A new approach to adaptive control: no nonlinearities

In adaptive control the goal is to design a controller to control an uncertain system whose parameters may be changing with time. Typically the controller consists of an identifier (or tuner) which is used to adjust the parameters of a linear time-invariant (LTI) compensator, and under suitable assumptions on the plant model uncertainty it is proven that good asymptotic behaviour is achieved, such as model matching (for minimum phase systems) or stability. However, a typical adaptive controller does not track time-varying parameters very well, and it is often highly nonlinear, which can result in undesirable behaviour, such as large transients or a large control signal. Furthermore, most adaptive controllers provide only asymptotic tracking, with no ability to design for a pre-specified settling time.Here we propose an alternative approach, which yields a linear periodic controller. Rather than estimating the plant or compensator parameters, instead we estimate what the control signal would be if the plant parameters were known. In this paper we argue the utility of this approach and then examine the first order case in detail, including a simulation. We also explore the benefits and limitations of the approach.     

An adaptive controller which provides an arbitrarily good transientand steady-state response

A model reference adaptive control problem is posed. In the problem, the objective is not the usual one of forcing the error between the plant output and the reference model output asymptotically to zero, but instead, it is that of forcing this error to be less than a (arbitrarily small) prespecified constant after a (arbitrarily short) prespecified period of time, with a (arbitrarily small) prespecified upper bound on the amount of overshoot. It is shown that to achieve this goal for a stabilizable and detectable, single-input single-output linear time-invariant (LTI) plant, it is necessary and sufficient that the plant be minimum phase. Knowledge of an upper bound on the plant order, of the relative degree, and of the sign of the high-frequency gain is not required. The controller proposed consists of an LTI compensator together with a switching mechanism to adjust the compensator parameters. If an upper bound on the relative degree is available, the compensator has dynamics of order equal to this upper bound less one; otherwise, the order of the compensator is adjusted as well as its parameters     

Multiobjective problems: Linear and nonlinear control

In this note, we compare the value of nonlinear and linear control for a multiobjective control problem with a linear time-invariant (LTI) plant. We show that if the performance of a nonlinear feedback controller is measured by the maximum incremental gain, the optimal achievable performance with nonlinear, time-varying control is identical to that achievable by LTI control.     

Adaptive control of linear time-varying plants: a new modelreference controller structure

The problem of developing a control law which can force the output of a linear time-varying plant to track the output of a stable linear time-invariant reference model is discussed. It is shown that the standard model reference controller, used for linear time-invariant plants, cannot guarantee zero tracking error in general when the plant is time-varying. A new model reference controller is proposed which guarantees stability and zero tracking error for a general class of linear time-varying plants with known parameters. When the time-varying plant parameters are unknown but vary slowly with time, it is shown that the new controller can be combined with a suitable adaptive law so that all the signals in the closed loop remain bounded for any bounded initial conditions and the tracking error is small in the mean. The assumption of slow parameter variations in the adaptive case can be relaxed if some information about the frequency or the form of the fast varying parameters is available a priori. Such information can be incorporated in an appropriately designed adaptive law so that stability and improved tracking performance is guaranteed for a class of plants with fast varying parameters     

Adaptive stabilization using a nonlinear time-varying controller

Byrnes et al. (1986) showed that there is no smooth, finite-dimensional, nonlinear time-invariant (NLTI) controller which asymptotically stabilizes every finite-dimensional, stabilizable and detectable, linear time-invariant (LTI) plant (with a fixed number of inputs and outputs). Here we construct a finite-dimensional nonlinear time-varying (NLTV) controller which does exactly that; we treat both the discrete-time and continuous-time cases. With p equal to one in the discrete-time case and the number of plant outputs in the continuous-time case, we first show that for every stabilizable and detectable plant, there exists a p-dimensional linear time-varying (LTV) compensator which provides exponential stabilization; we then construct a (p+1)-dimensional NLTV controller which asymptotically stabilizes every admissible plant by switching between a countable number of such LTV compensators     

Adaptive linear control of nonlinear systems

The theory of stable parameter adaptive control has advanced to allow linear time-varying plants. However, a more honest view of such systems is that they are often derived from inexact linearization about a trajectory of a nonlinear system. Standard adaptive control based on a linear model can then be interpreted as one way to realize a nonlinear controller for a nonlinear plant. The implications of this view are studied. Analytically, the stability problem is seen to be equivalent to showing robustness to time-varying parameters and a locally bounded model uncertainty. It is shown that if the trajectory is known to be within a bound, a parameter estimator with projection can ensure boundedness of departures from the trajectory     

An adaptive controller which provides Lyapunov stability

An adaptive controller that can provide exponential Lyapunov stability for an unknown linear time-invariant (LTI) system is presented. The only required a priori information about the plant is that the order of an LTI stabilizing compensator be known, although this can be reduced to assuming only that the plant is stabilizable and detectable at the expense of using a more complicated controller. This result extends the work of M. Fu and B.R. Barmish (see ibid., vol.AC-31, p.1097-1103, Dec. 1986) in which it is shown that there exists an adaptive controller which provides exponential Lyapunov stability if it is assumed that an upper bound on the plant order is known and that the plant lies in a known compact set; it shows that adaptive stabilization is possible under very mild assumptions without large state deviations     

Feedback based adaptive compensation of control system sensor uncertainties

In this paper, the problem of adaptively compensating sensor uncertainties is addressed in a feedback based framework. In this study, sensor characteristics are modeled as parametrizable uncertain functions and a compensator is constructed to adaptively cancel the effects of sensor uncertainties, to generate an adaptive estimate of the plant output. Such an estimated output is used for the feedback control law. Adaptive control schemes using a model reference approach with sensor uncertainty compensation are developed for LTI plants with either known or unknown plant dynamics. A new feedback controller structure is developed for the case when the plant dynamics is unknown, to handle the plant and sensor uncertainties. Simulation results are presented to show that the proposed adaptive sensor uncertainty compensation designs significantly improve system tracking performance.     

An efficient LMI approach for the quadratic stabilisation of a class of linear,uncertain, time-varying systems

The purpose of this article is to provide a numerically efficient method for the quadratic stabilisation of a class of linear, discrete-time, uncertain, time-varying systems. The considered class of systems is characterised by an interval time-varying (ITV) matrix and constant sensor and actuator matrices. It is required to find a linear time-invariant (LTI) static output feedback controller yielding a quadratically stable closed-loop system independently of the parameter variation rate. The solvability conditions are stated in terms of linear matrix inequalities (LMIs). The set of LMIs includes the stability conditions for the feedback connection of a unique suitably defined extreme plant with an LTI output controller and the positivity of a closed-loop extremal matrix. A consequent noticeable feature of the article is that the total number of LMIs is independent of the number of uncertain parameters. This greatly enhances the numerical efficiency of the design procedure.     

一阶伪线性系统的模型参考输出跟踪控制

针对伪线性系统的模型参考输出跟踪问题,设计伪线性系统的模型参考跟踪策略.控制器分为两部分,其一为反馈镇定控制器,保证闭环系统是渐近稳定的;另一为前馈补偿控制器,通过求解基于控制器存在条件建立的方程组得到控制器的参量矩阵,使得闭环系统的输出渐近跟踪参考系统的输出,且当系统中存在时变系数时方法仍是有效的,控制器中保有部分自由度可以进一步利用,以提高具体控制任务所需的系统性能.数值仿真验证了所提出方法的有效性.     

The robust control of a servomechanism problem for linear time-invariant multivariable systems

Necessary and sufficient conditions are found for there to exist a robust controller for a linear, time-invariant, multivariable system (plant) so that asymptotic tracking/regulation occurs independent of input disturbances and arbitrary perturbations in the plant parameters of the system. In this problem, the class of plant parameter perturbations allowed is quite large; in particular, any perturbations in the plant data are allowed as long as the resultant closed-loop system remains stable. A complete characterization of all such robust controllers is made. It is shown that any robust controller must consist of two devices 1) a servocompensator and 2) a stabilizing compensator. The servocompensator is a feedback compensator with error input consisting of a number of unstable subsystems (equal to the number of outputs to be regulated) with identical dynamics which depend on the disturbances and reference inputs to the system. The sorvocompensator is a compensator in its own right, quite distinct from an observer and corresponds to a generalization of the integral controller of classical control theory. The sole purpose of the stabilizing compensator is to stabilize the resultant system obtained by applying the servocompensator to the plant. It is shown that there exists a robust controller for "almost all" systems provided that the number of independent plant inputs is not less than the number of independent plant outputs to be regulated, and that the outputs to be regulated are contained in the measurable outputs of the system; if either of these two conditions is not satisfied, there exists no robust controller for the system.     

Stabilising compensators for linear time-varying differential systems

In this paper, we describe a constructive test to decide whether a given linear time-varying (LTV) differential system admits a stabilising compensator for the control tasks of tracking, disturbance rejection or model matching and construct and parametrise all of them if at least one exists. In analogy to the linear time-invariant (LTI) case, the ring of stable rational functions, noncommutative in the LTV situation, and the Ku?era–Youla parametrisation play prominent parts in the theory. We transfer Blumthaler's thesis from the LTI to the LTV case and sharpen, complete and simplify the corresponding results in the book ‘Linear Time-Varying Systems’ by Bourlès and Marinescu.     

Nonlinear controller tuning based on a sequence of identifications of linearized time-varying models

A novel algorithm for tuning controllers for nonlinear plants is presented. The algorithm iteratively minimizes a criterion of the control performance. In each iteration one experiment is performed with a reference signal slightly different from the previous reference signal. The input–output signals of the plant are used to identify a linear time-varying model of the plant which is then used to calculate an update of the controller parameters. The algorithm requires an initial feedback controller that stabilizes the closed loop for the desired reference signal and in its vicinity, and that the closed-loop outputs are similar for the previous and current reference signals. The tuning algorithm is successfully tested on a laboratory set-up of the Furuta pendulum.     

Robust multi-parametric model predictive control for LPV systems with application to anaesthesia

We present a multi-parametric model predictive controller (mpMPC) for discrete-time linear parameter-varying (LPV) systems based on the solution of the mpMPC problem for discrete-time linear time-invariant (LTI) systems. The control method yields a controller that adapts to parameter changes of the LPV system. This is accomplished by an add-on unit to the implementation of the mpMPC for LTI systems. No modification of the optimal mpMPC solution for LTI systems is needed. The mpMPC for LPV systems is entirely based on simple computational steps performed on-line. This control design method could improve the performance and robustness of a mpMPC for LPV systems with slowly varying parameters. We apply this method to process systems which suffer from slow variation of system parameters due, for example, to aging or degradation. As an illustrative example the reference tracking control problem of the hypnotic depth during intravenous anaesthesia is presented: the time varying system matrix mimics an external disturbance on the hypnotic depth. In this example the presented mpMPC for LPV systems shows a reduction of approximately 60% of the reference tracking error compared to the mpMPC for LTI systems.     

Robust stabilization of multivariable uncertain plants via switching control

In general, interval plants with a large uncertainty can not be robustly stabilized by a linear, time-invariant controller. The solution presented in this paper is in terms of an overall non linear control law given by the connection of a linear, deterministic, time-varying compensator and of a supervisor. The compensator may assume a finite number of possible different configurations, at least one of the which stabilizes the plant. The task of the supervisor is that of determining the switching instants of a scanning among the elements of the family, according to a suitably defined experimental test on the output of the system.     

Periodic Compensation of Continuous-Time Plants

This note presents a periodic compensator which achieves robust stability for single-input-single-output (SISO), linear time invariant (LTI) plants having both right-half plane (RHP) poles and zeros, a job LTI controllers fail to do. In addition, for strictly proper plants this controller achieves model matching ensuring at the same time that the periodic oscillations present in the plant output are insignificant in magnitude. The design steps are straightforward and linear algebraic in nature     

Adaptive asymptotic tracking of repetitive signals-a frequencydomain approach

The problem of designing an adaptive tracking controller for repetitive signals is considered. It is assumed that the controlled plant is linear time-invariant (LTI) and that its steady-state behavior can be described by its frequency response. The proposed controller, being able to identify the gain and phase of a plant at selected frequencies, adjusts the input/output (I/O) map at frequencies contained in the reference signal so as to achieve zero steady-state errors. Hence, no structural knowledge of the plant is needed. However, the proposed controller does not alter the transient property of the plant, e.g., locations of poles. The transient property can be improved by adding feedback control action. A combination of the adaptive controller and a feedback LTI controller is studied     

Linear periodic controller design for decentralized control systems

In the decentralized control of linear time-invariant (LTI) systems, a decentralized fixed mode (DFM) is a system mode which is immoveable using an LTI controller, while a quotient DFM (QDFM) is one which is immoveable using any form of nonlinear time-varying compensation. If a system has no unstable DFMs, there are well-known procedures for designing an LTI stabilizing controller; for systems which have unstable DFMs but no unstable QDFMs, we provide a simple design algorithm which yields a linear periodic sampled-data stabilizing controller.     

Robust backstepping output tracking control for SISO uncertain nonlinear systems with unknown virtual control coefficients

The output tracking controller design problem is dealt with for a class of nonlinear strict-feedback form systems in the presence of nonlinear uncertainties, external disturbance, unmodelled dynamics and unknown time-varying virtual control coefficients. A new method based on signal compensation is proposed to design a linear time-invariant robust controller, which consists of a nominal controller and a robust compensator. It is shown that the closed-loop control system with a controller designed by the proposed method has robust asymptotical practical tracking property for any bounded initial conditions and robust tracking transient property if all initial states are zero.