I have a linear system in the observable canonical form as \begin{align} \dot{x} &= \underbrace{\begin{pmatrix} -a_2 & 1&0\\ -a_1 & 0&1\\ -a_0 & 0&0 \end{pmatrix}}_{A}x + \underbrace{\begin{pmatrix} b_2\\b_1\\1 \end{pmatrix}}_{B}u\\ y&=\underbrace{\begin{pmatrix} 1&0&0 \end{pmatrix}}_{C}x \end{align} where $b_2,b_1>0$. Assuming $x$ is available for measurement, i would like to design a state-space adaptive pole placement controller so that the closed loop poles are all located at $c$, and hence $x$ and $y$ converge to zero.

As a first step, i designed a PPC as if the parameters were known. The question is: do I need to convert the system into the controllable canonical form? Doing this I would obtain \begin{align} \dot{x} &= \underbrace{\begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ -a_0 & -a_1 & -a_2 \end{pmatrix}}_{A_c} x + \underbrace{\begin{pmatrix} 0\\0\\1 \end{pmatrix}}_{B_c}u\\ y &= \underbrace{\begin{pmatrix} 1&b_1&b_2 \end{pmatrix}}_{C_c}x \end{align}

which makes the estimation of the unknown parameters more tricky as $b_2, b_1$ are disappearing from the B matrix. Indeed, considering this canonical form, I'm not sure if the we can structure a Static parameter model like $z(t)= \theta_p^{*,T}\phi(t)$ with \begin{equation} z = \frac{s^3}{\varLambda_p(s)}x, \quad \theta_p^{*,T}=\begin{pmatrix} \theta_b^*\\\theta_a^* \end{pmatrix}, \quad \phi= \begin{pmatrix} \frac{\alpha_2(s)}{\varLambda_p(s)}u\\-\frac{\alpha_2(s)}{\varLambda_p(s)}x \end{pmatrix} \end{equation} where $\varLambda_p(s)$ is a stable monic polynomial. An other possible way to estimate the parameters could be considering the transfer function \begin{equation}\label{tf5} G(s)=\frac{ b_2 s^2 + b_1 s+1}{s^3+ a_2 s^2 + a_1 s + a_0} \end{equation} and then building a SPM from that, but I'm not sure about this. Any insights?

  • Can you clarify what you mean with PPC and SPM? – Kwin van der Veen Oct 11 at 20:55
  • PPC is pole placement control and SPM is static parameter model – Betelgeuse Oct 12 at 8:06

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