Categorical LQR Control with Linear Relations

Last time, I described and built an interesting expansion of describing systems via linear functions, that of using linear relations. We remove the requirement that mappings be functional (exactly one output vector for any given input vector), which extends our descriptive capability. This is useful for describing “open” pieces of a linear system like electric circuits. This blog post is about another application: linear dynamical systems and control.

Linear relations are an excellent tool for reachability/controllability. In a control system, it is important to know where it is possible to get the system to. With linear dynamics x_{t+1} = Ax_{t} + B u_{t}, an easy controllability question is “can my system reach into some subspace”. This isn’t quite the most natural question physically, but it is natural mathematically. My first impulse would be to ask “given the state starts in this little region over here, can it get to this little region over here”, but the subspace question is a bit easier to answer. It’s a little funky but it isn’t useless. It can detect if the control parameter only touches 1 of 2 independent dynamical systems for example. We can write the equations of motion as a linear relation and iterate composition on them. See Erbele for more.

There is a set of different things (and kind of more interesting things) that are under the purview of linear control theory, LQR Controllers and Kalman filters. The LQR controller and Kalman filter are roughly (exactly?) the same thing mathematically. At an abstract mathematical level, they both rely on the fact that optimization of a quadratic objective x^T Q x + r^T x + c with a linear constraints Ax=b is solvable in closed form via linear algebra. The cost of the LQR controller could be the squared distance to some goal position for example. When you optimize a function, you set the derivative to 0. This is a linear equation for quadratic objectives. It is a useful framework because it has such powerful computational teeth.

The Kalman filter does a similar thing, except for the problem of state estimation. There are measurements linear related to the state that you want to match with the prior knowledge of the linear dynamics of the system and expected errors of measurement and environment perturbations to the dynamics.

There are a couple different related species of these filters. We could consider discrete time or continuous time. We can also consider infinite horizon or finite horizon. I feel that discrete time and finite horizon are the most simple and fundamental so that is what we’ll stick with. The infinite horizon and continuous time are limiting cases.

We also can consider the dynamics to be fixed for all time steps, or varying with time. Varying with time is useful for the approximation of nonlinear systems, where there are different good nonlinear approximation (taylor series of the dynamics) depending on the current state.

There are a couple ways to solve a constrained quadratic optimization problem. In some sense, the most conceptually straightforward is just to solve for a span of the basis (the “V-Rep” of my previous post) and plug that in to the quadratic objective and optimize over your now unconstrained problem. However, it is commonplace and elegant in many respects to use the Lagrange multiplier method. You introduce a new parameter for every linear equality and add a term to the objective \lambda^T (A x - b). Hypothetically this term is 0 for the constrained solution so we haven’t changed the objective in a sense. It’s a funny slight of hand. If you optimize over x and \lambda, the equality constraint comes out as your gradient conditions on \lambda. Via this method, we convert a linearly constrained quadratic optimization problem into an unconstrained quadratic optimization problem with more variables.

Despite feeling very abstract, the value these Lagrangian variables take on has an interpretation. They are the “cost” or “price” of the equality they correspond to. If you moved the constraint by an amount Ax - b + \epsilon, you would change the optimal cost by an amount \lambda \epsilon. (Pretty sure I have that right. Units check out. See Boyd https://www.youtube.com/watch?v=FJVmflArCXc)

The Lagrange multipliers enforcing the linear dynamics are called the co-state variables. They represent the “price” that the dynamics cost, and are derivatives of the optimal value function V(s) (The best value that can be achieved from state s) that may be familiar to you from dynamic programming or reinforcement learning. See references below for more.

Let’s get down to some brass tacks. I’ve suppressed some terms for simplicity. You can also add offsets to the dynamics and costs.

A quadratic cost function with lagrange multipliers. Q is a cost matrix associated with the x state variables, R is a cost matrix for the u control variables.

C = \lambda_0 (x_0 - \tilde{x}_0) + \sum_t x_t^T Q x_t + u_t^T R u_t + \lambda_{t+1}^T (x_{t+1} - A x_t - B u_t  )

Equations of motion results from optimizing with respect to \lambda by design.

\nabla_{\lambda_{t+1}} C = x_{t+1} - A x_t - B u_t  = 0.

The initial conditions are enforced by the zeroth multiplier.

\nabla_{\lambda_0} C = x_i - x_{0} = 0

Differentiation with respect to the state x has the interpretation of backwards iteration of value derivative, somewhat related to what one finds in the Bellman equation.

\nabla_{x_t} C = Q x_{t} + A^T \lambda_{t+1} - \lambda_{t} = 0 \Longrightarrow \lambda_{t} =  A^T \lambda_{t+1} + Q x_{t}

The final condition on value derivative is the one that makes it the most clear that the Lagrange multiplier has the interpretation of the derivative of the value function, as it sets it to that.

\nabla_{x_N} C = Q x_N - \lambda_{N} = 0

Finally, differentiation with respect to the control picks the best action given knowledge of the value function at that time step.

\nabla_{u_t} C = R u_{t} - B^T \lambda_t  = 0

Ok. Let’s code it up using linear relations. Each of these conditions is a separate little conceptual box. We can build the optimal control step relation by combining these updates together

The string diagram for a single time step of control

The following is a bit sketchy. I am confident it makes sense, but I’m not confident that I have all of the signs and other details correct. I also still need to add the linear terms to the objective and offset terms to the dynamics. These details are all kind of actually important. Still, I think it’s nice to set down in preliminary blog form.

Initial conditions and final conditions. Final conditions fix the final value derivative to Qx. Initial conditions set x to some value. Should there be a leg for lambda on the initial conditions?

Bits and Bobbles

  • The code in context https://github.com/philzook58/ConvexCat/blob/master/src/LinRel.hs
  • Some of the juicier stuff is nonlinear control. Gotta walk before we can run though. I have some suspicions that a streaming library may be useful here, or a lens-related approach. Also ADMM.
  • Reminiscent of Keldysh contour. Probably a meaningless coincidence.
  • In some sense H-Rep is the analog of (a -> b -> Bool) and V-rep [(a,b)]
  • Note that the equations of motion are relational rather than functional for a control systems. The control parameters u describe undetermined variables under our control.
  • Loopback (trace) the value derivative for infinite horizon control.
  • Can solve the Laplace equation with a Poincare Steklov operator. That should be my next post.
  • There is something a touch unsatisfying even with the reduced goals of this post. Why did I have to write down the quadratic optimization problem and then hand translate it to linear relations? It’s kind of unacceptable. The quadratic optimization problem is really the most natural statement, but I haven’t figured out to make it compositional. The last chapter of Rockafellar?

References

https://arxiv.org/abs/1405.6881 Baez and Erbele – Categories in Control

http://www.scholarpedia.org/article/Optimal_control

http://www.argmin.net/2016/05/18/mates-of-costate/ – interesting analogy to backpropagation. Lens connection?

https://courses.cs.washington.edu/courses/cse528/09sp/optimality_chapter.pdf

Linear Relation Algebra of Circuits with HMatrix

Oooh this is a fun one.

I’ve talked before about relation algebra and I think it is pretty neat. http://www.philipzucker.com/a-short-skinny-on-relations-towards-the-algebra-of-programming/. In that blog post, I used finite relations. In principle, they are simple to work with. We can perform relation algebra operations like composition, meet, and join by brute force enumeration.

Unfortunately, brute force may not always be an option. First off, the finite relations grow so enormous as to be make this infeasible. Secondly, it is not insane to talk about relations or regions with an infinite number of elements, such as some continuous blob in 2D space. In that case, we can’t even in principle enumerate all the points in the region. What are we to do? We need to develop some kind of finite parametrization of regions to manipulate. This parametrization basically can’t possibly be complete in some sense, and we may choose more or less powerful systems of description for computational reasons.

In this post, we are going to be talking about linear or affine subspaces of a continuous space. These subspaces are hyperplanes. Linear subspaces have to go through the origin, while affine spaces can have an offset from the origin.

In the previous post, I mentioned that the finite relations formed a lattice, with operations meet and join. These operations were the same as set intersection and union so the introduction of the extra terminology meet and join felt a bit unwarranted. Now the meet and join aren’t union and intersection anymore. We have chosen to not have the capability to represent the union of two vectors, instead we can only represent the smallest subspace that contains them both, which is the union closed under vector addition. For example, the join of a line and point will be the plane that goes through both.

Linear/Affine stuff is great because it is so computational. Most questions you cant to ask are answerable by readily available numerical linear algebra packages. In this case, we’ll use the Haskell package HMatrix, which is something like a numpy/scipy equivalent for Haskell. We’re going to use type-level indices to denote the sizes and partitioning of these spaces so we’ll need some helper functions.

Matrices are my patronum. They make everything good. Artwork courtesy of David

In case I miss any extensions, make typos, etc, you can find a complete compiling version here https://github.com/philzook58/ConvexCat/blob/master/src/LinRel.hs

In analogy with sets of tuples for defining finite relations, we partition the components of the linear spaces to be “input” and “output” indices/variables \begin{bmatrix} x_1 & x_2 & x_3 & ... & y_1 & y_2 & y_3 & ... \end{bmatrix}. This partition is somewhat arbitrary and easily moved around, but the weakening of strict notions of input and output as compared to functions is the source of the greater descriptive power of relations.

Relations are extensions of functions, so linear relations are an extension of linear maps. A linear map has the form y = Ax. A linear relation has the form Ax + By = 0. An affine map has the form y = Ax + b and an affine relation has the form Ax + By = b.

There are at least two useful concrete representation for subspaces.

  1. We can write a matrix A and vector b down that corresponds to affine constraints. Ax = b. The subspace described is the nullspace of A plus a solution of the equation. The rows of A are orthogonal to the space.
  2. We can hold onto generators of subspace. x = A' l+b where l parametrizes the subspace. In other words, the subspace is generated by / is the span of the columns of A'. It is the range of A'.

We’ll call these two representations the H-Rep and V-Rep, borrowing terminology from similar representations in polytopes (describing a polytope by the inequalities that define it’s faces or as the convex combination of it’s vertices). https://inf.ethz.ch/personal/fukudak/lect/pclect/notes2015/PolyComp2015.pdf These two representations are dual in many respects.

It is useful to have both reps and interconversion routines, because different operations are easy in the two representations. Any operations defined on one can be defined on the other by sandwiching between these conversion functions. Hence, we basically only need to define operations for one of the reps (if we don’t care too much about efficiency loss which, fair warning, is out the window for today). The bulk of computation will actually be performed by these interconversion routines. The HMatrix function nullspace performs an SVD under the hood and gathers up the space with 0 singular values.

These linear relations form a category. I’m not using the Category typeclass because I need BEnum constraints hanging around. The identity relations is x = y aka Ix - Iy = 0.

Composing relations is done by combining the constraints of the two relations and then projecting out the interior variables. Taking the conjunction of constraints is easiest in the H-Rep, where we just need to vertically stack the individual constraints. Projection easily done in the V-rep, where you just need to drop the appropriate section of the generator vectors. So we implement this operation by flipping between the two.

We can implement the general cadre of relation operators, meet, join, converse. I feel the converse is the most relational thing of all. It makes inverting a function nearly a no-op.

Relational inclusion is the question of subspace inclusion. It is fairly easy to check if a VRep is in an HRep (just see plug the generators into the constraints and see if they obey them) and by using the conversion functions we can define it for arbitrary combos of H and V.

It is useful the use the direct sum of the spaces as a monoidal product.

A side note: Void causes some consternation. Void is the type with no elements and is the index type of a 0 dimensional space. It is the unit object of the monoidal product. Unfortunately by an accident of the standard Haskell definitions, actual Void is not a BEnum. So, I did a disgusting hack. Let us not discuss it more.

Circuits

Baez and Fong have an interesting paper where they describe building circuits using a categorical graphical calculus. We have the pieces to go about something similar. What we have here is a precise way in which circuit diagrams can be though of as string diagrams in a monoidal category of linear relations.

An idealized wire has two quantities associated with it, the current flowing through it and the voltage it is at.

When we connect wires, the currents must be conserved and the voltages must be equal. hid and hcompose from above still achieve that. Composing two independent circuits in parallel is achieve by hpar.

Independent resistors in parallel.

We will want some basic tinker toys to work with.

A resistor in series has the same current at both ends and a voltage drop proportional to the current

Composing two resistors in parallel adds the resistance. (resistor r1) <<< (resistor r2) == resistor (r1 + r2))

A bridging resistor allows current to flow between the two branches

A bridging resistor

Composing two bridge circuits is putting the bridge resistors in parallel. The conductance G=\frac{1}{R} of resistors in parallel adds. hcompose (bridge r1) (bridge r2) == bridge 1 / (1/r1 + 1/r2).

parallel resistors compose

An open circuit allows no current to flow and ends a wire. open ~ resistor infinity

At branching points, the voltage is maintained, but the current splits.

This cmerge combinator could also be built using a short == bridge 0 , composing a branch with open, and then absorbing the Void away.

We can bend wires up or down by using a composition of cmerge and open.

Voltage and current sources enforce current and voltage to be certain values

Measurements of circuits proceed by probes.

Inductors and capacitors could be included easily, but would require the entries of the HMatrix values to be polynomials in the frequency \omega, which it does not support (but it could!). We’ll leave those off for another day.

We actually can determine that the rules suggested above are being followed by computation.

Bits and Bobbles

  • Homogenous systems are usually a bit more elegant to deal with, although a bit more unfamiliar and abstract.
  • Could make a pandas like interface for linear relations that uses numpy/scipy.sparse for the computation. All the swapping and associating is kind of fun to design, not so much to use. Labelled n-way relations are nice for users.
  • Implicit/Lazy evaluation. We should let the good solvers do the work when possible. We implemented our operations eagerly. We don’t have to. By allowing hidden variables inside our relations, we can avoid the expensive linear operations until it is useful to actually compute on them.
  • Relational division = quotient spaces?
  • DSL. One of the beauties of the pointfree/categorical approach is that you avoid the need for binding forms. This makes for a very easily manipulated DSL. The transformations feel like those of ordinary algebra and you don’t have to worry about the subtleties of index renaming or substitution under binders.
  • Sparse is probably really good. We have lots of identity matrices and simple rearrangements. It is very wasteful to use dense operations on these.
  • Schur complement https://en.wikipedia.org/wiki/Schur_complement are the name in the game for projecting out pieces of linear problems. We have some overlap.
  • Linear relations -> Polyhedral relations -> Convex Relations. Linear is super computable, polyhedral can blow up. Rearrange a DSL to abuse Linear programming as much as possible for queries.
  • Network circuits. There is an interesting subclass of circuits that is designed to be pretty composable.

https://en.wikipedia.org/wiki/Two-port_network Two port networks are a very useful subclass of electrical circuits. They model transmission lines fairly well, and easily composable for filter construction.

It is standard to describe these networks by giving a linear function between two variables and the other two variables. Depending on your choice of which variables depend on which, these are called the z-parameters, y-parameters, h-parameters, scattering parameters, abcd parameters. There are tables of formula for converting from one form to the others. The different parameters hold different use cases for composition and combining in parallel or series. From the perspective of linear relations this all seems rather silly. The necessity for so many descriptions and the confusing relationship between them comes from the unnecessary and overly rigid requirement of have a linear function-like relationship rather than just a general relation, which depending of the circuit may not even be available (there are degenerate configurations where two of the variables do not imply the values of the other two). A function relationship is always a lie (although a sometimes useful one), as there is always back-reaction of new connections.

The relation model also makes clearer how to build lumped models out of continuous ones. https://en.wikipedia.org/wiki/Lumped-element_model

https://en.wikipedia.org/wiki/Transmission_line https://en.wikipedia.org/wiki/Distributed-element_model

    null
  • Because the type indices have no connection to the actual data types (they are phantom) it is a wise idea to use smart constructors that check that the sizes of the matrices makes sense.
  • Nonlinear circuits. Grobner Bases and polynomial relations?
  • Quadratic optimization under linear constraints. Can’t get it to come out right yet. Clutch for Kalman filters. Nice for many formulations like least power, least action, minimum energy principles. Edit: I did more in this direction here http://www.philipzucker.com/categorical-lqr-control-with-linear-relations/
  • Quadratic Operators -> Convex operators. See last chapter of Rockafellar.
  • Duality of controllers and filters. It is well known (I think) that for ever controller algorithm there is a filter algorithm that is basically the same thing.
    • LQR – Kalman
    • Viterbi filter – Value function table
    • particle filter – Monte Carlo control
    • Extended Kalman – iLQR-ish? Use local approximation of dynamics
    • unscented kalman – ?

Refs

Failing to Bound Kissing Numbers

https://en.wikipedia.org/wiki/Kissing_number

Cody brought up the other day the kissing number problem.Kissing numbers are the number of equal sized spheres you can pack around another one in d dimensions. It’s fairly self evident that the number is 2 for 1-d and 6 for 2d but 3d isn’t so obvious and in fact puzzled great mathematicians for a while. He was musing that it was interesting that he kissing numbers for some dimensions are not currently known, despite the fact that the first order theory of the real numbers is decidable https://en.wikipedia.org/wiki/Decidability_of_first-order_theories_of_the_real_numbers

I suggested on knee jerk that Sum of Squares might be useful here. I see inequalities and polynomials and then it is the only game in town that I know anything about.

Apparently that knee jerk was not completely wrong

https://arxiv.org/pdf/math/0608426.pdf

Somehow SOS/SDP was used for bounds here. I had an impulse that the problem feels SOS-y but I do not understand their derivation.

One way the problem can be formulated is by finding or proving there is no solution to the following set of equations constraining the centers x_i of the spheres. Set the central sphere at (0,0,0,…) . Make the radii 1. Then\forall i. |x_i|^2 = 2^2 and \forall i j.  |x_i - x_j|^2 \ge 2^2

I tried a couple different things and have basically failed. I hope maybe I’ll someday have a follow up post where I do better.

So I had 1 idea on how to approach this via a convex relaxation

Make a vector x = \begin{bmatrix} x_0 & y _0 & x_1 & y _1 & x_2 & y _2 & ... \end{bmatrix} Take the outer product of this vector x^T x = X Then we can write the above equations as linear equalities and inequalities on X. If we forget that we need X to be the outer product of x (the relaxation step), this becomes a semidefinite program. Fingers crossed, maybe the solution comes back as a rank 1 matrix. Other fingers crossed, maybe the solution comes back and says it’s infeasible. In either case, we have solved our original problem.

Didn’t work though. Sigh. It’s conceivable we might do better if we start packing higher powers into x?

Ok Round 2. Let’s just ask z3 and see what it does. I’d trust z3 with my baby’s soft spot.

It solves for 5 and below. Z3 grinds to a halt on N=6 and above. It ran for days doin nothing on my desktop.

Ok. A different tact. Try to use a positivstellensatz proof. If you have a bunch of polynomial inequalities and equalities if you sum polynomial multiples of these constraints, with the inequalities having sum of square multiples, in such a way to = -1, it shows that there is no real solution to them. We have the distance from origin as equality constraint and distance from each other as an inequality constraint. I intuitively think of the positivstellensatz as deriving an impossibility from false assumptions. You can’t add a bunch of 0 and positive numbers are get a negative number, hence there is no real solution.

I have a small set of helper functions for combining sympy and cvxpy for sum of squares optimization. I keep it here along with some other cute little constructs https://github.com/philzook58/cvxpy-helpers

and here is the attempted positivstellensatz.

It worked in 1-d, but did not work in 2d. At order 3 polynomials N=7, I maxed out my ram.

I also tried doing it in Julia, since sympy was killing me. Julia already has a SOS package

It was faster to encode, but it’s using the same solver (SCS), so basically the same thing.

I should probably be reducing the system with respect to equality constraints since they’re already in a Groebner basis. I know that can be really important for reducing the size of your problem

I dunno.

Blah blah blah blah A bunch of unedited trash

https://github.com/peterwittek/ncpol2sdpa Peter Wittek has probably died in an avalanche? That is very sad.

These notes

https://web.stanford.edu/class/ee364b/lectures/sos_slides.pdf

Positivstullensatz.

kissing number

Review of sum of squares

minimimum sample as LP. ridiculous problem
min t
st. f(x_i) – t >= 0

dual -> one dual variable per sample point
The only dual that will be non zero is that actually selecting the minimum.

Hm. Yeah, that’s a decent analogy.

How does the dual even have a chance of knowing about poly airhtmetic?
It must be during the SOS conversion prcoess. In building the SOS constraints,
we build a finite, limittted version of polynomial multiplication
x as a matrix. x is a shift matrix.
In prpducing the characterstic polynomial, x is a shift matrix, with the last line using the polynomial
known to be zero to
eigenvectors of this matrix are zeros of the poly.

SOS does not really on polynomials persay. It relies on closure of the suqaring operaiton

maybe set one sphere just at x=0 y = 2. That breaks some symmettry

set next sphere in plane something. random plane through origin?

order y components – breaks some of permutation symmettry.

no, why not order in a random direction. That seems better for symmettry breaking

Learn Coq in Y

Edit: It’s up! https://learnxinyminutes.com/docs/coq/

I’ve been preparing a Learn X in Y tutorial for Coq. https://learnxinyminutes.com/

I’ve been telling people this and been surprised by how few people have heard of the site. It’s super quick intros to syntax and weirdness for a bunch of languages with inline code tutorials.
I think that for me, a short description of that mundane syntactic and programming constructs of coq is helpful.
Some guidance of the standard library, what is available by default. And dealing with Notation scopes, which is a pretty weird feature that most languages don’t have.
The manual actually has all this now. It’s really good. Like check this section out https://coq.inria.fr/refman/language/coq-library.html . But the manual is an intimidating documents. It starts with a BNF description of syntax and things like that. The really useful pedagogical stuff is scattered throughout it.

Anyway here is my draft (also here https://github.com/philzook58/learnxinyminutes-docs/blob/master/coq.html.markdown where the syntax highlighting isn’t so janked up). Suggestions welcome. Or if this gets accepted, you can just make pull requests

Bonus. An uneditted list of tactics. You’d probably prefer https://pjreddie.com/coq-tactics/