Discrete-Time State-Space Representation

Discrete-time state-space models are the cornerstone of modern digital control and estimation, enabling the analysis and design of systems that are monitored and controlled by computers. By capturing system dynamics at distinct sampling instants, these models bridge the gap between continuous physical processes and discrete computational algorithms. Mastering this representation is essential for implementing predictive controllers, state estimators like Kalman filters, and simulating complex sampled-data systems with precision.

From Continuous to Discrete-Time Models

A continuous-time linear time-invariant (LTI) system is described by the equations $\dot{x}(t)=Ax(t)+Bu(t)$ and $y(t)=Cx(t)+Du(t)$, where $x$ is the state vector, $u$ is the input, and $y$ is the output. However, digital computers cannot process this continuous flow of information; they operate on data sampled at regular intervals. A discrete-time state-space model translates this continuous behavior into a form usable by computers, described by matrix difference equations:

$$x[k+1]=A_{d}x[k]+B_{d}u[k]$$ $$y[k]=C_{d}x[k]+D_{d}u[k]$$

Here, $k$ is the discrete-time index (corresponding to time $t=kT$, where $T$ is the sampling period), $x[k]$ is the state vector at the $k$-th sample, and $A_d$, $B_d$, $C_d$, and $D_d$ are the discrete-time system matrices. The key challenge is deriving these discrete matrices from their continuous counterparts $(A,B,C,D)$ in a way that accurately represents the system's behavior at the sampling instants.

Discretization: The Matrix Exponential Method

The most common and rigorous method for converting a continuous model to a discrete one assumes that the input $u(t)$ is held constant between sampling instants using a Zero-Order Hold (ZOH). This is a standard model for digital-to-analog converters. Under this assumption, the discrete-time matrices are calculated using the matrix exponential.

The state transition matrix $A_d$ is given by the matrix exponential of the continuous $A$ matrix scaled by the sampling period: $$A_d=e^{AT}$$ This matrix governs how the system state evolves from one time step to the next in the absence of any new input.

The discrete input matrix $B_d$ accounts for the effect of the held input over the sampling interval: $$B_d={\int }_0^T{e}^{A\tau }d\tau \cdot B$$ For systems where $A$ is invertible, this simplifies to $B_d=A^{-1}(e^{AT}-I)B$.

The output matrices typically remain unchanged ($C_d=C$, $D_d=D$) because the output equation is memoryless and only depends on the current sampled state and input. The core of the discretization process is the evaluation of the matrix exponential $e^{AT}$, which can be performed using numerical tools or analytical methods for simple systems.

Preserved Properties: Controllability and Observability

A critical question is whether the useful properties of the continuous system survive the discretization process. Fortunately, for a sufficiently fast sampling rate, controllability and observability are preserved. A continuous system that is controllable (you can steer its state anywhere using an appropriate input) will yield a discrete-time model that is also controllable, provided that the sampling period $T$ does not create destructive interference in the system dynamics.

Mathematically, if the continuous pair $(A,B)$ is controllable, then the discrete pair $(A_d,B_d)$ is controllable for almost all choices of $T$. The same holds for observability. However, there are pathological sampling rates that can cause a loss of these properties. For instance, if you sample a continuous system at exactly twice its natural frequency, you might alias the dynamics and lose the ability to observe or control certain modes. Therefore, selecting an appropriate sampling period $T$ is not just a numerical step but a design choice that affects fundamental system properties.

Enabling Digital Controller and Observer Design

The primary utility of the discrete-time state-space form is that it enables direct design in the discrete domain. Once you have $A_d$ and $B_d$, you can use pole placement or Linear Quadratic Regulator (LQR) design to compute a state feedback gain $K_d$ for a digital controller. The control law $u[k]=-K_{d}x[k]$ is then directly implementable in a microcontroller.

Similarly, discrete-time models are necessary for designing digital state observers, such as the discrete Kalman filter. The predictor-corrector structure of the Kalman filter relies entirely on the discrete-time state-space equations to predict the next state ($A_{d}x[k]+B_{d}u[k]$) and then update the prediction with a new measurement. This direct digital design flow, from modeling to implementation, is more straightforward and accurate than designing in continuous time and then attempting to approximate the controller with a discrete equivalent.

Common Pitfalls

1. Ignoring the Zero-Order Hold Assumption: A frequent mistake is to naively approximate $A_d\approx I+AT$ and $B_d\approx BT$. This Euler approximation is only valid for very fast sampling rates and does not correctly account for the held input between samples. Always use the exact ZOH discretization formulas ($e^{AT}$ and its integral) unless you have explicitly validated that the approximation error is acceptable for your application.
2. Choosing an Inappropriate Sampling Period: Selecting a sampling time $T$ that is too slow (relative to the system's fastest dynamics) will lead to aliasing and a loss of controllability/observability. Conversely, an excessively fast sampling rate wastes computational resources. A good rule of thumb is to sample at least 10-30 times faster than the highest frequency of interest in the system's closed-loop response.
3. Assuming $C$ and $D$ Always Remain Unchanged: While $C_d=C$ and $D_d=D$ is standard for ZOH discretization, this is not universal. If your system model includes a computational delay or if you are using a different hold circuit (e.g., a first-order hold), the output and feedthrough matrices may require modification. Always verify the assumptions of your discretization method.
4. Mishandling Nonlinear or Time-Varying Systems: The matrix exponential method is exact only for LTI systems. For nonlinear systems, you must linearize around an operating point before discretizing, or use more advanced numerical integration techniques. Discretizing time-varying systems requires solving a time-varying differential equation, not simply evaluating $e^{A(t)T}$.

Summary

• Discrete-time state-space models, expressed as $x[k+1]=A_{d}x[k]+B_{d}u[k]$, are the fundamental representation for analyzing and designing computer-controlled systems.
• Discretization under a Zero-Order Hold assumption uses the matrix exponential: $A_d=e^{AT}$ and $B_d={\int }_0^T{e}^{A\tau }d\tau B$, which accurately maps continuous system behavior to sampling instants.
• Under appropriate sampling conditions, the critical system properties of controllability and observability are preserved from the continuous-time model to its discrete counterpart.
• This representation enables the direct digital design of state-feedback controllers ($u[k]=-K_{d}x[k]$) and discrete-time observers like the Kalman filter, creating a seamless path from model to implementation.
• Successful application requires careful selection of the sampling period and strict adherence to the assumptions of the discretization method to avoid errors like aliasing and loss of control authority.