Self-tuning control of nonminimum-phase systems

The instability that arises from the pole-zero cancellation of elementary self-tuners and most MRAC algorithms has given rise to a misapprehension that nonminimum-phase systems can pose insoluble problems with the application of adaptive control. Since most practical processes exhibit some form of nonminimum-phase behaviour (particularly when controlled digitally) this appeared to restrict the usefulness of self-tuning methods. However, during the last few years new algorithms have been suggested which partly overcome these difficulties and have been shown to be effective in industrial trials. The paper describes the practical features that a self-tuning controller must possess and discusses how plant dead-time and excess continuous-time poles can lead to discrete-time nonminimum-phase zeros. The source of instability of elementary self-tuners is analysed and various suggested approaches for overcoming this problem are reviewed. These include the use of a generalized minimum variance cost-function incorporating control weighting, factorization methods which avoid cancellation of the offending zeros, adaptive control based on state-space linear quadratic Gaussian (LQG) theory, and explicit pole-placement via solution of a Diophantine or Bezoutian equation. Gawthrop's hybrid controller, which avoids those nonminimum-phase zeros due to continuous-time pole excess, is briefly discussed. A series of simulations in which a nonminimum-phase plant is subjected to a pattern of set-point changes is presented to show some of the features of the algorithms. In applications the order of the process is unlikely to be known (or indeed be finite); the pole-placement algorithm which is particularly effective in the known-order case is shown to lack robustness to underspecified order. The use of a large control weighting however restores stability to the generalized minimum-variance controller. The conclusions are that with appropriate design and engineering judgement adaptive control is viable even for nonminimum-phase processes.     

On the convergence properties of adaptive pole-placementcontrollers with antiwindup compensators

To avoid the windup problem in adaptive pole-placement controllers in the presence of a saturating input, an antiwindup compensator based on a generalization to the conditioning technique is introduced to the controller. This modification to the controller also provides a unified approach to investigate the asymptotic properties of the adaptive controllers with a class of antiwindup compensators, at least, when applied to stable processes     

Self-convergence of weighted least-squares with applications tostochastic adaptive control

A recursive least-squares algorithm with slowly decreasing weights for linear stochastic systems is found to have self-convergence property, i.e., it converges to a certain random vector almost surely irrespective of the control law design. Such algorithms enjoy almost the same nice asymptotic properties as the standard least-squares. This universal convergence result combined with a method of random regularization then easily can be applied to construct a self-convergent and uniformly controllable estimated model and thus may enable us to form a general framework for adaptive control of possibly nonminimum phase autoregressive-moving average with exogenous input (ARMAX) systems. As an application, we give a simple solution to the well-known stochastic adaptive pole-placement and linear-quadratic-Gaussian (LQG) control problems in the paper     

Constrained predictive pole-placement control with linear models

Predictive pole-placement (PPP) control is a continuous-time MPC using a particular set of basis functions leading to pole-placement behaviour in the unconstrained case. This paper presents two modified versions of the PPP controller which are each shown to have desirable stability properties when controlling systems with input, output and state constraints.     

A robust indirect adaptive regulation algorithm withself-excitation capability

A robust version of the adaptive control algorithm established by G. Kreisselmeier (1989) is presented. If the conventional dead-zone adaptive law is used in the adaptive system, it is troublesome that the size of the unmodeled dynamics, denoted by ∈, must be within the chosen dead-zone size for robustness. In this robust version, by introducing a positive design parameter, the robust stability can be achieved without this constraint. It is also shown that the plant output and control input will converge asymptotically within a certain bound. Moreover, in the ideal case, the plant output and control input will converge to zero under some condition     

Adaptive RBFNN versus conventional self-tuning: comparison of two parametric model approaches for non-linear control

In this work a practical study evaluates two parametric modelling approaches — linear and non-linear (neural) — for automatic adaptive control. The neural adaptive control is based on a developed hybrid learning technique using an adaptive (on-line) learning rate for a Gaussian radial basis function neural network. The linear approach is used for a self-tuning pole-placement controller. A selective forgetting factor method is applied to both control schemes: in the neural case to estimate on-line the second-layer weights and in the linear case to estimate the parameters of the linear process model. These two techniques are applied to a laboratory-scaled bench plant with the possibility of dynamic changes and different types of disturbances. Experimental results show the superior performance of the neural approach particularly when there are dynamic changes in the process.     

Model reference adaptive control of a Wiener type non-linear system using only input and output measurements

Model reference adaptive control is applied to the control of non-linear systems represented by the Wiener model, wherein the non-linearity is any-order poly-nominal. The adaptive algorithms proposed in this paper guarantee the asymptotic stability of the error between the plant output and that of the reference model by using only input and output measurements. The concept of growing at the same rate is applied in the stability proof. Smoothing the output response is also considered. To illustrate the validity, the proposed adaptive algorithms are demonstrated by a numerical example.     

An indirect adaptive controller with a self-excitation capability

An indirect adaptive pole placement controller is presented which stabilizes and asymptotically regulates any discrete-time single-input, single-output linear time-invariant plant which is of known order n , is controllable, and observable, and has unknown parameters. To avoid singular points in the algorithm, the adaptive controller solves the pole-placement design equation asymptotically with time rather than trying to solve it exactly at each time instant. The stability of the adaptive control system and the asymptotic regulation of the plant output to zero are ensured by an additional self-excitation generated by the adaptive controller. A novel kind of an error signal to control the magnitude of the self-excitation is obtained by suitably filtering the self-exciting signal and monitoring changes of the controller parameters as they are generated by the adaptive algorithm     

具有角速度和输入约束的航天器姿态协同控制

提出了基于有向图的航天器姿态协同控制算法, 并且系统的角速度和控制输入满足有界性的约束. 当外部扰动存在时, 设计了自适应算法估计扰动的上界, 采用滤波器补偿的方法处理控制输入饱和问题, 并且设计了新的自适应姿态协同控制算法. 对于所设计的控制算法, 给出了稳定性分析, 证明了系统具有几乎全局渐近稳定性. 进一步把控制算法推广到时变通信时滞情况, 当控制器参数满足一定条件时, 仍然能够保证编队系统的几乎全局渐近稳定性. 通过数值仿真, 验证了所提出的控制方案的有效性.     

An algebraic approach to iterative learning control

In this paper discrete-time iterative learning control (ILC) systems are analysed from an algebraic point of view. The algebraic analysis shows that a linear-time invariant single-input–single-output model can always represented equivalently as a static multivariable plant due to the finiteness of the time-axis. Furthermore, in this framework the ILC synthesis problem becomes a tracking problem of a multi-channel step-function. The internal model principle states that for asymptotic tracking (i.e. convergent learning) it is required that an ILC algorithm has to contain an integrator along the iteration axis, but at the same time the resulting closed-loop system should be stable. The question of stability can then be answered by analysing the closed-loop poles along the iteration axis using standard results from multivariable polynomial systems theory. This convergence theory suggests that time-varying ILC control laws should be typically used instead of time-invariant control laws in order to guarantee good transient tracking behaviour. Based on this suggestion a new adaptive ILC algorithm is derived, which results in monotonic convergence for an arbitrary linear discrete-time plant. This adaptive algorithm also has important implications in terms of future research work—as a concrete example it demonstrates that ILC algorithms containing adaptive and time-varying components can result in enhanced convergence properties when compared to fixed parameter ILC algorithms. Hence it can be expected that further research on adaptive learning mechanisms will provide a new useful source of high-performance ILC algorithms.     

Robust discrete-time adaptive control subject to a control input amplitude constraint

Subject to a control input amplitude constraint, an adaptive algorithm is proposed to control discrete-time plants with unmodelled dynamics. By introducing an additional feedback signal, it is shown that the uniform boundedness of all signals in the adaptive loop can be guaranteed. The nominal plant is assumed to be stable but unnecessarily minimum phase. Various properties of this adaptive algorithm are analysed. It is shown that the performance which can be achieved with no control input amplitude constraint in the non-adaptive case (i.e. the case when the true nominal plant is known a priori) is asymptotically well maintained in the adaptive system. The analysis is supported by computer simulation results.     

Controllability is not necessary for adaptive pole placementcontrol

The key issue for adaptive pole-placement control of linear time-invariant systems is the possible singularity of the Sylvester matrix corresponding to the coefficient estimate. However, to overcome the difficulty, the estimate is modified by several methods which are either nonrecursive and with high computational load or recursive but with random search involved. All of the previous works are done under the assumption that the system is controllable. This paper gives the necessary and sufficient condition, which is weaker than controllability, for the system to be adaptively stabilizable. First, a nonrecursive algorithm is proposed to modify the estimates, and the algorithm is proved to terminate in finitely many steps. Then, with the help of stochastic approximation, a recursive algorithm is proposed for obtaining the modification parameters; it is proved that these modification parameters turn out to be a constant vector in a finite number of steps. This leads to the convergence of the modified coefficient estimates. For both algorithms the Sylvester matrices corresponding to the modified coefficient estimates are asymptotically uniformly nonsingular; thus, the adaptive pole-placement control problem can be solved, i.e., the system can be adaptively stabilized     

Adaptive control of the filling velocity of thermoplastics injection molding

Injection velocity, a key variable in injection molding, was controlled via an adaptive controller using a self-tuning regulator (STR) scheme. The pole-placement design was employed first, together with the performance enhancement techniques of anti-windup estimation, feed-forward control, and cycle-to-cycle adaptation. The pole-placement design with the enhancement techniques was found experimentally to work very well over different molding conditions. However, this design was also found to be sensitive to the model mismatch. To overcome this problem, a new adaptive controller based on a generalized predictive control (GPC) principle was designed to make the controller more robust. Experiments have shown that the adaptive GPC control of injection velocity has inherently good set-point tracking performance and excellent tolerance to model structure mismatch.     

Robust controller design for discrete unstable non-minimum-phase delayed stochastic processes

Control of unstable non-minimum-phase delayed stochastic processes is a challenging problem. In this work based on the Diophantine equation and using pole-placement technique, a discrete control scheme for such processes has been proposed. Robust stability of the suggested control structure has been shown. Advantages of the proposed scheme over the existing algorithms have been shown through computer simulations. It has been shown that performance of the proposed scheme for handling model mismatch and colored noise is superior to the previous work proposed in the literature.     

Persistency of excitation for continuous-time systems — Time-domain approach

This paper presents the time-domain approach to the analysis of the convergence of continuous-time adaptive control and estimation algorithms. The time-domain definition of persistently exciting signals is introduced and the convergence of estimation algorithms is established in the cases of open-loop and closed-loop systems. An application of the persistency of excitation theory to the design of a globally stable adaptive pole-placement controller is given.     

A new adaptive law for robust adaptation without persistent excitation

A new adaptive law motivated by the work of loannou and Kokotovic (1983) is proposed for the robust adaptive control of plants with unknown parameters. In this adaptive law the output error plays a dual role in the adjustment of the control parameter vector. The advantages of using the adaptive law over others proposed in the literature are discussed. In the ideal case the adaptive system has bounded solutions; in addition, the origin of the error equations is exponentially stable when the reference input is persistently exciting and has a sufficiently large amplitude. The adaptive system is also shown to be robust under bounded external disturbances. Finally, it is shown that, by suitably modifying the adaptive law, the overall system can be made robust in the presence of a class of unmodeled dynamics of the plant. Simulation results are presented throughout the paper to complement the theoretical developments.     

Robust controller synthesis for systems with input nonlinearities: a trade-off between gain variation and parametric uncertainty

A feedback control-system design problem involving input nonlinearities and structured plant parameter uncertainities is considered. Multivariable absolute stability theory is merged with the guaranteed cost control approach to robust stability and performance to obtain a theory of full- and reduced-order robust control design that accounts for input time-varying sector bounded nonlinearities. The principal result is a sufficient condition for characterizing dynamic controllers of a fixed dimension which are guaranteed to provide robust stability for plant parametric variations and absolute stabilization for input nonlinearities. The proposed framework provides a systematic design trade-off between classical robustness guarantees (i.e., gain and phase margins) versus parametric robustness. Furthermore, the framework is directly applicable to uncertain systems with saturating controls and provides fixed-order dynamic output feedback controllers with guaranteed domains of attraction. © 1998 John Wiley & Sons, Ltd.     

Robust stabilization for uncertain time-delay systems containingsaturating actuators

A class of uncertain time-delay systems containing a saturating actuator is considered. These systems are characterized by delayed state equations (including a saturating actuator) with norm-bounded parameter uncertainty (possibly time varying) in the state and input matrices. The delay is assumed to be constant bounded but unknown. Using a Razumikhin approach for the stability of functional differential equations, upper bounds on the time delay are given such that the considered uncertain system is robustly stabilizable, in the case of constrained input, via memoryless state feedback control laws. These bounds are given in terms of solutions of appropriate finite dimensional Riccati equations     

Mean-square stability of the Normalized Least-Mean Fourth algorithm for white Gaussian inputs

Normalized forms of adaptive algorithms are usually sought in order to obtain convergence properties independent of the input signal power. Such is the case of the well-known Normalized LMS (NLMS) algorithm. The Least-Mean Fourth (LMF) adaptive algorithm has been shown to outperform LMS in different situations. However, the LMF stability is dependent on both the signal power and on the adaptive weights initialization. This paper studies the behavior of two normalized forms of the LMF algorithm for Gaussian inputs. Contrary to what could be expected, the mean-square stability of both normalized algorithms is shown to be dependent upon the input signal power. Thus, the usefulness of the NLMF algorithm is open to question.     

A functional equations approach to nonlinear discrete-time feedback stabilization through pole-placement

The present work proposes a new approach to the nonlinear discrete-time feedback stabilization problem with pole-placement. The problem's formulation is realized through a system of nonlinear functional equations and a rather general set of necessary and sufficient conditions for solvability is derived. Using tools from functional equations theory, one can prove that the solution to the above system of nonlinear functional equations is locally analytic, and an easily programmable series solution method can be developed. Under a simultaneous implementation of a nonlinear coordinate transformation and a nonlinear discrete-time state feedback control law that are both computed through the solution of the system of nonlinear functional equations, the feedback stabilization with pole-placement design objective can be attained under rather general conditions. The key idea of the proposed single-step design approach is to bypass the intermediate step of transforming the original system into a linear controllable one with an external reference input associated with the classical exact feedback linearization approach. However, since the proposed method does not involve an external reference input, it cannot meet other control objectives such as trajectory tracking and model matching.