Preamble

This work, published in 1967, addresses the control of linear discrete-time systems. We consider a system of the form:
$$\begin{aligned} \dot{x} = Ax + bu_k \end{aligned}$$
where $x$ is an $n$-dimensional state vector, $b$ is a constant vector, and $u_k$ is a control signal that remains constant over the sampling interval $kT_0 \le t < (k+1)T_0$. Here, $T_0$ denotes the sampling period. Following the methodology established in \cite{1}, the continuous system (1) can be represented as a discrete-time difference equation:
$$\begin{aligned} x_{k+1} = Bx_k + du_k \end{aligned}$$
where $x_k = x(kT_0)$, $B = \exp(AT_0)$, and $d = (\exp(AT_0)) \int_0^{T_0} [\exp(-As)] b \, ds$. The objective is to analyze the properties of such systems and develop control laws $u_k = \Phi[x(kT_0)]$ that ensure stability and desired performance characteristics.

2. Canonical Transformations and Controllability

To simplify the analysis of the discrete system (3), we employ linear transformations. Let $y = T_1x$ be a transformation that maps the system into a canonical form. According to the criteria established in \cite{2}, the system is controllable if the vectors $d, Bd, \dots, B^{n-1}d$ are linearly independent. If the system is not fully controllable, it can be decomposed into a controllable subsystem of dimension $m < n$.

Specifically, we consider the case where the control enters through a reduced-order manifold. Let us define a transformation $T_2$ such that the system (4) is partitioned. For $m=2$, the vectors $d$ and $Bd$ define the controllable subspace. We assume that the components $d_n \neq 0$ and proceed with a recursive transformation of the state variables. The resulting system equations involve coefficients $b_{ij}$ and $d_i$ derived from the matrices $B$ and $d$.

The transformation process involves defining $q = (B - I)d$, where $I$ is the identity matrix. If $q_i \neq 0$ for $i \le n-1$, we can further reduce the system to a form suitable for applying specific control laws, such as relay or sliding mode control.

3. Asymptotic Analysis and Small Sampling Periods

When the sampling period $T_0$ is small, we can utilize asymptotic expansions. Let $v = T_0$ be a small parameter. The matrix $B$ can be expanded as $B = \exp(Av) = E + Av + O(v^2)$. As $v \to 0$, the discrete system (3) approaches its continuous counterpart. We introduce scaled variables $z$ and $v$ to capture the behavior of the system across different time scales.

The transformed system can be written as:
$$\begin{aligned} z_{k+1} &= Pz_k + p v_{1k} \ v_{1,k+1} &= Qz_k + R_0 v_k + d_n^0 u_k \end{aligned}$$
where $P, Q, R_0$ are matrices of appropriate dimensions and $d_n^0$ is the transformed control gain. Under the assumption that $v \to 0$, we can analyze the stability of the equilibrium point. We define a Lyapunov-like function to prove that the trajectories of the discrete system remain close to the trajectories of the continuous system (1). Specifically, for a given number of steps $N$, the difference between the discrete state $z_k$ and the continuous state $z(t)$ is of the order $O(v^s)$, where $s > 0$.

The control law $u_k = \text{sign}(\sigma_k)$ is often employed in these systems. We demonstrate that for sufficiently small $v$, the system enters a neighborhood of the switching manifold and exhibits behavior analogous to a sliding mode in continuous systems. The stability conditions involve the spectral radius of the matrix $(P + pR_0)$ and the properties of the gain $d_n^0$.

4. Numerical Example and Simulation

To illustrate the theoretical results, we consider a third-order system. Let the state variables be $y_1, y_2, y_3$. Applying the transformation $z = T_2 y$, we obtain a system where the control $u_k$ directly influences the higher-order derivatives. For a sampling period $T_0 = v$, the control law is defined as $u_k = \text{sign}(z_{1k} + \alpha z_{2k} + \beta z_{3k})$.

Simulation results indicate that the state trajectories $z_1, z_2$ converge to the origin. The phase plane analysis shows the existence of regions $S_1, S_2, S_3$ where the control signal remains constant. As $v$ decreases, the "chattering" effect near the switching surface is reduced, and the discrete-time system accurately tracks the ideal sliding manifold of the continuous system.

Conclusion

This paper has presented a method for the analysis and synthesis of control laws for discrete-time linear systems. By utilizing canonical transformations and asymptotic expansions for small sampling periods, we have established conditions for stability and convergence. The results are applicable to the design of digital controllers for physical systems where the sampling rate is high relative to the system dynamics.

References

1. Pospelov, G. S., "On the suppression of oscillations in sampled-data systems," Proceedings of the Academy of Sciences, 1966.
2. Gelig, A. Kh., "Stability of sampled-data systems with back-step dynamics," Automation and Remote Control, Vol. 3, No. 4, pp. 579–588, 1967.