Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Adaptive Control of PDEs

  • Henrik Anfinsen
  • Ole Morten Aamo
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_100022-1
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Abstract

The infinite-dimensional backstepping method has recently been used for adaptive control of partial differential equations (PDEs). We will in this article briefly explain the main ideas for the three most commonly used methods for backstepping-based adaptive control of PDEs: Lyapunov-based design, identifier-based design, and swapping-based design. Swapping-based design is also demonstrated on a simple, scalar hyperbolic PDE with an uncertain in-domain parameter, clearly showing all the steps involved in using this method.

Keywords

Partial differential equations Hyperbolic systems Stabilization Adaptive control Parameter estimation Infinite-dimensional backstepping 

Introduction

The backstepping method has for the last decade and a half been used with great success for controller and observer design of partial differential equations (PDEs). Starting with nonadaptive results for parabolic PDEs (Liu 2003; Smyshlyaev and Krstić 2004, 2005; Krstić and Smyshlyaev 2008c), the method was extended to hyperbolic PDEs in Krstić and Smyshlyaev (2008b), where a controller for a scalar 1-D system was designed. Extensions to coupled hyperbolic PDEs followed in Vazquez et al. (2011), Di Meglio et al. (2013), and Hu et al. (2016). The backstepping approach offers a systematic way of designing controllers and observers for linear PDEs. One of its key strengths is that state feedback control laws and state observers can be derived for the infinite-dimensional system directly, with all analysis carried out in the infinite-dimensional framework, avoiding any artifacts caused by spatial discretization.

Backstepping has turned out to be well suited for dealing with uncertain systems by adaptive designs. The first result appeared in Smyshlyaev and Krstić (2006) where a parabolic PDE with an uncertain parameter is stabilized by backstepping. Extensions in several directions subsequently followed (Krstić and Smyshlyaev 2008a; Smyshlyaev and Krstić 2007a, b), culminating in the book Adaptive Control of Parabolic PDEs (Smyshlyaev and Krstić 2010). Most of the ideas developed for parabolic PDEs carry over to the hyperbolic case, with the first result appearing in Bernard and Krstić (2014) for a scalar equation. Extensions to coupled PDEs followed in Anfinsen and Aamo (2017a, b, c) and Yu et al. (2017), to mention a few.

Methods for Adaptive Control of PDEs

In Smyshlyaev and Krstić (2010) and Anfinsen and Aamo (2019), three main types of control design methods for adaptive control of PDEs are mentioned. These are
  1. 1.

    Lyapunov-based design: This approach directly addresses the problem of closed-loop stability, with the controller and adaptive law designed simultaneously from Lyapunov functions. It has been used for adaptive control design for parabolic PDEs in Krstić and Smyshlyaev (2008a) and for a scalar hyperbolic PDE in Xu and Liu (2016). The Lyapunov method produces the simplest adaptive law in the sense of the controller’s dynamical order, but the stability proof quickly becomes complicated as the systems get more complex. For this reason, no results extending those of Xu and Liu (2016) to coupled PDEs have appeared in the literature.

     
  2. 2.

    Identifier-based design: An identifier is usually a copy of the system dynamics with uncertain system parameters replaced by estimates and with error signals in terms of differences between measured and computed signals injected. Adaptive laws are chosen so that boundedness of the identifier error is ensured, before a control law is designed with the aim of stabilizing the identifier. Boundedness of the original system follows from boundedness of the identifier and the identifier error. Identifier-based designs have been developed for parabolic PDEs in Smyshlyaev and Krstić (2007a) and hyperbolic PDEs in Anfinsen and Aamo (2016a) and Anfinsen and Aamo (2018). The control law is designed independently of the adaptive laws, and hence the identifier-based method is based on certainty equivalence (CE). Since the method employs a copy of the system dynamics along with the adaptive laws, the dynamical order of the controller is larger than the dynamical order of the system.

     
  3. 3.

    Swapping-based design: In swapping-based designs, filters are carefully constructed so that the system states can be statically expressed in terms of the filter states, the unknown parameters and some error terms that are proved to converge to zero. The key property of the static parameterization is that the uncertain parameters appear linearly, so that well-established parameter identification laws such as the gradient law or the least squares method can be used to generate estimates. It also allows for normalization, so that the update laws remain bounded regardless of the boundedness of the system states. Adaptive estimates of the states can then be generated from the filters and parameter estimates by substituting the uncertain parameters in the static parameterization with their respective estimates. A controller is designed for stabilization of the adaptive state estimates, meaning that this method, like the identifier-based method, is based on the certainty equivalence principle. The number of filters required when using this method typically equals the number of unknown parameters plus one, rendering the swapping-based adaptive controller of higher dynamical order than Lyapunov-based or identifier-based designs. Nevertheless, swapping-based design is the one most prominently used for adaptive control of hyperbolic PDEs since it extends to more complex systems in a natural way, as illustrated by the series of papers (Bernard and Krstić 2014; Anfinsen and Aamo 2016b, 2017a, d).

     

Application of the Swapping Design Method

To give the reader a flavor of what adaptive control designs for PDEs involve, we will demonstrate swapping-based design for stabilization of a simple linear hyperbolic PDE. First, we introduce some notation. We will by ||z|| and ||z|| denote the L2-norm and -norm respectively, that is \( ||z|| = \sqrt { \int _{0}^{1} z^2(x) dx } \) and ||z|| =supx ∈ [0,1]|z(x)| for some function z(x) defined for x ∈ [0, 1]. For a time-varying, real signal f(t), the following vector spaces are used: \( f \in \mathcal L_p \Leftrightarrow \left (\int _{0}^{\infty } |f(t)|{ }^p dt \right )^{\frac {1}{p}} < \infty \) for 1 ≤ p < , and \( f \in \mathcal L_\infty \Leftrightarrow \sup _{t \geq 0} |f(t)| < \infty . \) For two functions u(x), v(x) on x ∈ [0, 1], we define the operator ≡ as u ≡ v ⇔||u − v|| = 0, u ≡ 0 ⇔||u|| = 0. Lastly, the following space is defined \( \mathcal B([0, 1]) = \{ u(x) \mid ||u||{ }_\infty < \infty \} \).

Consider the linear hyperbolic PDE
$$\displaystyle \begin{aligned} u_t(x, t) - u_x(x, t) & = \theta u(0, t), \end{aligned} $$
(1a)
$$\displaystyle \begin{aligned} u(1, t) & = U(t), \end{aligned} $$
(1b)
$$\displaystyle \begin{aligned} u(x, 0) & = u_0(x) \end{aligned} $$
(1c)
where Open image in new window is an uncertain parameter, while \( u_0 \in \mathcal B([0, 1]) \). Although θ is uncertain, we assume that we have some a priori bound, as formally stated in the following assumption:

Assumption 1

A bound on θ is known. That is, we are in knowledge of a nonnegative constant\( \bar \theta \)so that\( |\theta | \leq \bar \theta \).

Nonadaptive Control

A finite-time convergent stabilizing nonadaptive controller is Krstić and Smyshlyaev (2008b)
$$\displaystyle \begin{aligned} U(t) = - \theta \int_{0}^{1} e^{\theta(1 - \xi)} u(\xi, t) d\xi. \end{aligned} $$
(2)
This can be proved using the backstepping transformation
$$\displaystyle \begin{aligned} w(x, t) &= T[k, u(t)](x){}\\ &= u(x, t) - \int_{0}^{x} k(x - \xi) u(\xi, t) d\xi \end{aligned} $$
(3)
with inverse
$$\displaystyle \begin{aligned} u(x, t) &=T^{-1}[\theta, w(t)](x){}\\ &= w(x, t) - \theta \int_{0}^{x} w(\xi, t) d\xi \end{aligned} $$
(4)
where
$$\displaystyle \begin{aligned} k(x) = - \theta e^{\theta x} \end{aligned} $$
(5)
is the kernel function. T maps system (1a) into the target system
$$\displaystyle \begin{aligned} w_t(x, t) - w_x(x, t)& = 0,\ w(x, 0) = w_0(x) {} \end{aligned} $$
(6a)
$$\displaystyle \begin{aligned} w(1, t)& =U(t)\\ &\quad +\theta \int_{0}^{1} e^{\theta(1 - \xi)} u(\xi, t) d\xi. {} \end{aligned} $$
(6b)
Choosing the control law U(t) as (2) yields w(1, t) = 0, for which it trivially follows that w ≡ 0 for t ≥ 1. From (4), it is clear that w ≡ 0 implies u ≡ 0.

Adaptive Control

Filter Design and Nonadaptive State Estimate

We introduce the input filter ψ and parameter filter ϕ as
$$\displaystyle \begin{aligned} \psi_t(x, t) - \psi_x(x, t)& =0,\\ \psi(1, t)& =U(t),\\ \psi(x, 0)& =\psi_0(x) {} \end{aligned} $$
(7a)
$$\displaystyle \begin{aligned} \phi_t(x, t) - \phi_x(x, t)& =0,\\ \phi(1, t)& =u(0, t),\\ \phi(x, 0)& =\phi_0(x). {} \end{aligned} $$
(7b)
Using the filters, a nonadaptive estimate\( \bar u \) of the system state u can be generated as
$$\displaystyle \begin{aligned} \bar u(x, t)& = \psi(x, t) + \theta F[\phi(t)](x),\\ F[\phi(t)](x)& =\int_{x}^{1} \phi(1 - \xi + x, t) d\xi. \end{aligned} $$
(8)
Notice that \( \bar u \) is linear in the uncertain parameter θ and that F, ϕ, and ψ are known.

Lemma 1

Consider system (1a), filters (7), and the variable\( \bar u \)generated from (8). For t ≥ 1, we have\(\bar u \equiv u.\)

Proof

Defining, \( e(x, t) = u(x, t) - \bar u(x, t) = u(x, t) - \psi (x, t) - \theta F[\phi (t)](x) \), it is straightforward to show, using the dynamics (1a) and (7), that e satisfies the dynamics
$$\displaystyle \begin{aligned} e_t(x, t) - e_x(x, t)& =0,\\ e(1, t)& =0, e(x, 0) =e_0(x) \end{aligned} $$
(9)
and hence e ≡ 0 and \( \bar u \equiv u \) for t ≥ 1. □

Adaptive Law

From the linear relationship \( u(x, t) = \bar u(x, t) = \psi (x, t) + \theta F[\phi (t)](x) \) for t ≥ 1 ensured by Lemma 1, we have u(0, t) = ψ(0, t) + θF[ϕ(t)](0) from which we propose the gradient adaptive law with normalization
$$\displaystyle \begin{aligned} \dot{\hat \theta}(t) & = \begin{cases} \text{proj}_{\bar \theta} \left \{ \gamma \dfrac{\hat e(0, t) \int_{0}^{1} \phi(x, t) dx}{1 + ||\phi(t)||{}^2}, \hat \theta(t) \right \}, & \text{for} \; t \geq 1 \\ 0 & \text{otherwise} \\ \end{cases} & \hat \theta(0), & = \hat \theta_0, & |\hat \theta_0| & \leq \bar \theta \end{aligned} $$
(10)
for some gain γ > 0, \( \hat \theta _0 \) chosen inside the feasible domain as given by Assumption 1 and where
$$\displaystyle \begin{aligned} \hat e(x, t) & = u(x, t) - \hat u(x, t), & \hat u(x, t) & = \psi(x, t) + \hat \theta(t) F[\phi(t)](x), \end{aligned} $$
(11)
and the projection operator is defined as
$$\displaystyle \begin{aligned} \text{proj}_{a}(\tau, \omega) = \begin{cases} 0 & \text{if} \; (\omega = -a \; \text{and} \; \tau \leq 0) \; \text{or} \; (\omega = a \; \text{and} \; \tau \geq 0) \\ \tau & \text{otherwise}. \\ \end{cases} \end{aligned} $$
(12)
Notice that the adaptive state estimate\( \hat u \) is obtained from the nonadaptive estimate by simply substituting the uncertain parameter θ with its adaptive estimate provided by (10). The following lemma states important properties regarding the adaptive law which are crucial in the closed-loop stability analysis.

Lemma 2

Subject to Assumption 1 , the adaptive law (10) with initial condition\( |\hat \theta _0| \leq \bar \theta \)provides the following signal properties
$$\displaystyle \begin{aligned} |\hat \theta(t)| & \leq \bar \theta, \forall t \geq 0 {} \end{aligned} $$
(13a)
$$\displaystyle \begin{aligned} \dot{\hat \theta}, \sigma & \in \mathcal L_2 \cap \mathcal L_\infty, & \sigma(t) & = \dfrac{\hat e(0, t)}{\sqrt{1 + ||\phi(t)||}}. {} \end{aligned} $$
(13b)

Proof

Property (13a) follows from the initial condition (10) and the projection operator (Anfinsen and Aamo 2019, Lemma A.1). Consider
$$\displaystyle \begin{aligned} V(t) & = \dfrac{1}{2 \gamma} \tilde \theta^2(t), & \tilde \theta(t) & = \theta - \hat \theta(t). \end{aligned} $$
(14)
For t ∈ [0, 1), we will trivially have \( \dot V(t) = 0 \). For t ≥ 1, we have, using the property \( -\tilde \theta \text{proj}_{\bar \theta } (\tau , \hat \theta ) \leq - \tilde \theta \tau \) (Anfinsen and Aamo 2019, Lemma A.1), that
$$\displaystyle \begin{aligned} \dot V(t) \leq - \tilde \theta(t) \dfrac{\hat e(0, t) \int_{0}^{1} \phi(x, t) dx}{1 + ||\phi(t)||{}^2}. \end{aligned} $$
(15)
From (8), (11) and Lemma 1, we have, for t ≥ 1 \( \hat e(0, t) = u(0, t) - \hat u(0, t) = \psi (0, t) + \theta F[\phi (t)](0) - \psi (0, t) - \hat \theta (t) F[\phi (t)](0) = \tilde \theta (t) \int _{0}^{1} \phi (x, t) dx, \) and substituting this into (15), we obtain
$$\displaystyle \begin{aligned} \dot V(t) \leq - \sigma^2(t), \end{aligned} $$
(16)
where we have used the definition of σ stated in (13b). This proves that V is bounded and nonincreasing and hence has a limit, V, as t →. Integrating (16) in time from zero to infinity gives
$$\displaystyle \begin{aligned} \int_0^\infty \sigma^2(t) dt & \leq-\int_0^\infty \dot V(t) dt\\ & = V(0) - V_\infty \,{\leq}\, V(0) \,{\leq}\, \dfrac{2}{\gamma} \bar \theta^2 < \infty, \end{aligned} $$
(17)
and hence \(\sigma \in \mathcal L_2.\) Moreover, using \(\hat e(0, t)=\tilde \theta (t) \int _{0}^{1} \phi (x, t) dx,\) we have
$$\displaystyle \begin{aligned} \sigma^2(t)& =\dfrac{\hat e^2(0, t)}{1 + ||\phi(t)||{}^2}\\ &\leq\tilde \theta^2(t) \dfrac{(\int_{0}^{1} \phi(x, t) dx)^2}{1 + ||\phi(t)||{}^2}\\ &\leq\tilde \theta^2(t) \dfrac{\int_{0}^{1} \phi^2(x, t) dx}{1 + ||\phi(t)||{}^2} \notag \\ & \leq \tilde \theta^2(t) \dfrac{||\phi(t)||{}^2}{1 + ||\phi(t)||{}^2} \leq \tilde \theta^2(t) \leq 4 \bar \theta^2 < \infty \end{aligned} $$
(18)
where we used Cauchy-Schwarz’ inequality and hence \( \sigma \in \mathcal L_\infty . \) For t ∈ [0, 1) or if the projection is active, the adaptive law (10) is zero, and hence boundedness of \( \dot {\hat \theta } \) follows. For t ≥ 1, we have for an inactive projection and inserting \( \hat e(0, t) = \tilde \theta (t) \int _{0}^{1} \phi (x, t) dx \), that
$$\displaystyle \begin{aligned} |\dot{\hat \theta}(t)|& =\gamma \left |\dfrac{\hat e(0, t) \int_{0}^{1} \phi(x, t) dx}{1 + ||\phi(t)||{}^2}\right |\\ &=\gamma \left |\dfrac{\hat e(0, t)}{\sqrt{1 + ||\phi(t)||{}^2}}\dfrac{\int_{0}^{1} \phi(x, t) dx}{\sqrt{1 + ||\phi(t)||{}^2}}\right |\\ &\leq\gamma |\sigma(t)| \end{aligned} $$
(19)
and since \( \sigma \in \mathcal L_2 \cap \mathcal L_\infty \), it follows that \( \dot {\hat \theta } \in \mathcal L_2\)\(\cap \, \mathcal L_\infty \). □

Adaptive Control Law

As mentioned, the swapping-based design employs certainty equivalence. An adaptive stabilizing control law is hence obtained from the nonadaptive control law (2) by simply substituting the uncertain parameter θ by its adaptive estimate \( \hat \theta \) and the unmeasured state u by its adaptive state estimate \( \hat u \) to obtain
$$\displaystyle \begin{aligned} U(t) & = - \hat \theta(t) \int_{0}^{1} e^{\hat \theta(t)(1 - \xi)} \hat u(\xi, t) d\xi. \end{aligned} $$
(20)
We obtain the following main result for the closed-loop dynamics.

Theorem 1

Consider system (1a), filters (7), the adaptive law (10), and the adaptive state estimate (11). The control law (20) ensures
$$\displaystyle \begin{aligned} ||u||, ||\psi||, ||\phi||, ||u||{}_\infty, ||\psi||{}_\infty, ||\phi||{}_\infty & {\in} \mathcal L_2 {\cap} \mathcal L_\infty \end{aligned} $$
(21a)
$$\displaystyle \begin{aligned} ||u||, ||\psi||, ||\phi||, ||u||{}_\infty, ||\psi||{}_\infty, ||\phi||{}_\infty & \rightarrow 0. \end{aligned} $$
(21b)
The proof of Theorem 1 involves several steps. First, a backstepping transformation with a time-varying kernel function brings the adaptive state estimate’s dynamics into a target system which involves the parameter estimate as well as it’s time derivatives (adaptive laws). Next, a relatively comprehensive Lyapunov analysis involving heavy use of Cauchy-Schwarz’ and Young’s inequalities, along with the properties of Lemma 2, follows, which establishes that the Lyapunov function
$$\displaystyle \begin{aligned} V_1(t) &= 4 \int_{0}^{1} (1 + x) \eta^2(x, t) dx \\ &\quad + \int_{0}^{1} (1 + x) \phi^2(x, t) dx \end{aligned} $$
(22)
satisfies
$$\displaystyle \begin{aligned} \dot V_1(t) & \leq - c V_1(t) + l_1(t) V_1(t) + l_2(t) \end{aligned} $$
(23)
where c is a positive constant, \(\eta (x,t) = T[\hat k, \hat u(t)](x) \) with \( \hat k \) given by (5) with θ replaced by \(\hat \theta \) and l1 and l2 are non-negative and integrable (i.e., \(l_1, l_2 \geq 0, l_1, l_2 \in \mathcal L_1\)). Finally, established stability and convergence results from the literature on adaptive control are applied.

Summary and Future Directions

It is apparent from the example that the swapping-based method is quite involved even for the simple scalar system. However, the method extends to more complex systems in a natural way and can be modified (see Anfinsen and Aamo 2019) to accommodate spatially varying parameters, any number of coupled hyperbolic equations carrying information in either direction, interface with ODE’s at the boundary or in the interior, etc. The backstepping transformation induces a complicated relationship between the uncertain parameters of the system in its original coordinates and the uncertainties appearing in the target system, often leading to vast over-parameterization accompanied by poor robustness properties of the closed loop. Robustness in general is a topic for further investigation before the method is ripe for application to practical problems (Lamare et al. 2018). How to deal with nonlinearities is another important direction of future research. While backstepping was invented for systems described by nonlinear ordinary differential equations, it has had little success on the nonlinear arena for PDEs. Some local stability results exist for linear backstepping designs (applied to semi-linear PDEs without uncertainty, Coron et al. 2013), but it is unclear if backstepping can be helpful in providing truly nonlinear control laws in the PDE case. For semi-linear systems without uncertainty, nonlinear state feedback laws and state observers have been designed by an approach not using backstepping (Strecker and Aamo 2017). Interestingly, though, the state feedback law reduces to the backstepping control law in the linear case.

Recommended Reading

The book (Krstić and Smyshlyaev 2008c) gives an introduction to both nonadaptive and adaptive control of PDEs using the backstepping method. For a more in-depth study of adaptive control of PDEs, the books (Smyshlyaev and Krstić 2010) and (Anfinsen and Aamo 2019) provide the state-of-the-art for the parabolic and hyperbolic case, respectively.

Cross-References

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Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  • Henrik Anfinsen
    • 1
  • Ole Morten Aamo
    • 1
  1. 1.Department of Engineering CyberneticsNTNUTrondheimNorway

Section editors and affiliations

  • Miroslav Krstic
    • 1
  1. 1.Department of Mechanical and Aerospace EngineeringUniversity of CaliforniaSan Diego, La JollaUSA