On using least-squares updates without regressor filtering in identification and adaptive control of nonlinear systems

In continuous-time system identification and adaptive control the least-squares parameter estimation algorithm has always been used with regressor filtering, which adds to the dynamic order of the identifier and affects its performance. We present an approach for designing a least-squares estimator that uses an unfiltered regressor. We also consider a problem of adaptive nonlinear control and present the first least-squares-based adaptive nonlinear control design that yields a complete Lyapunov function. The design is presented for linearly parametrized nonlinear control systems in ‘normal form’. A scalar linear example is included which adds insight into the key ideas of our approach and allows showing that, for linear systems, our Lyapunov-LS design with unfiltered regressor, presented in the note for unnormalized least-squares, can also be extended to normalized least-squares.     

Explicit convergence rates for MRAC-type systems

It is well known since many years ago that for the linear time-varying system , with A Hurwitz, and B(t) bounded and globally Lipschitz, it is necessary and sufficient for global exponential stability, that B(t) satisfy the so-called persistency of excitation condition. In this note, we revisit this question and provide explicit bounds for the convergence rate and on the overshoot of the transient behaviour of the solutions e(t), θ(t) as functions of the richness of B(·). We believe that the result that we present is useful since knowing convergence rates aids in the construction of converse Lyapunov functions. Moreover, the type of systems that we study here appear for instance in model reference adaptive control (MRAC).     

Persistency of excitation in continuous-time systems

For the class of linear time-invariant single-input continuous systems we find conditions on the input under which the state is persistently exciting for adaptive identification purposes. These conditions are expressed through time-domain properties of the filtered input. They are both necessary and sufficient and no prior constraints are placed on the structure of the input wave or its boundedness.     

A universality advantage of stochastic excitation signals for adaptive control

Adaptive control schemes usually depend on estimation of system parameters, which in turn depends excitation of modes associated with these parameters. Such excitation is supplied by plant noise, the adaptive control signal itself, and any external excitation such as a reference signal. In this note, it is shown that there is a universality advantage for any externally applied signal to be stochastic rather than deterministic. The crucial property of a stochastic signal exploited in this note is its unpredictability by any causal system, such as an adaptive control scheme. When such unpredictable signals excite an adaptive control scheme, there is no need to deliberately constrain the adaptation to be ‘slow’ or ‘excitation maintaining’ to ensure adequate identification.     

Persistency of excitation criteria for linear,multivariable, time-varying systems

For continuous-time, multiple-input, multiple-output, linear systems, we present conditions under which the persistency of excitation of one regression vector implies the persistency of another regression vector derived from the first via a linear, dynamical transformation. We then introduce a definition of sufficient richness for vector input signals in the form of a persistency of excitation condition on a basis regression vector. Finally we establish input conditions which guarantee the persistency of excitation of a large class of regression vectors obtained from both time-invariant and time-varying systems. The work reported was performed while both authors were at the Department of Systems Engineering, Research School of Physical Sciences, Australian National University, G.P.O. Box 4, A.C.T. 2601, Australia.     

Infinite horizon backward stochastic differential equation and exponential convergence index assignment of stochastic control systems

This paper studies exponential convergence index assignment of stochastic control systems from the viewpoint of backward stochastic differential equation. Like deterministic control systems, it is shown that the exact controllability of an open-loop stochastic system is equivalent to the possibility of assigning an arbitrary exponential convergence index to the solution of the closed-loop stochastic system, formed by means of suitable linear feedback of the states. As an application, a sufficient and necessary condition for the existence and uniqueness of the solution of a class of infinite horizon forward-backward stochastic differential equations is provided.     

On parameter convergence in least squares identification and adaptive control

Least squares estimation is appealing in performance and robustness improvements of adaptive control. A strict condition termed persistent excitation (PE) needs to be satisfied to achieve parameter convergence in least squares estimation. This paper proposes a least squares identification and adaptive control strategy to achieve parameter convergence without the PE condition. A modified modeling error that utilizes online historical data together with instant data is constructed as additional feedback to update parameter estimates, and an integral transformation is introduced to avoid the time derivation of plant states in the modified modeling error. On the basis of these results, a regressor filtering–free least squares estimation law is proposed to guarantee exponential parameter convergence by an interval excitation condition, which is much weaker than the PE condition. And then, an identification‐based indirect adaptive control law is proposed to establish exponential stability of the closed‐loop system under the interval excitation condition. Illustrative results considering both identification and control problems have verified the effectiveness and superiority of the proposed approach.     

衰减激励条件下递阶最小二乘辨识的均方收敛性

为减少递推辨识的计算量，提出了递阶辨识原理，它是将系统分解为多个维数较小的虚拟子系统进行辨识，从而获得递阶最小二乘辨识方法。在衰减激励条件下，针对时不变系统研究了递阶最小二乘法的收敛性，得到了参数估计误差均方收敛于零时衰减指数应满足的条件。递阶最小二乘具有良好的性能，其计算量比递推量小二乘辨识要小得多，并具有容易实现等优点。     

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.     

Exponential convergence of adaptive identification and control algorithms

Output and equation error adaptive identification algorithms are shown to be exponentially convergent under a deterministic or stochastic persistently exciting (or spanning) condition on the system inputs together with several other standard conditions. An adaptive control algorithm is shown to be exponentially convergent under a deterministic or stochastic persistently exciting condition on the reference trajectory together with some standard conditions.     

On robustness in control and LTI identification: Near-linearity and non-conic uncertainty

Robustness issues are studied in the context of linear models in linear controller design and in system identification when the true system is nonlinear. The notion of nearly linear system is generalized to include time-varying and open-loop unstable systems. This class of systems, although it presents the simplest possible (global) generalization of linear systems, is in many ways nontrivial for the purpose of linear controller design and linear model identification from input-output data. This is mainly due to the presence of non-conic uncertainty, which is not included in standard treatments of robust control theory and stochastic system identification theory. Signal distribution theory, a realistic non-stochastic signal analysis tool, is used to study the limiting least squares estimates of linear finite impulse response (FIR) and autoregressive with external input (ARX) model parametrizations for two classes of nonlinear systems. Some connections to worst-case analysis of linear model identification are also discussed.     

态空间系统的梯度优化辨识及收敛性分析*

为解决状态空间系统的预报误差与系统参数之间的非线性、非凸性给参数估计带来的困难,提出了状态空间系统的梯度优化辨识方法。分析了基于局部线性化的梯度辨识原理,给出了基于QR分解、奇异值分解(SVD)确定参数搜索方向的实现方案,得到了估计系统参数的迭代辨识算法。探讨了算法的收敛性、给出了算法收敛速度的解析表达式,最后进行了数值仿真,实验结果说明了所提出方法的有效性。     

非持续激励条件下系统辨识递推最小二乘最小范数算法

系统辨识中广泛应用的最小二乘算法需要输入向量序列满足持续激励性条件(PE条件); 但在大多情况下这是难以满足的. 本文提出了一种不依赖于PE条件的递推最小二乘、最小范数辨识算法. 首先分析了最小二乘算法解空间的结构, 并运用罚函数方法, 将参数辨识问题转化为无约束优化问题. 然后, 提出了将步长、罚因子等过程控制参数统一的迭代－递推形式的辨识算法, 证明了算法在给定的控制参数约束下收敛于唯一的最小二乘、最小范数解向量. 仿真实验表明在非PE条件下算法的有效性.     

Optimal finite-dimensional recursive identification in a polynomial output mapping class

A time-varying polynomial output equation expressing an implicit dependence of the observed signal on the parameter vector admits a finite-dimensional recursive representation for the optimal on-line estimator. Both continuous and discrete observations result in the differential systems which give the estimate. Deterministic exponentially weighted least-squares is applied. Exponential stability of the identifier and convergence to the true parameter value are shown. Two examples are presented. A continuous application originates in a pH-control process. The discrete identification is applied in a dynamic ARMA-type model which is quadratic in the parameter.     

On parameter convergence in adaptive control

It is well known that the parameter error as well as the model-plant mismatch error in a model reference adaptive scheme tends exponentially to zero iff a certain sufficient richness condition holds for signals inside the time-varying plant control loop. In this paper we give conditions on the reference signal (the exogenous input to the adaptive loop) — namely, that it have as many spectral lines as there are unknown parameters, in order to guarantee parameter convergence.     

How exciting can a signal really be?

The rate of parameter convergence in a number of adaptive estimation schemes is related to the smallest eigenvalue of the average information matrix determined by the regression vector. Using a very simple examples, we illustrate that the input signals that maximize this minimum eigenvalue may be quite different from the input signals that optimize more classical input design criteria, e.g. D-optimal criterion.     

Auxiliary model-based least-squares identification methods for Hammerstein output-error systems

The difficulty in identification of a Hammerstein (a linear dynamical block following a memoryless nonlinear block) nonlinear output-error model is that the information vector in the identification model contains unknown variables—the noise-free (true) outputs of the system. In this paper, an auxiliary model-based least-squares identification algorithm is developed. The basic idea is to replace the unknown variables by the output of an auxiliary model. Convergence analysis of the algorithm indicates that the parameter estimation error consistently converges to zero under a generalized persistent excitation condition. The simulation results show the effectiveness of the proposed algorithms.     

A note on persistency of excitation

We prove that if a component of the response signal of a controllable linear time-invariant system is persistently exciting of sufficiently high order, then the windows of the signal span the full system behavior. This is then applied to obtain conditions under which the state trajectory of a state representation spans the whole state space. The related question of when the matrix formed from a state sequence has linearly independent rows from the matrix formed from an input sequence and a finite number of its shifts is of central importance in subspace system identification.     

Parameter consistency and quadratically constrained errors-in-variables least-squares identification

In this article, we investigate the consistency of parameter estimates obtained from least-squares identification with a quadratic parameter constraint. For generality, we consider infinite impulse-response systems with coloured input and output noise. In the case of finite data, we show that there always exists a possibly indefinite quadratic constraint depending on the noise realisation that results in a constrained optimisation problem that yields the true parameters of the system when a persistency condition is satisfied. When the noise covariance matrix is known to within a scalar multiple, we prove that solutions of the quadratically constrained least-squares (QCLs) estimator with a semidefinite constraint matrix are both unbiased and consistent in the sense that the averaged problem and limiting problem produce, respectively, unbiased and true (with probability 1) estimators. In addition, we provide numerical results that illustrate these properties of the QCLS estimator.