Differential dynamic programming

Differential dynamic programming (DDP) is an optimal control algorithm of the trajectory optimization class. The algorithm was introduced in 1966 by Mayne and subsequently analysed in Jacobson and Mayne's eponymous book. The algorithm uses locally-quadratic models of the dynamics and cost functions, and displays quadratic convergence. It is closely related to Pantoja's step-wise Newton's method.

Finite-horizon discrete-time problems

The dynamics

$\mathbf{x}_{i+1} = \mathbf{f}(\mathbf{x}_i,\mathbf{u}_i)$

(1)

describe the evolution of the state $\textstyle\mathbf{x}$ given the control $\mathbf{u}$ from time $i$ to time $i+1$. The total cost $J_0$ is the sum of running costs $\textstyle\ell$ and final cost $\ell_f$, incurred when starting from state $\mathbf{x}$ and applying the control sequence $\mathbf{U} \equiv \{\mathbf{u}_0,\mathbf{u}_1\dots,\mathbf{u}_{N-1}\}$ until the horizon is reached:

$J_0(\mathbf{x},\mathbf{U})=\sum_{i=0}^{N-1}\ell(\mathbf{x}_i,\mathbf{u}_i) + \ell_f(\mathbf{x}_N),$

where $\mathbf{x}_0\equiv\mathbf{x}$, and the $\mathbf{x}_i$ for $i>0$ are given by Eq. 1. The solution of the optimal control problem is the minimizing control sequence $\mathbf{U}^*(\mathbf{x})\equiv \operatorname{argmin}_{\mathbf{U}} J_0(\mathbf{x},\mathbf{U}).$ Trajectory optimization means finding $\mathbf{U}^*(\mathbf{x})$ for a particular $\mathbf{x}_0$, rather than for all possible initial states.

Dynamic programming

Let $\mathbf{U}_i$ be the partial control sequence $\mathbf{U}_i \equiv \{\mathbf{u}_i,\mathbf{u}_{i+1}\dots,\mathbf{u}_{N-1}\}$ and define the cost-to-go $J_i$ as the partial sum of costs from $i$ to $N$:

$J_i(\mathbf{x},\mathbf{U}_i)=\sum_{j=i}^{N-1}\ell(\mathbf{x}_j,\mathbf{u}_j) + \ell_f(\mathbf{x}_N).$

The optimal cost-to-go or value function at time $i$ is the cost-to-go given the minimizing control sequence:

$V(\mathbf{x},i)\equiv \min_{\mathbf{U}_i}J_i(\mathbf{x},\mathbf{U}_i).$

Setting $V(\mathbf{x},N)\equiv \ell_f(\mathbf{x}_N)$, the dynamic programming principle reduces the minimization over an entire sequence of controls to a sequence of minimizations over a single control, proceeding backwards in time:

$V(\mathbf{x},i)= \min_{\mathbf{u}}[\ell(\mathbf{x},\mathbf{u}) + V(\mathbf{f}(\mathbf{x},\mathbf{u}),i+1)].$

(2)

This is the Bellman equation.

Differential dynamic programming

DDP proceeds by iteratively performing a backward pass on the nominal trajectory to generate a new control sequence, and then a forward-pass to compute and evaluate a new nominal trajectory. We begin with the backward pass. If

$\ell(\mathbf{x},\mathbf{u}) + V(\mathbf{f}(\mathbf{x},\mathbf{u}),i+1)$

is the argument of the $\min[]$ operator in Eq. 2, let $Q$ be the variation of this quantity around the $i$-th $(\mathbf{x},\mathbf{u})$ pair:

$\begin{align}Q(\delta\mathbf{x},\delta\mathbf{u})\equiv &\ell(\mathbf{x}+\delta\mathbf{x},\mathbf{u}+\delta\mathbf{u})&&{}+V(\mathbf{f}(\mathbf{x}+\delta\mathbf{x},\mathbf{u}+\delta\mathbf{u}),i+1) \\ -&\ell(\mathbf{x},\mathbf{u})&&{}-V(\mathbf{f}(\mathbf{x},\mathbf{u}),i+1) \end{align}$

and expand to second order

$\approx\frac{1}{2} \begin{bmatrix} 1\\ \delta\mathbf{x}\\ \delta\mathbf{u} \end{bmatrix}^\mathsf{T} \begin{bmatrix} 0 & Q_\mathbf{x}^\mathsf{T} & Q_\mathbf{u}^\mathsf{T}\\ Q_\mathbf{x} & Q_{\mathbf{x}\mathbf{x}} & Q_{\mathbf{x}\mathbf{u}}\\ Q_\mathbf{u} & Q_{\mathbf{u}\mathbf{x}} & Q_{\mathbf{u}\mathbf{u}} \end{bmatrix} \begin{bmatrix} 1\\ \delta\mathbf{x}\\ \delta\mathbf{u} \end{bmatrix}$

(3)

The $Q$ notation used here is a variant of the notation of Morimoto where subscripts denote differentiation in denominator layout. Dropping the index $i$ for readability, primes denoting the next time-step $V'\equiv V(i+1)$, the expansion coefficients are

$\begin{alignat}{2} Q_\mathbf{x} &= \ell_\mathbf{x}+ \mathbf{f}_\mathbf{x}^\mathsf{T} V'_\mathbf{x} \\ Q_\mathbf{u} &= \ell_\mathbf{u}+ \mathbf{f}_\mathbf{u}^\mathsf{T} V'_\mathbf{x} \\ Q_{\mathbf{x}\mathbf{x}} &= \ell_{\mathbf{x}\mathbf{x}} + \mathbf{f}_\mathbf{x}^\mathsf{T} V'_{\mathbf{x}\mathbf{x}}\mathbf{f}_\mathbf{x}+V_\mathbf{x}'\cdot\mathbf{f}_{\mathbf{x}\mathbf{x}}\\ Q_{\mathbf{u}\mathbf{u}} &= \ell_{\mathbf{u}\mathbf{u}} + \mathbf{f}_\mathbf{u}^\mathsf{T} V'_{\mathbf{x}\mathbf{x}}\mathbf{f}_\mathbf{u}+{V'_\mathbf{x}} \cdot\mathbf{f}_{\mathbf{u} \mathbf{u}}\\ Q_{\mathbf{u}\mathbf{x}} &= \ell_{\mathbf{u}\mathbf{x}} + \mathbf{f}_\mathbf{u}^\mathsf{T} V'_{\mathbf{x}\mathbf{x}}\mathbf{f}_\mathbf{x} + {V'_\mathbf{x}} \cdot \mathbf{f}_{\mathbf{u} \mathbf{x}}. \end{alignat}$

The last terms in the last three equations denote contraction of a vector with a tensor. Minimizing the quadratic approximation (3) with respect to $\delta\mathbf{u}$ we have

${\delta \mathbf{u}}^* = \operatorname{argmin}\limits_{\delta \mathbf{u}}Q(\delta \mathbf{x},\delta \mathbf{u})=-Q_{\mathbf{u}\mathbf{u}}^{-1}(Q_\mathbf{u}+Q_{\mathbf{u}\mathbf{x}}\delta \mathbf{x}),$

(4)

giving an open-loop term $\mathbf{k}=-Q_{\mathbf{u}\mathbf{u}}^{-1}Q_\mathbf{u}$ and a feedback gain term $\mathbf{K}=-Q_{\mathbf{u}\mathbf{u}}^{-1}Q_{\mathbf{u}\mathbf{x}}$. Plugging the result back into (3), we now have a quadratic model of the value at time $i$:

$\begin{alignat}{2} \Delta V(i) &= &{} -\tfrac{1}{2}Q_\mathbf{u} Q_{\mathbf{u}\mathbf{u}}^{-1}Q_\mathbf{u}\\ V_\mathbf{x}(i) &= Q_\mathbf{x} & {}- Q_\mathbf{u} Q_{\mathbf{u}\mathbf{u}}^{-1}Q_{\mathbf{u}\mathbf{x}}\\ V_{\mathbf{x}\mathbf{x}}(i) &= Q_{\mathbf{x}\mathbf{x}} &{} - Q_{\mathbf{x}\mathbf{u}}Q_{\mathbf{u}\mathbf{u}}^{-1}Q_{\mathbf{u}\mathbf{x}}. \end{alignat}$

Recursively computing the local quadratic models of $V(i)$ and the control modifications $\{\mathbf{k}(i),\mathbf{K}(i)\}$, from $i=N-1$ down to $i=1$, constitutes the backward pass. As above, the Value is initialized with $V(\mathbf{x},N)\equiv \ell_f(\mathbf{x}_N)$. Once the backward pass is completed, a forward pass computes a new trajectory:

$\begin{align} \hat{\mathbf{x}}(1)&=\mathbf{x}(1)\\ \hat{\mathbf{u}}(i)&=\mathbf{u}(i) + \mathbf{k}(i) +\mathbf{K}(i)(\hat{\mathbf{x}}(i) - \mathbf{x}(i))\\ \hat{\mathbf{x}}(i+1)&=\mathbf{f}(\hat{\mathbf{x}}(i),\hat{\mathbf{u}}(i)) \end{align}$

The backward passes and forward passes are iterated until convergence.

Regularization and line-search

Differential dynamic programming is a second-order algorithm like Newton's method. It therefore takes large steps toward the minimum and often requires regularization and/or line-search to achieve convergence . Regularization in the DDP context means ensuring that the $Q_{\mathbf{u}\mathbf{u}}$ matrix in Eq. 4 is positive definite. Line-search in DDP amounts to scaling the open-loop control modification $\mathbf{k}$ by some $0<\alpha<1$.

See also

• Optimal control

External links

• A Python implementation of DDP
• A MATLAB implementation of DDP