A globally convergent direct adaptive pole‐placement controller for nonminimum phase systems with relaxed excitation assumptions

Abstract

In this paper, we propose the first solution to the long-standing problem of designing a globally convergent direct adaptive pole-placement controller for linear, time-invariant discrete-time systems with arbitrary zeros that does not rely on persistency of excitation assumptions. As is well known, the main difficulty of this design is that it involves the estimation of parameters that enter nonlinearly in the regression model. This problem can be overcome introducing an overparameterized representation of the system, which imposes very strict persistency of excitation conditions to prove the parameter convergence. The latter is avoided here using a new version of the dynamic regressor extension and mixing parameter estimator recently proposed in the literature. The main feature of this estimator is that it generates, out of an m-dimensional vector regression, m scalar regression models. This property allows us to estimate only the controller parameters of interest for the adaptive implementation, whose convergence is ensured under assumptions that are strictly weaker than the classical persistency of excitation requirement. Simulation results that illustrate the performance of the proposed scheme are also presented.

Linear‐like properties arise naturally in the adaptive control setting

“…Furthermore, in the continuous‐time setting one is faced with additional technical challenges, since the obvious continuous‐time counterpart is non‐Lipschitz continuous, so even existence and uniqueness of the closed‐loop solution is not guaranteed. At present, when the classical estimation algorithm is used, all that is usually proven in the literature is that if the input is bounded, then the state is bounded, although some approaches provide crisper bounds, for example see Reference 50. It would be interesting to see if the methodology used herein can be leveraged to provide a more insightful bound ‐ perhaps a convolution bound plus an offset. Recently a new system identification technique, known as Dynamic Regressor Extension and Mixing, has been proposed in both the continuous‐time 52 and discrete‐time 53 settings: it has significantly better convergence properties than standard system ID algorithms. It will be interesting to see if the ideas of that approach can be applied to the PA considered here to yield a controller with even better performance.…”

“…Recently a new system identification technique, known as Dynamic Regressor Extension and Mixing, has been proposed in both the continuous‐time 52 and discrete‐time 53 settings: it has significantly better convergence properties than standard system ID algorithms. It will be interesting to see if the ideas of that approach can be applied to the PA considered here to yield a controller with even better performance.…”

Parameter Estimation of Nonlinearly Parameterized Regressions without Overparameterization nor Persistent Excitation: Application to System Identification and Adaptive Control

“…That is, the controller relocates the poles of the system in a desired position but preserves the open-loop zeros. For a lucid exposition of this problem see [12,Section 5.3] and [29] for a review of the recent literature.…”

“…and S(a i , b i ) ∈ R 2v×2v -called the Sylvester matrix-is linearly dependent on the coefficients a i , b i , and is full rank if and only if A(q −1 ) and B(q −1 ) are coprime. It is well-known that the adaptive version of the previous controller, called APPC, suffers from serious drawbacks [12,29]. In its indirect version-that is when we estimate the parameters of the plant a i , b i and then compute from them, via the solution of (61), the parameters of the controller l i , p i -the problem is that the Sylvester matrix with the estimated parameters âi (k), bi (k) may loose rank during the transient behavior.…”

“…In particular, we provide solutions to the, essentially open, problems of direct and indirect adaptive pole-placement control (APPC) without overparameterization nor persistency of excitation (PE) requirements. 2 It should be pointed out that a solution to the direct APPC problem using overparameterization, hence requiring some excitation conditions, has been recently reported in [29].…”