Software to approximately solve the quadratic-quadratic regulator (QQR) and polynomial-quadratic regulator (PQR) problems. The description of the algorithms are provided in the papers
- The Quadratic-Quadratic Regulator Problem: Approximating feedback controls for quadratic-in-state nonlinear systems, American Control Conference, 2020. https://arxiv.org/abs/1910.03396
by Jeff Borggaard and Lizette Zietsman
and
- On Approximating Polynomial-Quadratic Regulator Problems, Mathematical Theory of Networks and Systems, 2020 (accepted). https://arxiv.org/abs/2009.11068 and appeared in IFAC-PapersOnLine, 54(9), 329-334, 2021.
by Jeff Borggaard and Lizette Zietsman.
Download and untar, or clone this and the KroneckerTools repositories:
git clone https://www.github.com/jborggaard/KroneckerTools
git clone https://www.github.com/jborggaard/QQR
Optional: Get the Matlab functions for efficiently solving linear systems with a special Kronecker sum structure (laplace-like structure) at https://anchp.epfl.ch/index-html/software/misc and detailed in the preprint: Recursive blocked algorithms for linear systems with Kronecker product structure, by Minhong Chen and Daniel Kressner. Place the directory "tensor_recursive" within the kronecker directory.
Optional: Tests to match our Quadratic-Quadratic Regulator paper can be performed by requesting the Nonlinear Systems Toolbox from Art Krener. If this is the case, adjust the path in setNSTpath.m and set testNST=true inside the tests_ACC.m script.
The installation can be tested in Matlab (we used R2019b) by typing
>> examplesForACC
A stand-alone test (setting testNST=false) can be found in the examples directory The details of some of our functions and test examples are provided below.
For a quickstart, open the live script example10.mlx in the examples directory using Matlab2022b or later.
If A is n-by-n, B is n-by-m, Q is n-by-n, R is m-by-m, and N is n-by-n^2 , with [A,B] a controllable pair, we can compute the coefficients of the feedback laws and the value function in Matlab as
>> [k,v] = qqr(A,B,Q,R,N,degree);
The variable k is a cell array with k{1} being m-by-n and is the usual first-degree feedback law computed by lqr, k{2} is m-by-n^2 , up to k{degree} which is m-by-n^degree . The feedback control is then
>> u = k{1}*x + k{2}*kron(x,x) + ... + k{degree}*kron(kron(... ,x),x);
This can be simplified using the function kronPolyEval.m from the KroneckerTools repository as
>> u = kronPolyEval(k,x,degree);
The variable v is a cell array with v{2} being n-by-n^2 , up to v{degree+1} which is n-by-n^degree+1 . These are coefficients of the polynomial approximation to the value function. From an initial x0, we can compute the approximation to the value function as
>> J = v{2}*kron(x0,x0) + ... + v{degree+1}*kron(kron(... ,x0),x0);
or J = kronPolyEval(v,x0,degree+1); (kronPolyEval skips the empty matrix v{1}=[]).
For details on how to run qqr, one can also type
>> help qqr
for examples how to run qqr see the live script
>> addpath('examples')
>> open example10.mlx
as well as
example4.m and example5.m
in the examples directory.
With the same assumptions on A,B,Q,R as in qqr, but we now allow for higher degree polynomial terms by defining N to be a cell array N{2} is n-by-n^2, N{3} is n-by n^3, etc. (the N{1} term is ignored). The controlled polynomial system is then described as either
\dot{x} = A*x + B*u + N{2}*kron(x,x) + N{3}*kron(kron(x,x),x)
>> [k,v] = cqr(A,B,Q,R,N,degree);
or
\dot{x} = A*x + B*u + N{2}*kron(x,x) + N{3}*kron(kron(x,x),x) + ... N{p}*kron(...kron(x,x),x)
>> [k,v] = pqr(A,B,Q,R,N,degree);
The description of the feedback coefficients k and value function coefficients v are exactly as in qqr above.
For details on how to run cqr or pqr, one can also type
>> help cqr
>> help pqr
A special version of cqr is available when N{2}=0 (in this case, the feedback terms are all odd. This version, cqrOdd is called with N{3} (not the cell array).
>> [k,v] = cqrOdd(A,B,Q,R,N{3},degree)
(degree should be odd in this case).
Defines the path where functions for working with Kronecker product expressions as well as the optional tensor_recursive solver is stored. The default settings may work but can be changed if the solver directories are located elsewhere.
Defines the path where the Nonlinear Systems Toolbox by Krener is located. This is optional and only used for testing and debugging.
Builds and solves the full Kronecker product form of the polynomial approximation to the HJB equation. Schur decomposition of A+Bk{1} should be performed to produce an upper triangular system. This would still be an O(n^2degree ) algorithm and prohibitively expensive. This file is only included for archival reasons.
Solves the polynomial-quadratic regulator problem.
Solves the cubic-quadratic regulator problem (a front end to pqr).
Solves the quadratic-quadratic regulator problem (a front end to pqr).
A special case that exploits computation when the right-hand-size polynomial only includes odd degree terms.
A number of examples can be found in the examples directory. More are available in the PolynomialSystems repository.
Solves the control problem for randomly generated systems. Some systems are nearly uncontrollable, others have near zero coefficients, so relative errors could be large.
Solves a control problem using a discretization of the 1-dimensional Burgers equation (found in the included Burgers1DControl directory). The control inputs are spatially distributed uniform sources.
Similar to example1.m, except we force A to be negative-definite, symmetric.
Compare feedback strategies for the Lorenz system.
Similar to example2.m, except we consider a linear reaction term and use a better change-of-variables to convert the discretized system to an explicit system of controlled differential equations.
A simple first-order system where we can investigate convergence of the value function (by plotting it).
A first order cubic control system introduced in Sakamoto and van der Schaft, 2007.
A ring of van der Pol oscillators. This example can generate arbitrarily large systems of cubic equations. The region of attraction varies with the degree of feedback control.
@misc{borggaard2019quadraticquadratic,
title={The Quadratic-Quadratic Regulator Problem:
Approximating feedback controls for quadratic-in-state nonlinear systems},
author={Jeff Borggaard and Lizette Zietsman},
booktitle={Proceedings of the 2020 American Control Conference},
year={2020},
eprint={1910.03396},
archivePrefix={arXiv},
primaryClass={math.OC}
}
@misc{borggaard2020polynomialquadratic,
title={On Approximating Polynomial-Quadratic Regulator Problems},
author={Jeff Borggaard and Lizette Zietsman},
year={2020},
eprint={2009.11068},
archivePrefix={arXiv},
primaryClass{math.OC}
}
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