# What is the difference between disturbance and noise for dynamic systems?

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:

1. 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?

2. 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?

3. 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?

• Your distinction between continuous and discrete systems seems misguided. Disturbances and noise obviously apply to both. – Scott Seidman Nov 26 '14 at 13:25
• But in most references about control theory, only disturbance or noise is considered. For example, in Kalman filter, only noise is considered. But, in observer designing, such as sliding mode observer designing, only disturbance is considered. Maybe, you can read some references about control theory, then you can know what I state is true in references. – lovewinter Nov 26 '14 at 13:54

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".

• Thanks very much for your answer. So, no matter whether the system is continuous or discrete, we can always utilize $d$ to describe all the external uncertainly along to the system ? Then, the first question is solved. Now, what about question 2 and 3? – lovewinter Nov 26 '14 at 14:08
• @lovewinter question 2 and 3 are covered by the answer because there is no need to assume anything about the disturbances, including whether they are "differential" (what ever you may mean by that). – akellyirl Nov 26 '14 at 14:19
• Thanks your very much. Now, we assume w(t) is the uncertainty in systems (3) and (4). However, in Kalman filtering theory, for the systems (3) and (4), we must know the statistical property of the noise. Is this meaning that Kalman filtering can only tackle systems that only influenced by noise? However, disturbance may be always existing in a real system. So, for a real system, Kalman filtering is invalid ? – lovewinter Nov 26 '14 at 15:09
• Maybe, there exist few systems which only influenced by noise. Maybe, we can assume the influence from disturbance to systems is tiny. But, how can we assume the influence of disturbance is tiny to the studied systems? For questions 2 and 3, we may need to compute the derivative of the output y sometimes, then we must assume the uncertainty d(t) is differential. So, can we assume the uncertainty d(t) is differential? Although, no restriction to d(t) is best. Thanks again for your kind help. – lovewinter Nov 26 '14 at 15:10