In most references from dynamic system theory, the following linear continuous dynamic system is considered.
$$\frac{\text{d}x(t)}{\text{d}t}=Ax(t)+Bu(t)+Dd_{1}(t)\quad (1)$$
$$y(t)=Cx(t)+Ed_{2}(t) \quad (2)$$
where \$x\in \mathbb{R}^{n}, y\in {{\mathbb{R}}^{p}},d_{1}\in
{{\mathbb{R}}^{m}},d_{2}\in{{\mathbb{R}}^{q}}\$ represent the state vector, measurement output vector, process disturbance and measurement disturbance vector respectively. \$A, B, C, D, E\$ are constants matrices of appropriate dimension.
Again, the following linear discrete dynamic system is mostly studied in references.
$$x(k+1)=Ax(k)+Bu(k)+Dw_{1}(k)\quad (3)$$
$$y(k)=Cx(k)+Ew_{2}(k)\quad (4)$$
where \$x\in \mathbb{R}^{n}, y\in {{\mathbb{R}}^{p}},w_{1}\in
{{\mathbb{R}}^{m}},w_{2}\in{{\mathbb{R}}^{q}}\$ represent the state vector, measurement output vector, process noise and measurement noise vector respectively.
My questions are:
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Are the disturbance \$d\$ and noise \$w\$ the same thing? If not, why in continuous system, only disturbance is considered, and only noise is considered in discrete system?
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In the continuous system, when the disturbance \$d\$ is stated as a certain function, can the disturbance \$d\$ be assumed to be differential? Is this assumption reasonable?
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In the continuous system, when the disturbance \$d\$ can be stated as a stochastic process such as Gauss white noise, can the disturbance \$d\$ be assumed to be differential? Is this assumption reasonable?
Best Answer
Disturbances considered in state-space systems are not constrained to be of any particular type. Step, sinusoidal, stochastic, impulse, disturbances are all described in the literature. Whether the system under consideration is continuous time or discrete doesn't matter; there is no distinction regarding the type of disturbance that can be / is analysed.
Sometimes one type of disturbance is more relevant to the problem at hand because they model real world phenomena; e.g. a step in a control system or a stochastic in a communications channel.
Step disturbances are popular for control system analysis because you usually require a zero steady-state error.
Stochastic disturbances are popularly analysed in Communications channels; but their application to control systems is also a well studied field; e.g. "Discrete Time Stochastic Systems", T.Soderstrom, Springer, 2002
It is true that discrete time controllers have become popular in the same era as stochastic approaches to control systems. This is partly coincidental but may also be due to easier analysis in discrete time; e.g. Soderstrom states "discrete time stochastic processes are much easier to handle than their continuous time counterparts, which have certain mathematical subtleties that are far from trivial to handle in a stringent way".