Reduced-order control using low-rank dynamic mode decomposition

Abstract

In this work, we perform full-state LQR feedback control of fluid flows using non-intrusive data-driven reduced-order models. We propose a model reduction method called low-rank dynamic mode decomposition (lrDMD) that solves for a rank-constrained linear representation of the dynamical system. lrDMD is shown to have lower data reconstruction error compared to standard optimal mode decomposition (OMD) and dynamic mode decomposition (DMD), but with an increased computational cost arising from solving a non-convex matrix optimization problem. We demonstrate model order reduction in the complex linearized Ginzburg–Landau equation in the globally unstable regime and on the unsteady flow over a flat plate at a high angle of attack. In both cases, low-dimensional full-state feedback controller is constructed using reduced-order models constructed using DMD, OMD and lrDMD. It is shown that lrDMD stabilizes the Ginzburg–Landau system with a lower- order controller and is able to suppress vortex shedding from an inclined flat plate at a cost lower than either DMD or OMD. It is further shown that lrDMD yields an improved estimate of the adjoint system, for a given rank, relative to DMD and OMD.

Appendices

Appendix A: Numerical methods

In this section, we present the two methods to solve the lrDMD optimization problem.

A.1 Subspace projection method

In this method, we use iterative subspace projection to find a good approximation of the optimal solution. We first make the observation that $Q_R$ is an orthogonal projection matrix in the column space of $X^TR$. This means that there exists an orthogonal matrix $C_R$ such that $C_RC_R^T = Q_R$ and $\text {Im}(C_R) = \text {Im}(Q_R)$. Substituting $Q_R$ with $C_RC_R^T$, we get the following cost function,

$$\begin{aligned} G(L,R) = -\left||L^TYC_R\right||_F^2. \end{aligned}$$

with the constraint that $\text {Im}(C_R) = \text {Im}(Q_R)$. If $C_R$ is fixed, the optimal solution for L is given by the left singular vectors of $YC_R$. Finding $C_R$ for a fixed L under the given constraint is not as trivial. If the left singular vectors of $X^TR$ span the same space as the left singular vectors of $Y^TL$, then optimal $C_R$ under the constraint will just be the left singular vectors of $X^TR$. As an approximation, we try to find R that minimizes the distance between $X^TR$ and $Y^TL$ by solving the following ‘Orthogonal Procrustes Problem’ [54]

$$\begin{aligned} \min _R \left||Y^TL - X^TR\right||_F^2. \end{aligned}$$

The closed form optimal solution to this problem is $R = UV^T$ where U and V are the left and the right singular vector matrices, respectively, from the singular vector decomposition of $XY^TL$. $C_R$ is given by the orthogonal basis of the column space of $X^TR$ denoted by $\Pi (X^TR)$. Algorithm 1 describes all the steps for this method.

The solution provided by this algorithm relies heavily on the initial guess. We only need an initial guess for R or, effectively, $C_R$. In our study, we choose the r leading left singular vectors of X as the initial guess for R. Note that this initial guess is the same as the projection subspace used in DMD. Due to the optimal choice of L for a fixed $C_R$, the chosen initial guess ensures that the algorithm provides a solution with a reconstruction error at most as high as the error in DMD reconstruction of the same rank.

A.2 Gradient descent method

In this section, we describe the Riemannian gradient descent method employed to solve the optimization problem. For a thorough review on Riemannian optimization on the Grassmanian manifold, the reader is referred to [40]. For a function F(L) defined on the Grassmanian manifold where $L\in \mathcal {G}_{r,m}$, the gradient $\text {grad}\, F(L) \in \mathcal {T}_{L}\mathcal {G}_{r,m}$ (tangent space to the manifold at the point L) is defined as

$$\begin{aligned} \text {grad}\,F(L)&= \nabla F(L) - LL^T\nabla F(L) \end{aligned}$$

(17)

$$\begin{aligned} \text {where}\quad (\nabla F(L))_{ij}&= \frac{d F}{dL_{ij}} \end{aligned}$$

(18)

Similarly, the action of the Hessian on any tangent vector $dL \in \mathcal {T}_L\mathcal {G}_{r,m}$ is defined as

$$\begin{aligned} \text {Hess}\,F(L)[dL]&= (I_m - LL^T)\nabla ^2F(L)[dL] - dL(L^T\nabla F(L)) \end{aligned}$$

(19)

$$\begin{aligned} \text {where}\quad (\nabla ^2 F(L))_{ij,kl}&= \frac{d^2F}{dL_{ij}dL_{kl}}. \end{aligned}$$

(20)

Further details of the method and its implementation can be seen in [36]. We employ the trust-region algorithm [42] to solve the optimization problem using the MATLAB package ManOpt [41].

Appendix B: lrDMD with control

The lrDMD framework can easily be extended to incorporate the effects of control. Consider a dynamical system with state vector $x\in \mathbb {R}^m$ such that

$$\begin{aligned} x_{k+1} = f(x_k,u_k), \end{aligned}$$

where $u \in \mathbb {R}^p$ is the control input and the subscripts denote the time iteration. We have access to a sequence of time snapshots of the state vector which we represent as a data matrices $X, Y \in \mathbb {R}^{m\times n}$ formed by n pairs of data snapshots as follows,

$$\begin{aligned} X:=(x_1|\cdots | x_n), \quad Y:=(x_2|\cdots |x_{n+1}). \end{aligned}$$

We additionally have access to a sequence of control inputs, which we arrange in the following matrix

$$\begin{aligned} \Omega :=(u_1|\cdots | u_n). \end{aligned}$$

Our goal is to use matrices X, Y and $\Omega$ to obtain a low-order approximation of the function $f(\cdot )$. To this end, we construct a linear approximation of f in both the state x and u such that

$$\begin{aligned} f(x,u) = \hat{A}x + \hat{B}u. \end{aligned}$$

Along with a rank-constraint on the system dynamical matrix, we get the following optimization problem

$$\begin{aligned} \min _{L,D,R,B} \left||Y - LDR^TX - LB\Omega \right||_F^2, \end{aligned}$$

(21)

where $D\in \mathbb {R}^{r\times r}$, $B\in \mathbb {R}^{r\times p}$, $L,R\in \mathbb {R}^{m\times r}$ and $L^TL=R^TR=I_r$ ($r\times r$ identity matrix) such that $\hat{A} = LDR^T$ is the r-ranked matrix approximating the state dynamics and $\hat{B} = LB$ accounts for the effect of control input on the state evolution.

For a fixed L and R, we observe that the objective function is convex for both D and B. The optimality conditions for the solution of D and B for fixed L and R are

$$\begin{aligned} D(R^TXX^TR) + B(\Omega X^TR)&= L^TYX^TR, \\ B(\Omega \Omega ^T) + D(R^TX\Omega ^T)&= L^TY\Omega ^T, \end{aligned}$$

which can be rewritten as

$$\begin{aligned} \left[ D \;\;\;\; B \right] \left[ \begin{array}{cc} R^TXX^TR \;\;\;\;&{} \Omega X^TR \\ R^TX\Omega ^T \;\;\;\;&{} \Omega \Omega ^T \end{array}\right] = \left[ L^TYX^TR \;\;\;\; L^TY\Omega ^T \right] \end{aligned}$$

(22)

We propose two ways to solve this problem. Both methods work with an initial guess for the optimal L, R which can be obtained from leading singular vectors of the data matrices as shown in the main text. The first method is alternative minimization by solving for optimal (D, B) for fixed (L, R) by solving Eq. () followed by gradient-based minimization of the objective function () for fixed (D, B). The second method is to use Schur complements to get closed form expressions for the optimal $(D^*(L,R),B^*(L,R))$, substitute that in the objective function () and use gradient-based methods to find optimal (L, R) solution. We will explore these avenues in future studies.

Appendix C: Projection error comparison with ERA

ERA [18] is a system identification method proposed for linear systems. Consider a linear system with state variable $x \in \mathbb {R}^{m}$ and control input $u\in \mathbb {R}^{p}$ governed by the equation

$$\begin{aligned} x_{k+1} = Ax_{k} + Bu_{k}, \end{aligned}$$

where subscripts denote the time iteration, $A\in \mathbb {R}^{m \times m}$ is the state-transition matrix and $B\in \mathbb {R}^{m\times p}$ captures the effect of control on the state variable. The data matrix $X\in \mathbb {R}^{m\times p(m_c+1)}$ using impulse response of this system will be

$$\begin{aligned} X = \left[ B\,\,AB\,\,\cdots \,\,A^{m_c}B\right] \end{aligned}$$

where $m_c + 1$ is the number of snapshots.

The first step of the ERA method is to form Hankel matrices from the impulse response data of the system. We construct a generalized Hankel matrix $H\in \mathbb {R}^{m(m_o+1)\times p(m_c+1)}$

$$\begin{aligned} H = \left[ \begin{array}{cccc} B &{} AB &{} \cdots &{} A^{m_c}B \\ AB &{} A^2B &{} \cdots &{} A^{m_c + 1}B \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ A^{m_o}B &{} A^{m_o+1}B &{} \cdots &{} A^{m_o + m_c}B \end{array}\right] \end{aligned}$$

where $m_c$ and $m_o$ are chosen such that $m_c + m_o \le m$. We then compute the singular value decomposition of H to obtain left and right singular vectors $U\in \mathbb {R}^{m(m_o+1)\times p(m_c+1)}$ and $V\in \mathbb {R}^{p(m_c+1)\times p(m_c+1)}$, and the diagonal matrix with decreasing singular values $\Sigma \in \mathbb {R}^{p(m_c+1)\times p(m_c+1)}$, such that

$$\begin{aligned} H = U\Sigma V^T. \end{aligned}$$

The primal modes $\phi \in \mathbb {R}^{m\times p(m_c+1)}$ and adjoint modes $\psi \in \mathbb {R}^{p(m_c+1)}$ are given by

$$\begin{aligned} \phi&= XV\Sigma ^{-1/2} \\ \psi&= \phi (\phi ^T\phi )^{-1}. \end{aligned}$$

To compare the projection error for r-ranked reduced-order model, we consider the r leading columns $\phi$ and $\psi$ to get $\phi _r$ and $\psi _r$, respectively. The projection errors for eigenmode v and adjoint mode w are given by $\epsilon (v,\phi _r)$ and $\epsilon (w,\psi _r)$ where $\epsilon (\cdot )$ is defined in Equation (14). For consistency with Sect. 5.1.1 we use $m_c = 14$, so that we use the same number of snapshots in the data matrix X. $m_o$ is chosen to be 4. The comparison of projection error of lrDMD, OMD and DMD with ERA is shown in Fig. 9. Even though ERA has the same projection error for the unstable eigenmode compared to DMD, it shows significant improvement in the unstable adjoint mode projection error. As shown in Fig. 9b, lrDMD outperforms ERA only at ranks higher than 5. However, Fig. 10 shows that increasing $m_o$ improves ERA performance in adjoint projection error and even outperforms lrDMD although having the same higher error in eigenmode projection. This shows that adding delay coordinates by increasing $m_o$ can significantly decrease the adjoint projection error but does not affect the eigenmode projection.

Fig. 9

a Projection error for the unstable eigenmode and b projection error for the unstable adjoint mode for DMD (red, circle), OMD (blue, cross), lrDMD (green, square) and ERA (black, diamond) for different rank approximations (colour figure online)

Fig. 10

a Projection error for the unstable eigenmode and b projection error for the unstable adjoint mode for lrDMD (green, square), ERA with $m_o =4$ (black, diamond) and ERA with $m_o = 15$ for different rank approximations (colour figure online)

Fig. 11

$C_D$ versus time for $35^{\circ }$ inclined flat plate with the actuator activated at $t_0 = 210$ and control applied every 20 timesteps using DMD-, OMD- and lrDMD-based reduced-order controllers learned using snapshots 10 timesteps apart

Appendix D: Effect of different learning and control timesteps

In this section, examine what happens when the learning and control timesteps differ for DMD, OMD and lrDMD. Let the resulting reduced- order model using any one of the three methods be $\widehat{A}$. We generate controllers that would be applied to the flow every 20 timepsteps, and therefore, we build reduced-order controllers using the model $\widehat{A}^2$. We do this for DMD, OMD and lrDMD while keeping all the parameters the same as in Sect. 5.2.3. Figure 11 shows the controller performance when the controllers are switched on at $t_0 = 210$. This time, DMD and OMD outperform lrDMD in stabilizing the system and driving it to the steady state of the flow. This is because, while DMD and OMD keep dynamics confined to one low-dim subspace, lrDMD has different input and output subspaces that are optimal for dynamics discretized for $T = 10\Delta t$. Applying the reduced-order model for 20 timestep requires projection from subspace L to R since

$$\begin{aligned} (LDR^T)^2 = L(DR^TLD)R^T. \end{aligned}$$

We think this projection leads to a deterioration of the controller performance. In practice, the input and output subspace learned from snapshots that are 10 timesteps apart should be used as an initial condition to get optimal subspaces for dynamics discretized for $T = 20\Delta t$.