mathematics//dynamical systems//sufficient state
A sufficient state is a choice of state variables for a model that keeps all the information from the past the model needs to predict the future, given the future inputs, and it is what lets a model continue from the present alone. Throw a stone: to predict where it lands you need its position and velocity now, and it makes no difference whether it was thrown or bounced off a wall. A water tank needs one number, its level; a DC motor two, speed and current; a drone as a rigid body twelve. Acceleration is not in the list because it carries no memory: Newton gives it from the forces, which depend on the state and the inputs.
A sufficient state is a choice of state variables for a model that keeps all the information from the past the model needs to predict the future, given the future inputs, and it is what lets a model continue from the present alone. Throw a stone: to predict where it lands you need its position and velocity now, and it makes no difference whether it was thrown or bounced off a wall. A water tank needs one number, its level; a DC motor two, speed and current; a drone as a rigid body twelve. Acceleration is not in the list because it carries no memory: Newton gives it from the forces, which depend on the state and the inputs.
The state is memory, and it is chosen for prediction rather than for what happens to be measured.
It is the minimal summary of the past that, with the future inputs, is enough to predict; what the sensors read is usually a part of it, and the rest has to be estimated (state estimation).
The present has to carry the right memory. In x˙=u−λx\dot x=u-\lambda xx˙=u−λx, two different histories that produce the same x(t0)x(t_0)x(t0) share the same future under the same later input. Two memories with different rates read through their sum behave otherwise:
x˙1=u−λ1x1,x˙2=u−λ2x2,y=x1+x2.\dot x_1=u-\lambda_1x_1,\qquad \dot x_2=u-\lambda_2x_2,\qquad y=x_1+x_2.x˙1=u−λ1x1,x˙2=u−λ2x2,y=x1+x2.
Here knowing only yyy is not enough: the same total can hold different proportions of fast and slow memory. Mechanics is analogous: position alone cannot continue a second-order motion; position plus velocity can.
Several choices of state are equally valid, and they all have the same size. Position and momentum describe a mass on a spring as well as position and velocity, as does any invertible combination of the two; what does not change is how many independent numbers are needed, the system order. A parameter that drifts slowly (the mass of a drone that drops its payload) can be promoted to a state and estimated with the rest (state augmentation).
The Markov property states sufficiency for a discrete process: Pr(Sk+1∣Sk,…,S0)=Pr(Sk+1∣Sk)\Pr(S_{k+1}\mid S_k,\ldots,S_0)=\Pr(S_{k+1}\mid S_k)Pr(Sk+1∣Sk,…,S0)=Pr(Sk+1∣Sk). It is a modelling choice, obtained by enlarging the state until the relevant memory is inside it, and it does not mean independence between successive states. The state of a Markov chain or of a decision process is the same idea applied to choices; the state vector of a dynamical model is the idea applied to physics.
Sufficient does not mean minimal, observed or controllable. A state can be sufficient and redundant, sufficient and partly hidden, sufficient and contain modes no input can reach (controllability and observability). An RNN hidden state is a summary the network learns because nobody gave it one, and whether it is sufficient depends on what the future needs.
With a pure delay, the state may need a segment of history rather than a finite vector (delay and lag).