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I'm writing a feedback controller for the following SIMO system, where I want to give as input reference position and velocity $r_{ref}$, $v_{ref}$. The errors on position and velocity will be combined according to a control law in the form

$u = -k_p(r-r_{ref}) - k_d(v-v_{ref})$

where $u$ is the scalar control signal for the system.

My transfer functions from $u$ to $r$ is

$T_{u\rightarrow r} = \frac{a}{s^2+b}$

and since $v = \dot{r}$, I get

$T_{u\rightarrow v} = \frac{as}{s^2+b}$.

My question is: how do I understand the performance limits of this type of system? Given the maximum input constraint that I can shape by looking at the K*S transfer function, what else limits this type of system? For example, can I still get reduced steady-state error without violating input constraints and have good stability margins with this control structure, or should I necessarily move to a more complex control structure (e.g., PID or $H_\infty$)?

At the moment I'm getting GM = 6dB (which seems fine), but PM = 14°, which is really poor.

Thanks a lot!

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  • $\begingroup$ What are the values of $a, b, k_p, k_d$? And what kind of $r_{ref}$, $v_{ref}$ do you want to track (constant, sine, ...)? $\endgroup$
    – SampleTime
    Apr 2 '20 at 19:27
  • $\begingroup$ Hi @SampleTime a and b are equal to 4 and 6 respectively. Kp and Kd is what I'm computing (in the last example I got kp = 0.78, kd = 1.97. Finally, the reference signal r_{ref} is some sinusoidal with frequency of about 0.1 Hz, while v_{ref} is simply the derivative of that as I said. Do you think that if I add a lead-lag compensator I can drastically improve robustness and performance giving the maximum input? (the maximum input expected for a unitary step input is 2). Thanks! $\endgroup$
    – venom
    Apr 3 '20 at 0:57
  • $\begingroup$ @SampleTime please share some knowledge! :) $\endgroup$
    – venom
    Apr 4 '20 at 3:05
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For tracking of periodic references you can make use of the internal model principle. Your plant is:

$$ \begin{align} \dot{x} &= A x + B u \newline y &= C x \end{align} $$

with

$$ \begin{align} A &= \begin{bmatrix} 0 & 1 \newline -b & 0 \end{bmatrix}, B = \begin{bmatrix} 0 \newline a \end{bmatrix}, C = \begin{bmatrix} 1 & 0 \end{bmatrix} \newline x &= \begin{bmatrix} x_1 \newline x_2 \end{bmatrix} = \begin{bmatrix} r \newline v \end{bmatrix} \newline y &= x_1 = r \end{align} $$

By the internal model principle you should use the information about your reference signal in the controller. You can do that by using the following controller dynamics:

$$ \begin{align} \dot{x}_c &= A_c x_c + B_c e \newline y_c &= C_c x_c \end{align} $$

with $\omega_0 = 2 \pi f_0$ (the frequency of your reference signal),

$$ \begin{align} A_c &= \begin{bmatrix} 0 & 1 \newline -\omega_0^2 & 0 \end{bmatrix} , B_c = \begin{bmatrix} 0 \newline \omega_0 \end{bmatrix}, C_c = \begin{bmatrix} 1 & 0 \newline 0 & 1 \end{bmatrix} \newline x_c &= \begin{bmatrix} x_{c,1} \newline x_{c,2} \end{bmatrix} \newline e &= r_{ref} - y \end{align} $$

Finally let $u = -(K x + K_c y_c)$ where $K = \begin{bmatrix} k_1 & k_2 \end{bmatrix}$ and $K_c = \begin{bmatrix} k_{c,1} & k_{c,2} \end{bmatrix}$. Put everything together:

$$ \begin{align} \dot{z} &= A_z z + B_z u + B_r r_{ref} \newline y &= C_z z \end{align} $$

with

$$ \begin{align} A_z &= \begin{bmatrix} A & 0 \newline -B_c C & A_c \end{bmatrix}, B_z = \begin{bmatrix} B \newline 0 \end{bmatrix}, B_r = \begin{bmatrix} 0 \newline B_c \end{bmatrix}, C_z = \begin{bmatrix} C & 0 \end{bmatrix} \newline z &= \begin{bmatrix} z_1 \newline z_2 \newline z_3 \newline z_4 \end{bmatrix} = \begin{bmatrix} x_1 \newline x_2 \newline x_{c,1} \newline x_{c,2} \end{bmatrix} \newline u &= -K_z z \newline y &= z_1 = r \end{align} $$

Design a controller matrix $K_z = \begin{bmatrix} K & K_c \end{bmatrix} = \begin{bmatrix} k_1 & k_2 & k_{c,1} & k_{c,2} \end{bmatrix}$ for $(A_z, B_z)$, for example with LQR. This will give you the open loop transfer function from $r_{ref}$ to $y$:

$$ G_o(s) = \frac{b_1 s + b_0}{s^4 + a_3 s^3 + a_2 s^2 + a_1 s + a_0} $$

and

$$ \begin{align} b_1 &= a k_{c,2} \omega_0 \newline b_0 &= -a k_{c,1} \omega_0 \newline a_3 &= a k_2 \newline a_2 &= \omega_0^2 + b + a k_1 \newline a_1 &= a k_2 \omega_0^2 \newline a_0 &= \omega_0^2 (b + a k_1) \end{align} $$ And the closed loop transfer function $G_{cl} = G_o/(1 + G_o)$.


We can now insert values: $a = 4, b = 6, f_0 = 0.1$. I use the following weight matrices for the LQR design:

$$ Q = \begin{bmatrix} 1 & 0 & 0 & 0 \newline 0 & 1 & 0 & 0 \newline 0 & 0 & 1 & 0 \newline 0 & 0 & 0 & 1 \end{bmatrix}, R = 1 $$

That gives me $K_z = \begin{bmatrix} 0.9789 & 1.2204 & 0.0466 & -1.8782\end{bmatrix}$ using the Matlab lqr function. The overall open loop transfer function is

$$ G_o(s) = \frac{4.72 s - 0.1171}{s^4 + 4.882 s^3 + 10.31 s^2 + 6.648 s + 3.797} $$

which has a gain margin of $20$ dB and a phase margin of $65.7^\circ$. Finally, we can look at the tracking performance:

Tracking

You can see that both $r_{ref}$ and $v_{ref}$ are tracked successfully, with the error converging asymptotically to zero and the control input $u$ is in the allowed interval of $-2 \leq u \leq 2$. Of course that also depends on the amplitude of your reference signal, if it gets larger, $u$ will also get larger so this gives you a limit on how large the amplitudes of your reference signal can be.

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  • $\begingroup$ Hi @SampleTime thanks a lot! Just a last question. How does the PM change if I have to use the velocity directly as feedback signal together with the position? shall I look at the magnitude MS of the 2x2 S sensitivity transfer matrix and then compute the new margin as PM = asin(1/(2*MS)) ? Because in the design above you're assuming the plant to be purely a SISO, right? $\endgroup$
    – venom
    Apr 4 '20 at 23:44
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    $\begingroup$ @venom The design above already uses both the position and the velocity for feedback, as it is a full-state feedback controller: $k_1$ is the gain for the position and $k_2$ the gain for the velocity, $k_{c,1}, k_{c_2}$ are gains for the controller dynamics. $\endgroup$
    – SampleTime
    Apr 5 '20 at 9:32
  • $\begingroup$ cool, thanks a lot! $\endgroup$
    – venom
    Apr 6 '20 at 1:10

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