## Adaptive controller design for tracking and disturbance attenuation in parametric strict-feedback nonlinear systems.(English)Zbl 0957.93046

The authors consider a class of single-input/single-output nonlinear systems in the noise-prone parametric strict-feedback form: \begin{aligned}\dot x_1=&x_2+f_1(x_1)+\phi_1'(x_1) \theta_1+ h_1'(x_1) w_1\\ &\dots\\ \dot x_{n-1}=&x_n+f_{n-1}(x_1,\dots,x_{n-1}) +\phi_{n-1}'(x_1,\dots,x_{n-1}) \theta_{n-1}\\ &+h_{n-1}'(x_1,\dots,x_{n-1}) w_{n-1}\\ \dot x_n=&f_n(x_1,\dots,x_n) +\phi_n'(x_1,\dots,x_n) \theta_n\\ &+h_n'(x_1,\dots,x_n) w_n+b(x_1,\dots,x_n) u\\ y=&x_1.\end{aligned} Here $$x=(x_1,\dots,x_n)'$$ is the state vector, with initial state $$x(0)$$; $$u$$ is the scalar control output; $$(w_1',\dots,w_n')'$$ is the $$q$$-dimensional exogenous input (disturbance) where $$w_i$$ is of dimension $$q_i$$; $$y$$ is the scalar output; $$(\theta_1',\dots,\theta_n')$$ is an $$r$$-dimensional vector of unknown parameters of the system where $$\theta_i$$ is of dimension $$r_i$$; and the nonlinear functions $$f_i,\phi_i,h_i$$ and $$b$$ are known. Design tools that lead to an explicit construction for a class of robust adaptive controllers that asymptotically track a given reference signal and achieve prespecified disturbance attenuation levels with respect to exogenous system inputs, are developed. The design procedure is based on worst case identification, the integrator backstepping methodology, and singular perturbations analysis. The closed-loop system is shown to admit a closed-form value function that satisfies an associated Hamilton-Jacobi-Isaacs inequality, thereby guaranteeing a desired level of performance for the adaptive controller. It is shown that the certainty-equivalence principle holds in the strict sense only for first-order systems, whereas for higher-order nonlinear systems it holds only asymptotically, as the confidence in the parameter estimates reaches infinity. A numerical example involving a third-order system clearly demonstrates the superior performance of the controller designed.

### MSC:

 93C40 Adaptive control/observation systems 93C10 Nonlinear systems in control theory 93B36 $$H^\infty$$-control 93C70 Time-scale analysis and singular perturbations in control/observation systems
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