Adaptive 2

CRM MRAC of a First-Order Nonlinear System

Stability Proof

Prove stability of the resulting closed-loop and prove convergence of the control error to zero The idea behind Closed-loop reference models is to force the trajectories of the closed loop reference model to move towards the plant model. The differential equations are given as \[\begin{aligned} \dot x_m^o &= a_m x_m^o + k_mr,& a_m <0 \\ \dot x_m^c &= a_m x_m^c +k_mr - le^c, \qquad&l, a_m < 0 \\ e^c&=x_p-x_m^c \end{aligned}\]

The closed loop dynamics are described by \(x_m^c\) while \(e^c\) denotes the feedback from control error. The additional term \(e^c\) in the dynamics of the closed loop reference model can be interpreted as the reference model moving towards the plant dynamics. Since the open loop and the closed loop dynamics differ, convergence of the closed to the open loop has to be ensured. While this is beyond the scope of the exercise, a proof for convergence of the error dynamics \(e^c\) is given.

Considering the control input \(u\) \[\begin{aligned} u &= \theta_a x_p + kr\end{aligned}\] the dynamics of the closed loop for the plant can be derived as follows \[\begin{aligned} \dot x_p &= a_p x_p + k_p u \\ &= a_p x_p + k_p ( \theta_a x_p + kr ) \\ &= ( a_p + k_p\theta_a^* ) x_p + k_p(\theta_a-\theta_a^*)x_p + k_p(k-\frac{k_m}{k_p})r +k_mr \\ &= a_mx_p + k_p\tilde \theta_a x_p + k_p\tilde k r + k_mr \\ &= a_mx_p + k_mr + k_p \tilde \theta^T \phi\end{aligned}\]

where \(\tilde \theta = \begin{bmatrix} \tilde \theta_a & \tilde k\end{bmatrix}\). In combination with the closed loop dynamics \(x_m^c\) from equation (\ref{eq:xmc}) the error dynamics can be evaluated \[\begin{aligned} \dot e_c &= \dot x_p - \dot x_m^c \\ &= a_m x_p + k_mr + k_p \tilde \theta^T\phi - a_mx_m^c - k_mr + le^c \\ &= a_m ( x_p - x_m^c ) + le^c + k_p \tilde \theta^T\phi \\ &= \underbrace{( a_m + l )}_{< 0} e^c + k_p\tilde \theta^T \phi\end{aligned}\]

which leads to \[\begin{aligned} e^{c}=\frac{1}{k^{*}}\frac{k_m}{s-(a_m+l)}\tilde{\theta}^T\phi=\frac{1}{k^{*}}M(s)[\tilde{\theta}^T\phi]\end{aligned}\] where \(M(s)\) is SPR. Thus \(e^c\) converges to zero because M(s) is SPR and therefore Hurwitz.

Using the MKY-Lemma, we know, that a Lyapunov function exits s.t. the derivative is negative definite. Hence the dynamics of the closed loop errror is asymptotically stable.

Investigation of a Non-minimal Plant Structure

In the following a system with a non-minimal structure (shown in figure \ref{fig:plant}), i.e. zero-pole-cancellation is allowed, will be analyzed and discussed.

The parameters \(k,a \in \R^1\) and \(\theta_u, \theta_y \in \R^{n-1}\) of the plant are to be determined in case of adaptive control.

The filters are Hurwitz given as

\[\begin{aligned} F(s) &= \frac{1}{\lambda(s)}\underbrace{\begin{bmatrix}1\\s\\\vdots\\s^{n-2}\end{bmatrix}}_{=:\xi(s)} \\\lambda(s) &= a_{n-1}s^{n-1} + a_{n-2}s^{n-2} + ... + a_1s+a_0\end{aligned}\]

Given Figure \ref{fig:plant} we determine the transfer function \(G_{nm}(s)\) as follows. \[G_{nm}(s)=\frac{k+\theta_u^TF}{s-a-\theta_y^TF}=\frac{\lambda k+\theta_u^T\xi(s)}{\lambda s-\lambda a-\theta_y^T\xi(s)}\]

System analysis

The relative degree is defined as the difference between the degree of the denominator’s polynomial order and the numerator’s polynomial order. Since \(\lambda\) is given to be of degree \(n-1\), the numerator’s polynomial of \(G_{nm}\) has degree \(n-1\) while the denominator’s polynomial has degree \(n\). Therefore the relative degree of the transfer function \(G_{nm}\) equals one.

2. What is the (full, i. e., non-minimal) order of \(G_{nm}\) , assuming non-zero parameters?

The order of a transfer function is denoted by the highest power of the denominator’s polynomial. The highest power of the denominator of \(G_{nm}\) is n. Therefore the order of \(G_{nm}\) is n.

3. Is F controllable?

A system can only lose its controlability or observability if a zero-pole-cancellation occurs (citation not found: lunze2). Conversely if no cancellation takes place controllability and observability is ensured.

Therefore we have to show that the transfer function \(F(s)\) is coprime to guarantee controllability and observability. Since \(\xi\) only contains potencies of \(s\) the only possible case in which a zero-pole-cancellation occurs is for \(a_0 = 0\). Due to the fact that \(\lambda(s)\) is Hurwitz all eigenvalues are less than zero. Hence \(a_0 \neq 0\), controllability and observability is ensured

Draw the linear part of the adaptive controller for \(G_{nm}\) . Assume the filters \(F\) and parameters to be known.

The differential equation of the plant \(G_{nm}(s)\) in the frequency domain \[\begin{aligned} G_{nm}:\; sX_p&=(a + \underbrace{\theta_y^TF}_{=:\Upsilon(s)})X_p + (\underbrace{\theta_u^TF}_{=:\zeta(s)} + k)U\end{aligned}\] The dynamics of the closed loop are required to behave similiar to the dynamics of an arbitrary reference model: \[\begin{aligned} G_m:\; sX_m=a_mX_m+k_mR \end{aligned}\] Since all parameters are assumed to be known an adaptive controller can be designed such that the resulting dynamics of the closed loop behaves like the dynamics of the reference model \[\begin{aligned} U&=-\frac{(a+\Upsilon)X_p}{\zeta+k}+\frac{a_mX_p+k_mR}{\zeta+k} \\ &=\frac{(-a-\Upsilon+a_m)X_p+k_mR}{\zeta+k}\end{aligned}\]

The block diagram of the adaptive controller is displayed in the following figure.