samples/texts/3314066/page_4.md

an exact factor of two, if $k_{\text{BW}}^{\text{T}}$ has been designed for a bandwidth $\omega_{\text{CL}} = \alpha$:

$$k_{\alpha }^{\text{T}}=\frac{1}{2}\cdot k_{\text{BW}}^{\text{T}}=\frac{1}{2}\cdot (k_{\text{BW},1}\cdots k_{\text{BW},n})\mathrm{.}(12)k\_\{\backslash alpha\}^\{\backslash text\{T\}\}\; =\; \backslash frac\{1\}\{2\}\; \backslash cdot\; k\_\{\backslash text\{BW\}\}^\{\backslash text\{T\}\}\; =\; \backslash frac\{1\}\{2\}\; \backslash cdot\; (k\_\{\backslash text\{BW\},1\}\; \backslash cdots\; k\_\{\backslash text\{BW\},n\}).\; \backslash quad\; (12)$$

Proof. We start by rewriting (11) as follows,

$$\alpha \mathbf{P}=-({\mathbf{A}}^{\mathrm{T}}\mathbf{P}+\mathbf{P}\mathbf{A})+\mathbf{S},(13)\backslash alpha\; \backslash mathbf\{P\}\; =\; -(\backslash mathbf\{A\}^\{\backslash mathrm\{T\}\}\; \backslash mathbf\{P\}\; +\; \backslash mathbf\{P\}\; \backslash mathbf\{A\})\; +\; \backslash mathbf\{S\},\; \backslash quad\; (13)$$

where, using (10) and (12),

$$\mathbf{S}=\frac{1}{2}{\mathbf{k}}_{\text{BW}}{\mathbf{k}}_{\text{BW}}^{\text{T}}=\frac{1}{2}\cdot (k_{\text{BW},1}{\mathbf{k}}_{\text{BW}}\cdots k_{\text{BW},n}{\mathbf{k}}_{\text{BW}})\mathrm{.}(14)\backslash mathbf\{S\}\; =\; \backslash frac\{1\}\{2\}\backslash mathbf\{k\}\_\{\backslash text\{BW\}\}\backslash mathbf\{k\}\_\{\backslash text\{BW\}\}^\{\backslash text\{T\}\}\; =\; \backslash frac\{1\}\{2\}\; \backslash cdot\; (k\_\{\backslash text\{BW\},1\}\backslash mathbf\{k\}\_\{\backslash text\{BW\}\}\; \backslash cdots\; k\_\{\backslash text\{BW\},n\}\backslash mathbf\{k\}\_\{\backslash text\{BW\}\}).\; \backslash quad\; (14)$$

Since A is an upper shift matrix, PA will result in P's columns pᵢ being shifted:

$$\mathbf{P}\mathbf{A}=(\mathbf{0}{\mathbf{p}}_1\cdots {\mathbf{p}}_{n-1})\mathrm{.}(15)\backslash mathbf\{P\}\backslash mathbf\{A\}\; =\; (\backslash mathbf\{0\}\; \backslash \; \backslash mathbf\{p\}\_1\; \backslash \; \backslash cdots\; \backslash \; \backslash mathbf\{p\}\_\{n-1\}).\; \backslash quad\; (15)$$

From the first column of (13) we obtain p₁, and, as an abbreviation, introduce Φ:

For all other columns ($i=2, \dots, n$):

We now recursively expand (17) for the final ($n$-th) column:

$$p_n=\sum _{i=1}^n(-1{)}^{(n-i)}\cdot {\mathrm{\Phi }}^{-(n-i+1)}\cdot \frac{k_{\text{BW},i}}{2}{k}_{\text{BW}}\mathrm{.}(18)p\_n\; =\; \backslash sum\_\{i=1\}^\{n\}\; (-1)^\{(n-i)\}\; \backslash cdot\; \backslash Phi^\{-(n-i+1)\}\; \backslash cdot\; \backslash frac\{k\_\{\backslash text\{BW\},i\}\}\{2\}\; k\_\{\backslash text\{BW\}\}.\; \backslash quad\; (18)$$

$\mathbf{p}_n^\mathrm{T}$ is the gain vector of the $\alpha$-controller, since, recalling (10) with (3), $\mathbf{k}_\alpha^\mathrm{T} = \mathbf{b}^\mathrm{T}\mathbf{P} = \mathbf{b}^\mathrm{T}\mathbf{P}^\mathrm{T} = \mathbf{p}_n^\mathrm{T}$. Multiplying (18) with $\mathbf{\Phi}^n$ one obtains:

$${\mathbf{\Phi }}^n\cdot {\mathbf{p}}_n=(\sum _{i=1}^n(-1{)}^{(n-i)}\cdot {\mathbf{\Phi }}^{(i-1)}\cdot k_{\text{BW},i})\cdot \frac{1}{2}{k}_{\text{BW}}\mathrm{.}(19)\backslash mathbf\{\backslash Phi\}^n\; \backslash cdot\; \backslash mathbf\{p\}\_n\; =\; \backslash left(\; \backslash sum\_\{i=1\}^\{n\}\; (-1)^\{(n-i)\}\; \backslash cdot\; \backslash mathbf\{\backslash Phi\}^\{(i-1)\}\; \backslash cdot\; k\_\{\backslash text\{BW\},i\}\; \backslash right)\; \backslash cdot\; \backslash frac\{1\}\{2\}\; k\_\{\backslash text\{BW\}\}.\; \backslash quad\; (19)$$

The characteristic polynomial of $\mathbf{\Phi}$ is:

$$\det (\lambda \mathbf{I}-\mathbf{\Phi })=\det (\lambda \mathbf{I}-(\alpha \mathbf{I}+{\mathbf{A}}^{\mathrm{T}}))=(\lambda -\alpha {)}^n\mathrm{.}(20)\backslash det\; (\backslash lambda\; \backslash mathbf\{I\}\; -\; \backslash mathbf\{\backslash Phi\})\; =\; \backslash det\; (\backslash lambda\; \backslash mathbf\{I\}\; -\; (\backslash alpha\; \backslash mathbf\{I\}\; +\; \backslash mathbf\{A\}^\{\backslash mathrm\{T\}\}))\; =\; (\backslash lambda\; -\; \backslash alpha)^n.\; \backslash quad\; (20)$$

Comparing (20) with (4) and (5) when $\alpha = \omega_{\text{CL}}$ we find the characteristic polynomial to be:

$$\det (\lambda \mathbf{I}-\mathbf{\Phi })={\lambda }^n-\sum _{i=1}^n(-1{)}^{(n-i)}\cdot k_{\text{BW},i}\cdot {\lambda }^{i-1}\mathrm{.}(21)\backslash det\; (\backslash lambda\backslash mathbf\{I\}\; -\; \backslash mathbf\{\backslash Phi\})\; =\; \backslash lambda^n\; -\; \backslash sum\_\{i=1\}^\{n\}\; (-1)^\{(n-i)\}\; \backslash cdot\; k\_\{\backslash text\{BW\},i\}\; \backslash cdot\; \backslash lambda^\{i-1\}.\; \backslash quad\; (21)$$

This allows us to apply the Cayley-Hamilton theorem to (19), with $\mathbf{\Phi}^n = \sum_{i=1}^n (-1)^{(n-i)} \cdot (\mathbf{\Phi}^{(i-1)}) \cdot k_{\text{BW},i}$ we finally obtain:

$$p_n=k_{\alpha }=\frac{1}{2}{k}_{\text{BW}}\mathrm{.}(22)p\_n\; =\; k\_\backslash alpha\; =\; \backslash frac\{1\}\{2\}\; k\_\{\backslash text\{BW\}\}.\; \backslash quad\; (22)$$

This concludes the proof. As the analytical solution of the algebraic Riccati equation (11), it provides a link between optimal control and pole placement for linear ADRC. □

Remark 2. Due to the duality of the design problem, a proof of the half-gain relation for the extended state observer design (with $k_{\text{ESO}} \cdot \omega_{\text{CL}} = \alpha$) can be constructed in the same manner.

5. EXAMPLES

Aim of this section is to provide visual insights into an ADRC-based control loop when using half-gain tuning for the controller, the extended state observer, or both. For this purpose we can restrict ourselves to a second-order plant with normalized gain and eigenfrequency:

$$P(s)=\frac{1}{s^2+2s+1}\mathrm{.}(23)P(s)\; =\; \backslash frac\{1\}\{s^2\; +\; 2s\; +\; 1\}.\; \backslash quad\; (23)$$

Since ADRC is almost insensitive to the damping ratio, especially of underdamped systems, cf. Herbst (2013), the informative value of our example will not be compromised by the particular choice of critical damping in $P(s)$.

Bandwidth parameterization is applied to a second-order ADRC ($n=2$) using $\omega_{\text{CL}} = 1$ rad/s, $k_{\text{ESO}} = 10$, and $b_0 = 1$. Four cases are being compared: (1) unmodified bandwidth tuning, (2) applying half-gain tuning only to the outer control loop (“K/2 controller”), (3) applying half-gain tuning only to the ESO (“L/2 observer”), and (4) half-gain tuning for both controller and observer.

5.1 Impact on Open-Loop Characteristics

For stability and dynamics, the feedback controller part of an ADRC control loop is essential. In Figure 4 the transfer functions from controlled variable $y$ to control signal $u$ are compared for the four possible cases. Additionally, the loop gain transfer functions are being compared in Figure 5.

Fig. 4. Comparison of the feedback controller transfer functions with or without half-gain tuning.

The most interesting result might be that half-gain observer tuning (“L/2” case) provides significantly improved high-frequency damping while having almost no impact on the lower frequencies up to and including the crossover frequency. On the other hand one has to expect some low-frequency performance penalty when (additionally or solely) applying half-gain controller tuning (“K/2” cases).