# Tag Info

5

As you pointed out, there are many state-space realizations of one particular transfer function. The reason is that a transfer function only represents the input-output behavior of a system (observable and controllable dynamics) and not the internal states. That being said, you can directly write state-space realizations from a transfer function with the so-...

3

You have $$f\left(\mathbf x, u\right) = \begin{bmatrix}\frac{-1}{T}\tau+\frac{K}{T} u \\ \frac{\tau}{mr} \\ 0 \end{bmatrix} \tag a$$ From which you (eventually) derive $$\mathbf {A}_d=\begin{bmatrix} 1-\frac{\Delta T}{T} & 0 &0 \\ \frac{\Delta T}{m_{op} r} & 1 & \frac{-\tau_{op} \Delta T}{m_{op}^{2} r}\\ 0& 0 & 1 \end{bmatrix} \tag ... 2 Try looking at the error term$$e(k) = \mathbf{y}(k) - \mathbf{C}_d\cdot\hat{\mathbf{x}}(k)$$and testing it for whiteness. If the state estimate is good, then all the predictable component will be predicted and the remainder will be white noise (unpredictable). 1 Since you don't have any inputs, x_n is only a function of past values of itself, your B an D should indeed be zero. You don't have your output defined either. You could pick the most recent x_n as output, but it could also be something else. A transfer functions describe an input output relation. However, you don't have inputs, so also no input output ... 1 I think you may do one of the following: Given a Parametric Model of the Signal You may use least squares. In case the model is Linear you may use linear least squares (For instance, polynomial regression). If the model is not linear, then a non linear least squares. Given a Dynamic Model of the Signal If you have a model which connect the signal u[t] to u[... 1 The Tustin approximation is concerned with transfer functions, i.e. relations between inputs and outputs. In state space representation$$ \dot{\mathbb{x}}(t) = A \mathbb{x}(t) + B \mathbb{u}(t)  \mathbb{y} = C \mathbb{x}(t) + D \mathbb{u}(t) $$for continuous time or$$ \mathbb{w}[(k+1)T] = A \mathbb{w}[kT] + B \mathbb{u}[kT]  \mathbb{y}[kT] = C \...

1

It describes a SIMO (single input, multiple outputs) system. In your case you have two outputs, described by the two different numerator polynomials.

1

First, typically when you're exerting a force on something and getting a position, the acceleration varies instantaneously with force. A more or less universal equation of motion for a single-axis linear system would be $m \ddot x = f_v(\dot x) + f_p(x)$. For a mass-spring-damper system, it'd be $m \ddot x = b \dot x + k x$. So you can easily express that ...

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