Optimal Control Theory
The theory • Optimal control theory is a mature mathematical discipline which provides algorithms to solve various control problems • The elaborate mathematical machinery behind optimal control models is rarely exposed to computer animation community • Most controllers designed in practice are theoretically suboptimal • This lecture closely follows the excellent tutorial by Dr. Emo Todorov (http://www.cs.washington.edu/homes/todorov/ papers/optimality_chapter.pdf)
• Discrete control: Bellman equations • Continuous control: HJB equations • Maximum principle • Linear quadratic regulator (LQR)
Standard problem • Find an action sequence ( u 0 , u 1 , ..., u n -1 ) and corresponding state sequence ( x 0 , x 1 , ..., x n- 1 ) minimizing the total cost � � • The initial state ( x 0 ) and the destination state ( x n ) are given
Discrete control $500 $120 $450 $150 $350 $200 $120 $250 $150 $350 $200 next( x , u ) cost( x , u ) $300 $250
Dynamic programming • Bellman optimality principle: • If a given state-action sequence is optimal and we remove the first state and action, remaining sequence is also optimal • The choice of optimal actions in the futures is independent of the past actions which led to the present state • The optimal state-action sequences can be constructed by starting at the final state and extending backwards
Optimal value function • v ( x ) = “minimal total cost for completing the task starting from state x ” • Find optimal actions: 1. Consider every action available at the current state 2. Add its immediate cost to the optimal value of the resulting next state 3. Choose an action for which the sum is minimal
Optimal control policy • A mapping from states to actions is called control policy or control law • Once we have a control policy, we can start at any state and reach the destination state by following the control policy • Optimal control policy satisfies � � • Its corresponding optimal value function satisfies
Value iteration • Bellman equations cannot be solved in a single pass if the state transitions are cyclic • Value iteration starts with a guess v (0) of the optimal value function and construct a sequence of improved guesses:
• Discrete control: Bellman equations • Continuous control: HJB equations • Maximum principle • Linear quadratic regulator (LQR)
Continuous control • State space and control space are continuos • Dynamics of the system: • Continuous time • Discrete time • Objective function:
HJB equation • HJB equation is a nonlinear PDE with respect to unknown function v � u ∈ U ( x ) ( l ( x , u , t ) + f ( x , u ) T v x ( x , t )) − v t ( x , t ) = min � • An optimal control π ( x , t ) is a value of u which achieves the minimum in HJB equation u ∈ U ( x ) ( l ( x , u , t ) + f ( x , u ) T v x ( x , t )) π ( x , t ) = arg min
Numerical solution • Non-linear differential equations do not always have classic solutions which satisfy them everywhere • Numerical methods guarantee convergence, but they rely on discretization of the state space, which grows exponentially in the state space dimension • Nevertheless, the HJB equations have motivated a number of methods for approximate solution
Parametric value function • Consider an approximation to the optimal value function � • The derivative function with respect to x � • Choose a large enough set of states and evaluate the right hand side of HJB using the approximated value function • Adjust theta such that get closer to target values
• Discrete control: Bellman equations • Continuous control: HJB equations • Maximum principle • Linear quadratic regulator (LQR)
Maximum principle • Maximum principle solves the optimal control for a deterministic dynamic system with boundary conditions • Can be derived via HJB equations or Lagrange multipliers • Can be generalized to other types of optimal control problems: free final time, intermediate constraints, first exit time, control constraints, etc
Derivation via HJB • The finite horizon HJB: � • If an optimal control policy, π ( x , t ) is given, we can set u = π ( x , t ) and drop the min operator in HJB
Maximum principle • The remarkable property of the maximum principle is that it is an ODE, even though we derived it starting from a PDE • An ODE is a consistency condition which singles out specific trajectories without reference to neighboring trajectories • Extremal trajectories which solve the above optimization remove the dependence on neighboring trajectories
Hamiltonian function • The maximum principle can be written in more compact and symmetric form with the help of the Hamiltonian function � � • Maximum principle can be redefined as
• Discrete control: Bellman equations • Continuous control: HJB equations • Maximum principle • Linear quadratic regulator (LQR)
Linear quadratic regulator • Most optimal control problems do not have closed-form solutions. One exception is LQR case • LQR is a class of problems which dynamic function is linear and cost function is quadratic • dynamics: • cost rate: • final cost • R is symmetric positive definite, and Q and Q f are symmetric • A , B , R , Q can be made time-varying
Optimal value function • For a LQR problem, the optimal value function is quadratic in x and can be expressed as where V ( t ) is a symmetric matrix � • We can obtain the ODE of V ( t ) via HJB equation
Discrete LQR • LQR is defined as follows when time is discretized • dynamics • cost rate • final cost • Let n = t f / Δ , the correspondence to continuous-time problem is
Optimal value function • We derive optimal value function from Bellman equation • Again, the optimal value function is quadratic in x and changes over time � • Plugging in Bellman equation, we obtain a recursive relation of V k � • The optimal control law is linear in x
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