Adaptive control designs for control-based continuation of periodic orbits in a class of uncertain linear systems

Abstract

This paper proposes two novel adaptive control designs for the feedback signals used in the control-based continuation paradigm to track families of periodic orbits of periodically excited dynamical systems, including black box simulation models and physical experiments. The proposed control designs rely on modifications to the classical model reference adaptive control framework and the more recent ${\mathscr {L}}_1$ adaptive control architecture, in which an additional low-pass filter is used to ensure guaranteed transient performance and robustness to time delays in the control input even in the limit of arbitrarily large adaptive gains. In contrast to the proportional control formulations that have been used in the literature on control-based continuation, the proposed control designs achieve stable performance with a minimum of parameter tuning. In the context of a class of linear systems with matched uncertainties, the paper demonstrates the successful integration of adaptive control feedback in control-based continuation. Specifically, the control designs are shown to ensure that the control input stabilizes the sought periodic orbits of the uncontrolled system and vanishes along these orbits, provided that an a priori unknown reference input is chosen appropriately. Numerical results obtained using the coco software package demonstrate how the combination of a nonlinear solver (Newton’s method) with the pseudo-arclength parameter continuation scheme can be used to trace the correct choice for the reference input under variations in an excitation parameter.

Appendices

Bounded inputs

Recall the model reference adaptive control design from Sect. 2.2. As shown there, $e=x-x_m$ and ${\tilde{\theta }}={\hat{\theta }}-\theta$ are bounded signals. Since this holds also for r and g, it follows from Eq. (11) and the fact that A is Hurwitz that $x_m$ is bounded, and consequently, that x is bounded. Equation (9) then implies that u is bounded.

For the ${\mathscr {L}}_1$ adaptive control design in Sect. 2.3, the boundedness of x, ${\hat{x}}$, and u follows under an additional condition on the bandwidth k. Here, ${\hat{\eta }}(t)\triangleq {\hat{\theta }}^\top (t) x(t)$ and ${\hat{\theta }}\in {\mathscr {B}}$ imply that

$$\begin{aligned} \left\| {\hat{\eta }}_\tau \right\| _{{\mathscr {L}}_\infty }\le L\left\| x_\tau \right\| _{{\mathscr {L}}_\infty } \end{aligned}$$

(56)

for any $\tau <\infty$ and some $L>0$. It follows from (30) that

$$\begin{aligned} \left\| {\hat{\eta }}_\tau \right\| _{{\mathscr {L}}_\infty }\le L\left( \left\| {\hat{x}}_\tau \right\| _{{\mathscr {L}}_\infty }+\frac{2R}{\sqrt{\lambda _\mathrm {min}(P)\varGamma }}\right) . \end{aligned}$$

(57)

Next, let $\zeta (t)\triangleq {\tilde{\theta }}^\top (t)r(t)$ and denote the Laplace transform of g(t) by g(s). It follows from Eq. (24) that

$$\begin{aligned} \Vert {\hat{x}}_\tau \Vert _{{\mathscr {L}}_{\infty }}&\le \Vert G(s)\Vert _{{\mathscr {L}}_1}\Vert {\hat{\eta }}_\tau \Vert _{{\mathscr {L}}_{\infty }}+\Vert H(s)\Vert _{{\mathscr {L}}_1}\big \Vert \zeta _\tau \big \Vert _{{\mathscr {L}}_{\infty }}\nonumber \\&\quad +\Vert (s{\mathbb {I}}-A)^{-1}\left( g(s)+x(0)\right) \Vert _{{\mathscr {L}}_{\infty }}, \end{aligned}$$

(58)

where $G(s)\triangleq H(s)(1-C(s))$, $H(s)\triangleq (s{\mathbb {I}}-A)^{-1}b$, and $C(s)=k/(s+k)$. Substitution of (57) in (58) and reorganization yields

$$\begin{aligned}&\left( 1-\Vert G(s)\Vert _{{\mathscr {L}}_1}L\right) \Vert {\hat{x}}_{\tau }\Vert _{{\mathscr {L}}_{\infty }}\le \Vert G(s)\Vert _{{\mathscr {L}}_1} \frac{2RL}{\sqrt{\lambda _{\text {min}}(P)\varGamma }}\nonumber \\&\quad +\Vert H(s)\Vert _{{\mathscr {L}}_1}\big \Vert \zeta _\tau \big \Vert _{{\mathscr {L}}_{\infty }}\nonumber \\&\quad +\Vert (s{\mathbb {I}}-A)^{-1}\left( g(s)+x(0)\right) \Vert _{{\mathscr {L}}_{\infty }} \end{aligned}$$

(59)

and the boundedness of ${\hat{x}}$ follows provided that k satisfies the stability condition

$$\begin{aligned} \Vert G(s)\Vert _{{\mathscr {L}}_1}L<1. \end{aligned}$$

(60)

It is straightforward to show that this holds provided that k exceeds some threshold. In this case, the bound on ${\tilde{x}}$ implies the boundedness of x. Since ${\hat{\theta }}$ is bounded by construction, it follows that u is bounded.

In the case of the linear system given by (34), the ${\mathscr {L}}_1$-norm of G(s) as a function of the filter bandwidth k is shown in Fig. 25. With ${\mathscr {B}}={\mathscr {B}}(0,2)$, it follows that $L=2\sqrt{2}$ and, consequently, that $\Vert G(s)\Vert _{{\mathscr {L}}_1}$ must be less than $1/(2\sqrt{2})$ for the stability condition to hold. In the main text, we use $k=10$ for which $\Vert G(s)\Vert _{{\mathscr {L}}_1}\approx 0.23$.

For the linear system given by (44), and with ${\mathscr {B}}={\mathscr {B}}(0,2)$, the stability condition again holds as long as $\Vert G(s)\Vert _{{\mathscr {L}}_1}<1/(2\sqrt{3})$. With $k=20$, $\Vert G(s)\Vert _{{\mathscr {L}}_1}\approx 0.26$.

Fig. 25

${\mathscr {L}}_1$-norm of G(s) as a function of the filter bandwidth k for the system defined in Eq. (34)

Transient performance

Suppose that k is chosen so that the stability condition (60) holds. The ideal dynamics of the ${\mathscr {L}}_1$ adaptive control design is represented by the nonadaptive reference system

$$\begin{aligned} {\dot{x}}_\mathrm {ref}&=Ax_\mathrm {ref}+b\left( u_\mathrm {ref}+\theta ^\top x_\mathrm {ref}\right) +g,\, x_\mathrm {ref}(0)=x(0), \end{aligned}$$

(61)

$$\begin{aligned} {\dot{u}}_\mathrm {ref}&=-k\left( u_\mathrm {ref}+\theta ^\top x_\mathrm {ref}\right) ,\,u_\mathrm {ref}(0)=0. \end{aligned}$$

(62)

It follows from the analysis below that

$$\begin{aligned}&\Vert x_\mathrm {ref}-x\Vert _{{\mathscr {L}}_{\infty }}\le \frac{\gamma _1}{\sqrt{\varGamma }}+\gamma _2\Vert r\Vert _{{\mathscr {L}}_{\infty }},\nonumber \\&\Vert u_\mathrm {ref}-u\Vert _{{\mathscr {L}}_{\infty }}\le \frac{\gamma _3}{\sqrt{\varGamma }}+\gamma _4\Vert r\Vert _{{\mathscr {L}}_{\infty }}\nonumber \\ \end{aligned}$$

(63)

and

$$\begin{aligned} \lim _{t\rightarrow \infty }\Vert x_\mathrm {ref}(t)-x(t)\Vert =0, \, \lim _{t\rightarrow \infty }\Vert u_\mathrm {ref}(t)-u(t)\Vert =0, \end{aligned}$$

(64)

where

$$\begin{aligned} \gamma _1&\triangleq \frac{\Vert C(s)\Vert _{{\mathscr {L}}_1}}{1-\Vert G(s)\Vert _{{\mathscr {L}}_1}L}\frac{2R}{\sqrt{\lambda _{\text {min}}(P)}}, \end{aligned}$$

(65)

$$\begin{aligned} \gamma _2&\triangleq \frac{4\Vert C(s)\Vert _{{\mathscr {L}}_1}}{1-\Vert G(s)\Vert _{{\mathscr {L}}_1}L}R^2\Vert H(s)\Vert _{{\mathscr {L}}_1}, \end{aligned}$$

(66)

$$\begin{aligned} \gamma _3&\triangleq \Vert H_1(s)\Vert _{{\mathscr {L}}_1}\frac{2R}{\sqrt{\lambda _{\text {min}}(P)}}+\Vert C(s)\Vert _{{\mathscr {L}}_1}L\gamma _1, \end{aligned}$$

(67)

$$\begin{aligned} \gamma _4&\triangleq 4\Vert C(s)\Vert _{{\mathscr {L}}_1}R^2+\Vert C(s)\Vert _{{\mathscr {L}}_1}L\gamma _2, \end{aligned}$$

(68)

and $H_1(s)\triangleq C(s)\frac{1}{{c}_0^\top H(s)}{c}_0^\top$ is a proper and BIBO stable transfer function for some $c_0$ (whose existence follows from the controllability of the pair (A, b), cf. Lemma A.12.1 in [13]).

Indeed, in the frequency domain

$$\begin{aligned} x_\mathrm {ref}(s)-x(s)&=H(s)C(s){\tilde{\eta }}(s)+G(s)\theta ^\top \nonumber \\&\quad \big (x_\mathrm {ref}(s)-x(s)\big ), \end{aligned}$$

(69)

where ${\tilde{\eta }}(s)$ is the Laplace transform of ${\tilde{\eta }}(t)\triangleq {\tilde{\theta }}(t)x(t)$. It follows that

$$\begin{aligned} \Vert (x_\mathrm {ref}-x)_{\tau }\Vert _{{\mathscr {L}}_{\infty }}\le \frac{\Vert C(s)\Vert _{{\mathscr {L}}_1}}{1-\Vert G(s)\Vert _{{\mathscr {L}}_1}L}\Vert H(s){\tilde{\eta }}(s)\Vert _{{\mathscr {L}}_{\infty }}. \end{aligned}$$

(70)

The error dynamics in (26) now imply that

$$\begin{aligned} H(s){\tilde{\eta }}(s)={\tilde{x}}(s)-H(s){\tilde{\xi }}(s), \end{aligned}$$

(71)

where ${\tilde{\zeta }}(s)$ is the Laplace transform of ${\tilde{\zeta }}(t)\triangleq {\tilde{\theta }}(t)r(t)$. By the triangle inequality,

$$\begin{aligned} \Vert H(s){\tilde{\eta }}(s)\Vert _{{\mathscr {L}}_{\infty }}&\le \frac{2R}{\sqrt{\lambda _{\text {min}}(P)\varGamma }}+4R^2\Vert H(s)\Vert _{{\mathscr {L}}_1}\Vert r_{\tau }\Vert _{{\mathscr {L}}_{\infty }} \end{aligned}$$

(72)

and the first part of (63) follows since the upper bounds are independent of $\tau$.

Similarly,

$$\begin{aligned} u_\mathrm {ref}(s)-u(s)=C(s){\tilde{\eta }}(s)-C(s)\theta ^\top \big (x_\mathrm {ref}(s)-x(s)\big ), \end{aligned}$$

(73)

where

$$\begin{aligned} C(s){\tilde{\eta }}(s)=H_1(s){\tilde{x}}(s)-C(s){\tilde{\zeta }}(s) \end{aligned}$$

(74)

and the second part of (63) again follows since the upper bounds are independent of $\tau$.

Finally, from (69), we obtain

$$\begin{aligned} x_\mathrm {ref}(s)-x(s)=({\mathbb {I}}-G(s)\theta ^\top )^{-1}{H(s)C(s){\tilde{\eta }}(s)}, \end{aligned}$$

(75)

where $({\mathbb {I}}-G(s){\theta }^\top )^{-1}$ is stable. Since $\lim _{s\rightarrow 0}s{\tilde{\eta }}(s)=\lim _{t\rightarrow \infty }{\tilde{\eta }}(t)=0$, the claims of asymptotic convergence in (64) follow from the final-value theorem.

The tightness of the bounds in (63) depends on the values of $\varGamma$ and the radius R. As suggested in the main text, the ${\mathscr {L}}_1$ framework remains robust to time delays with arbitrarily large values of the adaptive gain $\varGamma$, limited only by the available hardware. Similarly, an initially conservative estimate of R may be significantly reduced during control-based continuation, provided that ${\hat{\theta }}(t)\rightarrow \theta$ during the initial set of iterations of the Newton solver. Since $x(t)\rightarrow 0$, it follows that $x_\mathrm {ref}(0)$ is close to 0 in each subsequent iteration, ensuring that x(t) and u(t) remain close to 0 throughout the analysis.