Preamble

This work addresses the stabilization of dynamical systems under the influence of external disturbances. We consider a system of the form:
$$\frac{dy}{dt} = Y(y, t, v) + g(t) \tag{1.1}$$
where $y$ is an $n$-dimensional state vector, $Y$ is a vector function, $g(t)$ is an $n$-dimensional vector representing external disturbances, and $v$ is an $r$-dimensional control vector.

Following the methodology established by E. A. Barbashin \cite{1, 2, 3}, we first consider the nominal system in the absence of disturbances:
$$\frac{dy}{dt} = Y(y, t, v) \tag{1.2}$$
with initial and boundary conditions $y(0) = y^0$ and $y(t_1) = y^1$. Let the optimal control for this nominal system be denoted by $v = v^(t, y^0, y^1)$, which generates the optimal trajectory $y^(t, y^0, y^1)$.

When the disturbance $g(t)$ is present, we define the deviation from the optimal trajectory as $x = y - y^$ and the control deviation as $u = v - v^$. The dynamics of the error system can then be expressed as:
$$\frac{dx}{dt} = f(x, t, u) + g(t) \tag{1.6}$$
where the function $f$ is defined as:
$$f(x, t, u) = Y(y^ + x, t, v^ + u) - Y(y^, t, v^) \tag{1.7}$$
Our objective is to maintain the state $x = 0$ over the interval $[0, t_1]$. To achieve this, we discretize the time domain into intervals of length $T$, such that $t_k = kT$ for $k = 1, 2, \dots, N$. Within each interval $[(k-1)T, kT]$, we seek a control $u_k$ that minimizes the deviation $x(t)$.

Section 2. Estimation of the Disturbance

To compensate for the disturbance $g(t)$, we must first estimate its integral effect over the current interval. Integrating the system equation (2.1) over the $k$-th interval $[(k-1)T, kT]$, we obtain:
$$x_k - x_{k-1} = \int_{(k-1)T}^{kT} f(x(t), t, u_k) dt + \int_{(k-1)T}^{kT} g(t) dt \tag{2.2}$$
where $x_k = x(kT)$. From this, the integral of the disturbance can be isolated:
$$\int_{(k-1)T}^{kT} g(t) dt = x_k - x_{k-1} - \int_{(k-1)T}^{kT} f(x(t), t, u_k) dt \tag{2.3}$$
Using numerical integration techniques, specifically the trapezoidal rule as discussed in \cite{5}, the integral of $f$ can be approximated as:
$$\int_{(k-1)T}^{kT} f(x(t), t, u_k) dt \approx \frac{T}{2} [f(x((k-1)T), (k-1)T, u_k) + f(x(kT), kT, u_k)] + R(f) \tag{2.5}$$
where $R(f)$ is the approximation error. Let $F(t, T)$ represent the average value of the disturbance over an interval $T$:
$$F(t, T) = \frac{1}{T} \int_{t-T}^{t} g(\tau) d\tau \tag{2.8}$$
By applying a Taylor series expansion to $F(t, T)$ and neglecting terms of $O(T^2)$, we can derive a predictive formula for the disturbance in the subsequent interval. Specifically, the predicted disturbance for the $(k+1)$-th step, denoted as $F^0((k+1)T, T)$, is calculated using values from preceding steps:
$$F^0((k+1)T, T) = F(kT, T) + [F(kT, T) - F((k-1)T, T)] \tag{2.10}$$

Section 3. Control Synthesis

For the $(k+1)$-th interval, we aim to determine the control $u_{k+1}$ that minimizes the predicted deviation. The state at the end of the interval is approximated using a Runge-Kutta type scheme:
$$x_{k+1} = x_k + \frac{1}{6}(z_1 + 4z_2 + z_3) + O(T^4) \tag{3.3}$$
where the coefficients $z_i$ depend on the system dynamics $f$ and the predicted disturbance $\xi_{k+1} = F^0((k+1)T, T)$. This leads to a discrete-time transition mapping:
$$x_{k+1} = A(x_k, T, \xi_{k+1}, u_{k+1}) \tag{3.5}$$
The optimal control $u_{k+1}$ is found by minimizing the quadratic norm of the terminal error:
$$|x_{k+1}|^2 = \min_{u} \sum (x_i)^2 \tag{3.6}$$
This minimization yields a system of equations for the control components $u_j$:
$$\frac{\partial}{\partial u_j} \sum (x_i)^2 = 0 \tag{3.9}$$
For a linear system of the form $\dot{x} = Lx + Mu + g(t)$, the control law can be expressed in a feedback form:
$$u_{k+1} = P x_k + Q \xi_{k+1} \tag{3.13}$$
where $P$ and $Q$ are gain matrices determined by the system parameters and the sampling period $T$.

Section 4. Numerical Example

Consider a second-order system:
$$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= x_1 + u + g(t) \end{aligned} \tag{4.1}$$
Applying the discretization and control synthesis described above, we obtain the iterative relations for $x_1$ and $x_2$. The disturbance is assumed to be a harmonic function $g(t) = \sin t$.

[FIGURE: 1]
[FIGURE: 2]

Figures 1 and 2 illustrate the trajectories of the system starting from different initial conditions. In the first case, $x_1(0) = 1, x_2(0) = 0$, and in the second case, $x_1(0) = 0, x_2(0) = 0$. The results demonstrate that the proposed predictive control effectively compensates for the external disturbance, maintaining the system state near the origin.

References

1. Barbashin, E. A., Introduction to the Theory of Stability, Nauka, Moscow, 1967.
2. Barbashin, E. A., "On the realization of motions along a given trajectory," Avtomatika i Telemekhanika, Vol. 21, No. 7, 1960.
3. Barbashin, E. A., "Towards a theory of linear systems with variable parameters," Trudy Ural. Politekh. Inst., Vol. 102, 1960.
4. Demidovich, B. P., Maron, I. A., Fundamentals of Computational Mathematics, Fizmatgiz, 1962.