Nonlinear robust control of linear plants (Q1883747)

One of the control methods for plants with parametric uncertainty subject to exogenous disturbances suggests the use of specific nonlinear feedbacks which guarantee a reasonable performance (in terms of a certain criterion) for all plants in the specified class. In contrast to the methods of adaptive control, the implementation of such nonlinear feedbacks does not require any identification of unknown parameters of the plant or automatic adjustment of the coefficients of the controller. The authors propose a new solution to the problem of nonlinear robust output control of a linear plant. A class of linear time-invariant control plants of the form \[ y(t)=\frac{\beta(p)}{\alpha(p)}\left(u(t)+f(t)\right)\tag{1} \] is considered, where \(y\) is the controlled variable, \(u\) is the control signal, \(f\) is the exogenous unmeasurable disturbance, \(p = d/dt\) is the differentiation operator, and \(\alpha(p)\) and \(\beta(p)\) are polynomials in \(p\): \[ \alpha(p)= p^n+a_{n-1}p^{n-1}+\dots+a_1p+a_0,\quad \beta(p)= b_mp^m+ b_{m-1}p^{m-1}+\dots+b_1p+b_0. \] The control plant (1) is considered parametrically uncertain, which means that all or a part of the coefficients of the polynomials \(\alpha(p)\) and \(\beta(p)\) are not known. Additionally, the following conditions are assumed to be valid: the polynomial \(\beta(p)\) is Hurwitz; the order \(n\) of the plant and the relative degree \(\rho = n- m\geqslant1\) are assumed to be known; the sign of the coefficient \(b_m\) is known (for definiteness, \(b_m > 0\)); the immeasurable disturbance \(f\) is bounded. Let the desired behavior of the controlled variable be specified by the reference model \(y_r(t)= \frac {k_r} {\alpha_r(p)} r(t)\), where \(y_r\) is the output reference signal, \(r\) is the input reference signal (known bounded piecewise-continuous function of time), \(\alpha_r(p)=p^\rho+a^\ast_{\rho-1}p^{\rho-1}+\dots+a_1^\ast p +a_0^\ast\) is a Hurwitz polynomial of degree \(\rho\), and \(k_r > 0\) is the constant coefficient. The problem is to design a control law \(u=u(y,t)\) which does not contain differentiation operations and for any initial conditions ensures the boundedness of all signals in the closed-loop system along with arbitrarily small steady-state tracking error \(\varepsilon_y ={\overline\lim}_{t\to\infty}| y(t)-y_r(t)| <\Delta,\) where \(\Delta>0\) is arbitrarily small. The desired control is designed in two stages. First, the main control loop is synthesized under the condition that all parameters of the plant are known; next, additional feedbacks are introduced which ensure the compensation of parametric and signal disturbances of model (1). An illustrative example is considered.