to the nominal model in terms of frequency content, and then we propose a control design method which guarantees the stability of the closed loop system in the face of uncertainties. Last, we consider a generalization of a classical continuous time approach in representing the multiplicative uncertainty to sampled data systems by using a slightly modified sampler and present a method to calculate a frequency bound on the uncertainty in sampled data systems.

Steady-state, harmonic response and moments of linear systems with periodic jumps

This paper investigates the robust stabilization of uncertain time-delayed systems via delta operator approach. By introducing model transportation, the linear delta operator system with time-varying delays is reformulated into an interconnected system for which the uncertainties can become easy to deal with. Based on a two-term approximation and scaled small gain theorem, a new delay-dependent sufficient condition of state feed-back stabilization for an uncertain delta operator time-delayed system is established by using a novel Lyapunov-Krasovskii functional. The condition obtained can unify some previously suggested relevant methods seen in literature for achieving asymptotic stabilization of both continuous and discrete systems into the delta operator framework. Numerical examples presented explicitly demonstrate the advantages and effectiveness of the proposed method.