Discrete-Time Negative Imaginary Systems
1. Motivation: From Continuous-Time Theory to Digital Implementation
Negative imaginary (NI) systems theory is inherently formulated in continuous time, whereas almost all practical controllers are implemented digitally.
In a digital control loop, a computer
- samples plant outputs at discrete time instants,
- computes control actions according to a discrete-time control law,
- reconstructs a continuous-time control input via a zero-order hold (ZOH) device.
As a consequence, the resulting discrete-time dynamics are only an approximation of the underlying continuous-time system and, in general, do not preserve the same physical properties.
A key result of my work is that the NI property is an exception: it can be preserved under the sample-and-hold process.
2. Zero-Order Hold and NI Preservation
Consider a continuous-time system \[\dot x = f(x,u), \qquad y = h(x),\] which is negative imaginary. By definition, there exists a continuously differentiable, positive definite storage function ( V(x) ) (here, we consider the case without free body motion) such that \[\begin{equation} \dot V(x) \leq u^\top \dot y. \tag{1} \end{equation}\]
Under zero-order hold, the input is held constant over each sampling interval. Integrating both sides of (1) over the time interval \([t_k,t_{k+1}]\) yields \[V(x(t_{k+1}))-V(x(t_k))\leq u(t_k)^\top \bigl(y(t_{k+1})-y(t_k)\bigr).\tag{2}\]
Suppose the discrete-time system obtained from the sample and hold process has the model: \[x_{k+1} = f_d(x_k,u_k),\qquad y_k = h(x_k).\]
Then inequality (2) implies that the discrete-time system satisfies \[V(x_{k+1})-V(x_k)\leq u_k^\top \bigl(y_{k+1}-y_k\bigr). \tag{3}\]
We call a discrete-time system satisfying (3) a ZOH negative imaginary (ZOH-NI) system.
Importantly, (3) is a purely discrete-time dissipation inequality, expressed solely in terms of variables at the sampling instants.
An important implication of this result is that, unlike passivity, the NI property can be preserved under zero-order hold sampling, whereas passivity is in general not preserved under ZOH discretization.
3. Stability of ZOH-NI Systems
Once a continuous-time plant is implemented via sample and hold, stability must be analyzed and enforced at the discrete-time closed-loop level.
This requires controller design and stability theory that are intrinsically discrete time, rather than relying on continuous-time analysis followed by numerical discretization.
4. My Contribution
I developed a discrete-time control framework for ZOH-NI systems and established corresponding stability results. This framework enables direct digital controller design for sampled-data NI systems, avoiding the commonly used approach of first designing a continuous-time controller and subsequently discretizing it.
Also, there is no approximation involved in this control design method. Hence, the stabilizing controller is reliable.