Stability Analysis and Pole Placement Control for Discrete-Time Systems

Stability Analysis of Discrete-Time Systems

In control system design, system stability is the most fundamental and important characteristic. While in continuous-time systems, stability is determined by whether the roots (poles) of the characteristic equation are in the left half of the complex plane, discrete-time systems have different stability criteria. This article explains the stability of discrete-time systems and control system design using pole placement.

Stability Analysis of Discrete-Time Systems

Author Information: Hiroshi Okajima (Associate Professor, Department of Computer Science and Electrical Engineering, Kumamoto University, Web, YouTube) I have been a faculty member for about 20 years. My research in control engineering includes model error suppression compensators, state estimation, quantized control, and more.

Continuous-Time Linear Time-Invariant Systems and Their Discretization: Shift Form Representation and Delta Form Representation - Control Engineering Blog

Stability of Discrete-Time Systems

The state-space representation of a discrete-time system is generally expressed as follows:

\begin{eqnarray} x[k+1] &=& A_d x[k] + B_d u[k] \\ y[k] &=& C_d x[k] + D_d u[k] \end{eqnarray}

Here, $x[k] \in \mathbb{R}^n$ is the state vector, $u[k] \in \mathbb{R}$ is the input, and $y[k] \in \mathbb{R}$ is the output. To understand the characteristics of discrete-time systems, it is important to examine the roots of the characteristic equation, called the system poles.

Relationship Between Poles and Stability

The stability of a discrete-time system is characterized by the eigenvalues of the system matrix A_d (i.e., the poles of the system). Unlike continuous-time systems, the necessary and sufficient condition for a discrete-time system to be asymptotically stable is that the absolute values of all poles are less than 1. In other words, all poles must exist inside the unit circle on the complex plane.

For the poles $\lambda_i$ ( $i = 1, 2, \ldots, n$ ) of a discrete-time system:

If $|\lambda_i| \lt 1$ for all , the system is asymptotically stable.
If $|\lambda_i| \leq 1$ for all , and poles with $|\lambda_i| = 1$ are simple roots (multiplicity of 1), the system is stable (but not asymptotically stable).
If $|\lambda_i| \gt 1$ for at least one , or if poles with $|\lambda_i| = 1$ are multiple roots, the system is unstable.

Pole Positions in the Z-Plane

While poles of continuous-time systems are plotted on the plane, poles of discrete-time systems are plotted on the plane. On the plane:

Inside the unit circle ( $|z| \lt 1$ ): Stable region
On the unit circle (): Marginally stable region
Outside the unit circle ( $|z| \gt 1$ ): Unstable region

The following figure shows the stable region in the plane.

Control System Design by Pole Placement

In discrete-time systems, pole placement is also one of the important control system design methods. Similar to continuous-time systems, closed-loop system poles can be placed in desired positions using state feedback.

Pole Placement by State Feedback

When a discrete-time system is controllable, applying state feedback

\begin{equation} u[k] = -K x[k] + r[k] \end{equation}

results in a closed-loop system

\begin{equation} x[k+1] = (A_d - B_d K) x[k] + B_d r[k] \end{equation}

Here, is the feedback gain matrix, and r[k ] is the reference input.

The poles of the closed-loop system are determined by the eigenvalues of A_d - B_d K . If the system is controllable, it is possible to place the poles of the closed-loop system in arbitrary positions by selecting an appropriate gain matrix .

Design Methods for Feedback Gains in Discrete-Time Systems

The main methods for designing gains in discrete-time systems are as follows:

Direct method: Using coefficient comparison of characteristic polynomials
Ackerman's formula: Using the controllability matrix and desired characteristic polynomial
Discrete-time LQR (Linear Quadratic Regulator): Calculating optimal gains by minimizing a cost function

As an example, the method for calculating gains using Ackerman's formula is shown below.

\begin{equation} K = \begin{bmatrix} 0 & 0 & \cdots & 0 & 1 \end{bmatrix} W_c^{-1} \phi(A_d) \end{equation}

Here, W_c is the controllability matrix, and $\phi(A_d)$ is the desired characteristic polynomial $\phi(z) = (z - \lambda_1)(z - \lambda_2)\cdots(z - \lambda_n)$ evaluated at A_d .

Comparison of Continuous-Time and Discrete-Time Design

There are several important differences in design by pole placement between continuous-time and discrete-time systems.

Stable region of poles: Left side of the imaginary axis ( $\text{Re}(s) \lt 0$ ) for continuous-time, inside the unit circle ( $|z| \lt 1$ ) for discrete-time
Response characteristics: Relationship between continuous-time poles $s = \sigma \pm j\omega$ and discrete-time poles $z = e^{(\sigma \pm j\omega)T_s}$
Effect of sampling period: In discrete-time systems, control performance may deteriorate if the sampling period is too long
Changes in zeros due to discretization: Discretization of continuous-time systems may result in unstable zeros

Example of Pole Placement

As an example, let's consider the following discrete-time system.

\begin{eqnarray} x[k+1] &=& \begin{bmatrix} 1.2 & 0.5 \\ 0 & 0.8 \end{bmatrix} x[k] + \begin{bmatrix} 1 \\ 0.5 \end{bmatrix} u[k] \\ y[k] &=& \begin{bmatrix} 1 & 0 \end{bmatrix} x[k] \end{eqnarray}

The poles of this system (eigenvalues of A_d ) are $\lambda_1 = 1.2$ and $\lambda_2 = 0.8$ . Since $\lambda_1 \gt 1$ , this system is unstable.

Suppose we want to place the poles of the closed-loop system at $\lambda_1' = 0.6$ and $\lambda_2' = 0.7$ .

First, we calculate the controllability matrix.

\begin{equation} W_c = \begin{bmatrix} B_d & A_d B_d \end{bmatrix} = \begin{bmatrix} 1 & 1.2 + 0.5 \cdot 0.5 \\ 0.5 & 0.8 \cdot 0.5 \end{bmatrix} = \begin{bmatrix} 1 & 1.45 \\ 0.5 & 0.4 \end{bmatrix} \end{equation}

Since $\det(W_c) = 1 \cdot 0.4 - 1.45 \cdot 0.5 = 0.4 - 0.725 = -0.325 \neq 0$ , the system is controllable.

The desired characteristic polynomial is

\begin{equation} \phi_d(z) = (z - 0.6)(z - 0.7) = z^2 - 1.3z + 0.42 \end{equation}

The characteristic polynomial of the system matrix A_d is

\begin{equation} \phi(z) = (z - 1.2)(z - 0.8) = z^2 - 2z + 0.96 \end{equation}

Using these, we calculate the feedback gain $K = \begin{bmatrix} k_1, k_2 \end{bmatrix}$ . The characteristic polynomial of the closed-loop system is

\begin{equation} \det(zI - (A_d - B_d K)) = z^2 - (2 - k_1)z + (0.96 - 1.2k_1 - 0.5k_2) \end{equation}

Comparing coefficients with the desired characteristic polynomial, we get

\begin{eqnarray} 2 - k_1 &=& 1.3 \\ 0.96 - 1.2k_1 - 0.5k_2 &=& 0.42 \end{eqnarray}

Solving these equations

\begin{eqnarray} k_1 &=& 0.7 \\ k_2 &=& 0.6 \end{eqnarray}

Therefore, the feedback gain matrix is $K = \begin{bmatrix} 0.7, 0.6 \end{bmatrix}$ .

Summary

This article explained the stability of discrete-time systems and control system design by pole placement. The conditions for stability differ between discrete-time and continuous-time systems; in discrete-time systems, the condition for stability is that the poles are inside the unit circle.

In the design of discrete-time systems, the selection of the sampling period is also an important factor. A shorter sampling period yields a response closer to that of a continuous-time system, but if it is too short, problems such as the effect of variations in sampling intervals and increased computational load may arise.

The pole placement method is a powerful tool in the control of discrete-time systems, but in implementation, changes in zeros due to discretization and sampling effects must also be considered. In addition to pole placement by state feedback, a more practical control system can be constructed by combining it with state estimation using observers.

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