11

Use Google's define:<item to look up> feature. block diagram noun: block diagram; plural noun: block diagrams a diagram showing in schematic form the general arrangement of the parts or components of a complex system or process, such as an industrial apparatus or an electronic circuit. flow chart noun: flow chart; plural noun: flow charts; noun: ...


9

To a large extend the answer depends on the industry. For example if you work for a large aerospace company it might take a while before you can get chance to get involved in designing a control systems. In this case you will be most likely a team member design a control system. On the contrary if you happen to work for smaller company you will mostly get to ...


9

I hope I can help you in some way, your question bounces around a bit so I'm guessing a little on what you are wanting to know. This is how you come up with a controller that is implementable from scratch: 1) Make the model ("plant") The #1 preliminary skill a controls engineer needs is to be able to make a mathematical model of a system, whether it be ...


8

One way would be to implement some form of adaptive control. If your range of time constants is small and known, you could use something called "gain scheduling" where you determine before hand all the time constants you'll be dealing with (hopefully it is finite) and use if/then logic to define P I and D. It can be challenging to make sure you have covered ...


8

One way of doing this is using the Kalman decomposition. For this you need the reachable and unobservable subspaces. These subspaces can be constructed using the image of the controllability matrix and the kernel of the observability matrix respectively. The controllability matrix and its image, the observability matrix and its kernel are, $$ \mathcal{C} = \...


8

"Linear" imposes a set of restrictions. "Non-linear" simply means there are no restrictions. Many non-linear control schemes can be faster than linear ones. Linear control schemes are restricted to "smoothly" transitioning. Non-linear control can be implemented by suddenly slamming a digital value, for example. A good example of a fast-responding non-...


8

An example of control system with non oscillatory response. $$ \frac{d x}{d t} = 0 + u\\ u = x $$ Closed loop equation of this unstable system is $$ \frac{d x}{d t} = x $$ and its response is $$ x(t) = x(0) \cdot e^t $$ which is unstable response and does not oscillate. In general, for an unstable, linear system, the contribution of a pole on the positive ...


7

Because with the form: $C(s) = K_ds + K_p$ you have a non-proper transfer function. In process engineering, you always need to have proper transfer functions, where the degree of the numerator is less than or equal to the degree of the denominator. A work around is to introduce an "approximate derivative", which is the second form: $C(s) = (K_ds + K_p)/(\...


7

Convert the nonlinear model to state-space form, $x'=f(x,u)$, and linearize it to get a linear state-space or transfer function representation. You can use this to design a PID controller and simulate it that together with the original nonlinear model to see how the controller performs. As noted, the performance of the linear controller will most likely ...


7

Observability means that you can estimate the complete state using only the output, without knowing the initial state. In other words, you have to figure out where you are without knowing where you were initially. A more practical reason why this rarely works is that when you are limited by non-perfect sensors and non-zero sampling time, taking the ...


7

You are absolutely right. Since the magnitude is the absolute value of the complex function $F(j\omega)$ the plot does not change with the sign of the pole or zero. The Phase on the other hand does change with the sign of the pole making the system unstable. You can see this very good at this simple example (made with WolfamAlpha) This is the stable system. ...


6

For some systems, the salient criterion is settling time to within some error band. Sometimes you can get faster settling by allowing earlier overshoot. If you need a system to get to within some minimum error of before starting a process, you probably don't care what the system is doing before the process is started, only that it gets there as quickly as ...


6

Only the poles in the LHP are necessary for stability. This is because the transient response of a LTI system will consists of a linear combinations of $e^{p_i t}$. If a pole is complex, $p_i=\rho_i+i \sigma_i$, you can use Euler's formula, such that the contribution to the transient response can be written as, $$ e^{\rho_i t} \left(\cos(\sigma_i t)+i \sin(\...


6

Anti-windup is a concept for feedback controllers with integral terms, e.g. PID, to keep the integral term from „overcharging“ when regulating a large set point error. It basically saturates the integral term to keep the system from overshooting the set point. The classic form of anti-windup, as described above, does not actually ensure satisfaction of ...


5

There is a mistake in your expressions. The coefficient of $s$ is $2+k_p$. The conditions are: $$\frac{1}{3} \left(3 \left(k_p+2\right)-k_i\right)>0$$ $$k_i>0$$ $$k_p>0$$ This simplifies to: $$k_p>0$$ $$0<k_i<3 k_p+6$$ Update: If the transfer function is $$ \frac{1}{s^2+3 s + c} $$ where c is some positive constant, then the conditions ...


5

The solution is fairly straightforward. Short answer The system is controllable without any modifications. You made a small mistake calculating the new controllability matrix: you are missing the first column of $\bar{M}$, which should be $\begin{bmatrix}0 & I\end{bmatrix}^T$. Full proof Let's consider the second system, and define some augmented ...


5

I think there must be some other restrictions placed on your professor's comment. Adding a pole to a general second order system makes it a third order system and the dynamics can change dramatically. One example I can think of in which adding a pole or zero does not change the system dynamics dramatically is when the pole or zero is at much higher ...


5

To understand why proportional gain won't drive the error point to zero, it is best just to look at the math. Consider the PID loop shown in the image below. The loop algebra in the $s$ domain comes out to $$ \begin{align} e(s)&=r(s)-y(s)\\ y(s)&=P\ u(s)\\ u(s)&=\left(k_p+\frac{k_i}{s}+k_ds\right)e(s), \end{align} $$ where I have used $P$ ...


5

Without any more info, I think your problem arises from the values of $a_0$ and $a_1$. The answer is a little involved, so a bit of systems background is necessary. The short answer Your system should be stable, but I don't think you are simulating your system long enough to see it stabilize. Calculate your settling time and simulate it for at least that ...


5

A controller is built around a physical system. What is the open-loop behavior of the system? You have to sample the physical (continuous-time!) system in order to provide feedback to your discrete controller. What sampling rate should you be using for that microcontroller? Are you sure? Once you have the physical system modeled and the sampling mechanism ...


5

I thought I would expand a little on the answer offered by Karlo. Long story short, I would not try to calculate the analytical time response of a system to a square wave. That would be a serious pure-maths exercise, and not necessary for most engineering applications. Instead, I would suggest using the step response analytical vs. simulated to validate ...


5

In nonlinear control theory, you will recognize most concepts such as controllability and observability where the linear case is often a special case of the nonlinear case. I would highly recommend digging into linear control theory first if you have not done so. Depending on the course you take, concepts such as Lyapunov stability are discussed here, which ...


5

Suppose we have a third order polynomial in the form : $$ s^3+a_2s^2+a_1s+a_0 = 0$$ There is nice caveat for third order systems which is derived from the Routh-Hurwitz stability criterion. In order for this polynomial to be stable the following three conditions have to be met (trying to derive the Routh-Hurwitz table will be a total mess for this particular ...


5

Well as a fresh start: net(n) is the value before the node, this can be seen by equation (1). The value after the node is the following (lets call this $x$ for simplicity): $$x(n) = net(n) - a_1x(n-1) - a_2x(n-2)$$ Therefore: $$\tilde{y}(n) = b_0x(n)+b_1x(n-1)+b_2x(n-2)$$ Now substituting the first equation into the second one yields the following: $$\tilde{...


5

Let me phrase this a bit more elaborate. Suppose we have a state space system, ie: $$\dot{x} = Ax + Bu$$ Where the dynamics of the system are presented through $A$. For instance, the poles of the system are equal to the eigenvalues of $A$. The steady-state value can be computed by solving $0 = Ax^{ss}$. Now we have a full state feedback controller $u = -K(x-...


4

There are various levels of plant models you can develop, depending on what you want to do. The model you mention is probably suitable for most of the cases in terms of designing a controller. However, it seems that in your case the level of abstraction is too high, so my suggestion would to break the plant model further into sensors + plant + actuators, ...


4

Think for a moment about what it means for $\eta(t)$ to have reached its steady state. It means precisely that $\dot{\eta}(t)=0$. If you plug that into your first equation and solve for $\eta(t)$ you get your solution as you've already observed.


4

A professor of mine, long ago, once quipped: "Give me a word... any word at all... and I'll show you how the root of that word is a PDE!" A touch of Philosophy All physical systems are modeled by some set of mathematical equations, be they differential, algebraic, integral, stochastic, etc. The simplest models of physical systems only consider ...


4

PI - Proportional - Integral The output is a combination of how far you are from the goal and the integral of your distance from the goal (total error over time). This means that it will track small changes well but in the event of a large change it will be prone to overshooting. Good for systems which are inherently heavily damped. PD - Proportional - ...


4

Euler angles are easier to understand and use. Imagine a airtrafic controller getting a aircraft heading info as quaternion data. Euler angles are significantly easier to unerstan interpret and interpolate. While quaternions do have benefits they are also conceptually more complex to work with. In many applications the downsides of euler angles and matrices ...


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