Slow periodic motions in variable structure systems

Singularly perturbed relay control systems (SPRCS) with stable periodic motion in reduced systems are studied here. The slow motions integral manifold of such systems consists of parts that correspond to different values of control and the solutions of SPRCS contain the jumps from one part of the slow manifold to the other. The theorems about existence and stability of the slow periodic solutions are proved. An algorithm of asymptotic representation for this periodic solutions using a boundary layer method is suggested.     

FREQUENCY DOMAIN ANALYSIS OF FAST AND SLOW MOTIONS IN SLIDING MODES

A frequency domain approach to analysis of fast and slow motions in sliding modes is proposed in the paper. The proposed approach is based on the construction of a relay system that is equivalent to the original sliding mode control system and on the application of the locus of a perturbed relay system method. A precise model of the fast motions and a non‐reduced order model of the slow motions for the SISO linear time‐invariant case are obtained as a result of the proposed approach.     

Slow periodic motions with internal sliding modes in variable structure systems

Singularly perturbed relay systems (SPRS) in which the reduced systems have the stable periodic motions with internal sliding modes are studied. The slow motion integral manifold of such systems consists of the parts which correspond to the different values of relay control and the solutions may contain the jumps from one part of the slow manifold to another. For such systems a theorem about existence and stability of the periodic solutions is proved. An algorithm of asymptotic representation for this periodic solutions using boundary layer method is presented. It is proved that in the neighbourhood of the break away point the asymptotic representation starts with the first order boundary layer function.     

Limit cycles with chattering in relay feedback systems

Relay feedback has a large variety of applications in control engineering. Several interesting phenomena occur in simple relay systems. In the paper, scalar linear systems with relay feedback are analyzed. It is shown that a limit cycle where part of the limit cycle consists of fast relay switchings can occur. This chattering is analyzed in detail and conditions for approximating it by a sliding mode are derived. A result on existence of limit cycles with chattering is given, and it is shown that the limit cycles can have arbitrarily many relay switchings each period. Limit cycles with regular sliding modes are also discussed. Examples illustrate the results.     

Analysis of sliding modes in the frequency domain

A frequency-domain approach to the analysis of real and ideal sliding modes (SM) in single input single output (SISO) systems with linear plants is proposed. The real SM is analysed as a superposition of fast oscillations and slow motions in a relay system. The approach is based on the notion of the locus of a perturbed relay system (LPRS). The proposed LPRS is defined as a characteristic of the response of a linear part to an unequally spaced pulse control of variable frequency in a closed-loop. Formulas for computing the LPRS are derived. It is demonstrated that beside the well know chattering phenomenon, real SM also exhibits a non-ideal disturbance rejection, which can be quantitatively assessed with the use of the proposed approach. An example of the LPRS analysis of a SM system is given.     

Analysis of Chattering in Systems With Second-Order Sliding Modes

A systematic approach to the chattering analysis in systems with second-order sliding modes is developed. The neglected actuator dynamics is considered to be the main cause of chattering in real systems. The magnitude of oscillations in nonlinear systems with unmodeled fast nonlinear actuators driven by second-order sliding-mode control generalized suboptimal (2-SMC G-SO) algorithms is evaluated. Sufficient conditions for the existence of orbitally stable periodic motions are found in terms of the properties of corresponding Poincare maps. For linear systems driven by 2-SMC G-SO algorithms, analysis tools based on the frequency-domain methods are developed. The first of these techniques is based on the describing function method and provides for a simple approximate approach to evaluate the frequency and the amplitude of possible periodic motions. The second technique represents a modified Tsypkin's method and provides for a relatively simple, theoretically exact, approach to evaluate the periodic motion parameters. Examples of analysis and simulation results are given throughout this paper.     

Maximum periodic solutions of the Volterra integrodifferential equations in the critical case of a pair of pure imaginary roots

Systems with aftereffect are considered, which are described by integrodifferential equations of the Volterra type in the presence of a small perturbation prescribed by the periodic (or maximum periodic) time function. In the critical case of a pair of pure imaginary roots of a characteristic equation, the question is solved as to existence in the system of maximum periodic motions (i.e., motions tending at the unlimited increase of time to periodic modes) provided that the frequency of the periodic part of disturbance coincides with the natural frequency of a linearized homogeneous system. It is shown that in the analytic case the equations of motion of the system possess a set of the maximum periodic solutions representable by power series in a small parameter specifying a value of the disturbance, and in small arbitrary initial values of uncritical variables of the problem. The conditions of the existence of such solutions are indicated, which are defined by the terms up to the third order of equations.     

广义非线性系统的变结构控制理论

利用几何方法,从广义非线性系统本身出发,研究了广义非线性控制的变化结构控制理论,给出了系统存在变结构控制的充分实际没动模式的近似定理。从所得结论可知,滑动条件仅能保证实际滑动模的慢变状态赵近于理想滑动模的慢变状态,而不能保证实际滑动模的状态趋近于理想滑动模的快变状态,研究正常非线性系统的方法已不能简单地被利用到广义非线性系统。     

Generating self-excited oscillations for underactuated mechanical systems via two-relay controller

A tool for the design of a periodic motion in an underactuated mechanical system via generating a self-excited oscillation of a desired amplitude and frequency by means of the variable structure control is proposed. First, an approximate approach based on the describing function method is given, which requires that the mechanical plant should be a linear low-pass filter–the hypothesis that usually holds when the oscillations are relatively fast. The method based on the locus of a perturbed relay systems provides an exact model of the oscillations when the plant is linear. Finally, the Poincaré map's design provides the value of the controller parameters ensuring the locally orbitally stable periodic motions for an arbitrary mechanical plant. The proposed approach is shown by the controller design and experiments on the Furuta pendulum.     

Optimal control of ε-coupled and singularly perturbed distributed-parameter systems

Many distributed-parameter systems consist of interconnected subsystems involving fast and slow physical phenomena or reducing to a number of independent subsystems when a scalar parameter ε is zero. The purpose of this paper is to treat the control of such systems by invoking the ε-coupling and singular perturbation approaches developed by Kokotovic and his co-workers for lumped-parameter large-scale systems. In the case of ε- coupled distributed-parameter systems it is shown that the optimal state feedback matrix can be approximated by a Volterra-MacLaurin series with coefficients determined by solving two lower-order decoupled Riccati and linear equations. By using an mth-order approximation of the optimal feedback matrix, one obtains a (2m+1)th order approximation of the optimal performance function. In the singular perturbation approach the result is that for an O(ε²) suboptimal control one must solve two decoupled Riccati equations, one for the fast and one for the slow subsystem, and then construct appropriately the composite control law. By using only the Riccati equation for the slow subsystem, one obtains an O(ε) suboptimal control. The singular perturbation technique is then used to treat interconnected distributed-parameter systems involving may strongly coupled slow subsystems and weakly coupled fast subsystems.     

Chattering analysis in sliding mode systems with inertial sensors

Singularly perturbed relay control systems with second order sliding modes (SP2SM) are considered for modeling of sliding mode control systems with inertial sensors. It is shown that the asymptotically stable slow-motion integral manifold of a smooth singularly perturbed system, describing the motion of an original SP2SM in the second order sliding domain, is the asymptotically stable slow-motions integral manifold of the original SP2SM. For sliding mode control systems with inertial sensors sufficient conditions for the exponential decreasing of the amplitude of chattering and unlimited growth of frequency are found. A formula for asymptotic representation of 'ideal' switching surface oscillations is suggested for sliding mode systems with inertial sensors.     

In-the-large stability of relay and saturating control systems with linear controllers

It is shown that when the plant of a relay control system consists of a pure integrators, with n ≥ 3, and the control signal proceding the relay is a linear combination of state coordinates, the system is unstable in-the-large for all values of the controller coefficients. The same result holds if the relay is replaced by a. saturator.

The method used is to prove that (except for a degenerate case) state trajectories exist which go to infinity. This method enables more general systems to be treated than those discussed in a previous paper, which demonstrated the existence of a periodic solution.

Plants not consisting of pure integrators are also considered. The question of when it is worthwhile to incorporate non-linearities in the controller is discussed.     

On periodic motions in relay systems containing blocks with limiters

A method of studying periodic motions in relay systems having a block with limiters is proposed. The method is based on the phase locus of the relay system. The means for constructing the phase locus, for determining with its help periodic motions arising in the system, and for studying their stability are considered.     

Boundary Control for A Flexible Inverted Pendulum System Based on A Pde Model

A partial differential equation (PDE) model for a flexible inverted pendulum system (FIPS) is derived by the use of the Hamilton principle. To solve the coupling system model, a singular perturbation method was adopted. The PDE model was divided into a fast subsystem and a slow subsystem using the singular perturbation method. To stabilize the fast subsystem, a boundary control force was applied at the free end of the beam. It then was proven that the closed‐loop subsystem is appropriate and exponentially stable. For the slow subsystem, a sliding mode control method was employed to design a controller and the Linear Matrix Inequality (LMI) method was used to design the sliding surface. It then was shown that the slow subsystem is exponentially stable.     

Boundary Control for a Flexible Inverted Pendulum System Based on a PDE Model with Input Saturation

This research considers the control problem of a flexible inverted pendulum system (FIPS) in the presence of input saturation. The model for a flexible inverted pendulum system (FIPS) is derived via the Hamilton principle. The FIPS model is divided into a fast subsystem and a slow subsystem via the singular perturbation method. We introduce an auxiliary system to deal with the input saturation of a fast subsystem. To stabilize the fast subsystem, a boundary anti‐windup control force is applied at the free end of the beam. It is proven that the closed‐loop subsystem is stable. For the slow subsystem, a sliding mode control method is employed to design a controller and a new decoupling method to design the sliding surface. Then it is shown that the slow subsystem is stable. Finally, simulation results are provided to confirm the efficacy of the proposed controller.     

Sliding surface design for singularly perturbed systems

The equilibrium manifold of a singularly perturbed system has a close relationship with the sliding surface of a variable structure system (VSS). The fast time and slow timeresponses has a similar behaviour to the 'reaching mode' and 'sliding mode', respectively. This paper aims to equip the powerful composite control method with robustness through variable structure control design. The major bridge in between is a Lyapunov function. It is found that a singularly perturbed system in sliding mode may preserve two-time-scale attribute, in which a new equilibrium manifold exists on the sliding surface. Sliding motions that are attracted to the manifold can therefore be referred to as 'sliding mode in sliding mode'.     

Analysis of periodic motions in relay feedback systems with saturation in plant dynamics

In this article a method of analysis of possible periodic motions in relay feedback systems that have saturation in plant dynamics is presented. Also, a methodology of analysis of local orbital stability of those periodic motions is proposed. The given analysis is based on the concept of the state locus of a relay feedback system. Techniques for computing the state locus are proposed. Also, a methodology of finding a periodic motion that involves the state locus is given. Finally, the proposed method is illustrated by an example.     

Containment Control of General Linear Multi-agent Systems with Multiple Dynamic Leaders: a Fast Sliding Mode Based Approach

In this paper, distributed containment control problems of general linear multi-agent systems are investigated. The objective is to make the followers in a multi-agent network converge to the convex hull spanned by some leaders whose control inputs are nonzero and not available to any followers. Sliding mode surfaces are defined for the cases of reduced order and non-reduced order, respectively. For each case, fast sliding mode controllers are designed. It is shown that all the error trajectories exponentially reach the sliding mode surfaces in a finite time if for each follower, there exists at least one of the leaders who has a directed path to the follower, and the leaders' control inputs are bounded. The control Lyapunov function for exponential finite time stability, motivated by the fast terminal sliding mode control, is used to prove reachability of the sliding mode surfaces. Simulation examples are given to illustrate the theoretical results.      

Sliding mode control for Markovian jumping systems with actuator nonlinearities

This article considers the design of sliding mode control for stochastic Markovian jumping systems with actuator nonlinearities. In the design of integral sliding surfaces, a set of specified matrices are employed such that the connections among sliding surfaces corresponding to each mode are established. And then, a sliding mode controller depending on the transition rates is synthesised such that the reachability of the specified sliding surface can be ensured. Moreover, sufficient conditions for the stability of the sliding motions are derived. It is shown that the effect of Markovian switching can be coped with by the present sliding mode control method. Finally, the simulation results illustrating the proposed method are provided.