Post on 13-Jan-2022
Convex OptimizationMay 25, 2012
Dynamic Walking 2012
Convex OptimizationMay 25, 2012
Dynamic Walking 2012
Examples from Dynamic Walking
● Fit return maps with linear model despite noisy sensors / animals
● Approximate inverse dynamics (w/ constraints)○ Find muscle activations to match data in
musculoskeletal model○ Linear optimal control○ Nonlinear stability analysis○ Region of attraction○ Nonlinear control design
Tools● Linear programming● Quadratic programming● Semidefinite programming
○ Sum of squares Optimization
Software● SeDuMi: Solver of semidefinite programs
○ SDP○ LP, QP, etc.
● YALMIP○ Matlab wrapper for many solvers
Referenceshttp://www.stanford.edu/~boyd/cvxbook/
Optimization
Problemsf(x) has local minimaslow convergence
Decision Variables
Scalar Objective
Convex programming
No local minima (only 1 global minimum)
Definition of convex functions
Examples: linear function, quadratic, etc.x
f(x)
Convex programming
Constraints from a convex set
Linear half-space also a convex setCombinations of linear constraints result in convex sets
x1
x2any straight line intersecting set always has only one contiguous region inside set, never crosses out and then back in
Convex Optimization, Stephen BoydConvex Analysis, Rockafellar
Example: Fit model to data
t
y
trip
noisy data of recovery from trip
Quadratic Objectives
Linear Constraints
Example: linear_guassian.m [sic]sdpvar Ahat % defines the unknown variableerr = ...objective = err'*err;constraints = [Ahat<=1, Ahat>=-1]; % defines allowable bounds on Ahat% yalmip is pretty smart and allows all sorts of convex constraintssolvesdp(constraints, objectives) % picks an appropriate solver and runs it% returns a structure showing how well it did
Very noisy data being fit with a first-order model
Dynamics
Approximate inverse dynamics as QP
inertial matrixCoriolis, gravity Applied
torqueq1
q2
q3
Example: approximate inverse dynamics
ground reaction forces
Quadratic program
Solve arm muscle forces
muscle model based on:http://www.springerlink.com/content/j750t73548076000/?MUD=MP
Solve arm muscle forcesload meas_arm_data; % previously provided data, including q_meas, qd_meas, etc.
n=length(t);
ahat = sdpvar(6,n); % define 6 x n matrix to be solved for
A = MuscleMapping; % matrix of moment arms of muscles
con =[];
for i=1:n
[H,C,B] = ArmDynamics(q_meas(:,i),qd_meas(:,i));
% ArmDynamics returns mass matrix, other terms numerically
con = [con; H*qdd_meas(:,i) + C - B*A*ahat(:,i) == 0];
end
obj = norm (ahat(:),2); % define quadratic objective
con = [con;ahat(:)>=0]; % with non-negativity constraints
solvesdp(con,obj) % puts soln in ahat
load true_arm_data;
plot(t,double(ahat(1,:)),t,a(1,:));
% note ahat is an object that can be cast into a
% double matrix
legend('ahat','a')
Example: min max problem
convert to
Turn into a convex program by introducing new variable cThere are many ways to convert particular optimization problems (that don't necessarily look convex) into convex programs
This often arises when we have a piecewise linear objective function. We can write it like:
Semidefinite programming
The set of semidefinite matrices is convex.Semidefinite matrix:
Convex combination of semidefinite matrices:
Stability
Another option: Lyapunov stability
Convert into an SDP by optimizing a positive semi-definite matrix S that defines a Lyapunov function.Hard to check eigenvalues of A, but easier to find an S
meaning that we need an S such that
suggestion: use ^{\top} instead of ^T for transpose
Could check stability using
Robust stabilitySuppose A is not known precisely, but that it has possible perturbations that can be expressed as a weighted sum of matrices.Now find an S (i.e. Lyapunov function) that yields a stable closed-loop system despite any weights/perturbations within defined allowable bounds.
Robust stability via Common-Lyapunov function
Polynomial dynamics
Approximate ODE's state-derivative function with a polynomial in xAlso make V a polynomial in xWrite polynomial as sum of squares
Example: Test stability of a simple system%% Example 1
%% prove that \dot{x} = -x^3 is globally stable
sdpvar x;
f = -x^3;
V = x^2;
constraint = sos(-jacobian(V,x)*f);
sol = solvesos(constraint)
%% Example 2
%% prove that \dot(X) = -x + x^3 is stable for x in (-1,1)
sdpvar x;
f = -x + x^3;
V = x^2;
[lambda,c] = polynomial(x,2);
constraint = [sos(-jacobian(V,x)*f + lambda*(V-1)), sos(lambda)];
sol = solvesos(constraint,[],[],c)
x
xdot
1-1
S-procedure
To write as a SOS program, replace positivity with SOS where lambda is a polynomial in x of a specified degree
Example: van der Pol Oscillator
Find region of attraction represented as a sum of squares
Polynomial representation of equations of motion
H(s,c) sines and cosinesH(s,c) qddot - ... = B us^2 + c^2 = 1
Apply as a constraint!
Examples from Dynamic Walking
● Fit return maps with linear model despite noisy sensors / animals
● Approximate inverse dynamics (w/ constraints)○ Find muscle activations to match data in
musculoskeletal model○ Linear optimal control○ Nonlinear stability analysis○ Region of attraction○ Nonlinear control design
Tools● Linear programming● Quadratic programming● Semidefinite programming
○ Sum of squares Optimization
Software● SeDuMi: Solver of semidefinite programs
○ SDP○ LP, QP, etc.
● YALMIP○ Matlab wrapper for many solvers
Referenceshttp://www.stanford.edu/~boyd/cvxbook/
Optimization
Problemsf(x) has local minimaslow convergence
Decision Variables
Scalar Objective
Convex programming
No local minima (only 1 global minimum)
Definition of convex functions
Examples: linear function, quadratic, etc.x
f(x)
Convex programming
Constraints from a convex set
Linear half-space also a convex setCombinations of linear constraints result in convex sets
x1
x2any straight line intersecting set always has only one contiguous region inside set, never crosses out and then back in
Convex Optimization, Stephen BoydConvex Analysis, Rockafellar
Example: Fit model to data
t
y
trip
noisy data of recovery from trip
Quadratic Objectives
Linear Constraints
Example: linear_guassian.m [sic]sdpvar Ahat % defines the unknown variableerr = ...objective = err'*err;constraints = [Ahat<=1, Ahat>=-1]; % defines allowable bounds on Ahat% yalmip is pretty smart and allows all sorts of convex constraintssolvesdp(constraints, objectives) % picks an appropriate solver and runs it% returns a structure showing how well it did
Very noisy data being fit with a first-order model
Dynamics
Approximate inverse dynamics as QP
inertial matrixCoriolis, gravity Applied
torqueq1
q2
q3
Example: approximate inverse dynamics
ground reaction forces
Quadratic program
Solve arm muscle forces
muscle model based on:http://www.springerlink.com/content/j750t73548076000/?MUD=MP
Solve arm muscle forcesload meas_arm_data; % previously provided data, including q_meas, qd_meas, etc.
n=length(t);
ahat = sdpvar(6,n); % define 6 x n matrix to be solved for
A = MuscleMapping; % matrix of moment arms of muscles
con =[];
for i=1:n
[H,C,B] = ArmDynamics(q_meas(:,i),qd_meas(:,i));
% ArmDynamics returns mass matrix, other terms numerically
con = [con; H*qdd_meas(:,i) + C - B*A*ahat(:,i) == 0];
end
obj = norm (ahat(:),2); % define quadratic objective
con = [con;ahat(:)>=0]; % with non-negativity constraints
solvesdp(con,obj) % puts soln in ahat
load true_arm_data;
plot(t,double(ahat(1,:)),t,a(1,:));
% note ahat is an object that can be cast into a
% double matrix
legend('ahat','a')
Example: min max problem
convert to
Turn into a convex program by introducing new variable cThere are many ways to convert particular optimization problems (that don't necessarily look convex) into convex programs
This often arises when we have a piecewise linear objective function. We can write it like:
Semidefinite programming
The set of semidefinite matrices is convex.Semidefinite matrix:
Convex combination of semidefinite matrices:
Stability
Another option: Lyapunov stability
Convert into an SDP by optimizing a positive semi-definite matrix S that defines a Lyapunov function.Hard to check eigenvalues of A, but easier to find an S
meaning that we need an S such that
suggestion: use ^{\top} instead of ^T for transpose
Could check stability using
Robust stabilitySuppose A is not known precisely, but that it has possible perturbations that can be expressed as a weighted sum of matrices.Now find an S (i.e. Lyapunov function) that yields a stable closed-loop system despite any weights/perturbations within defined allowable bounds.
Robust stability via Common-Lyapunov function
Polynomial dynamics
Approximate ODE's state-derivative function with a polynomial in xAlso make V a polynomial in xWrite polynomial as sum of squares
Example: Test stability of a simple system%% Example 1
%% prove that \dot{x} = -x^3 is globally stable
sdpvar x;
f = -x^3;
V = x^2;
constraint = sos(-jacobian(V,x)*f);
sol = solvesos(constraint)
%% Example 2
%% prove that \dot(X) = -x + x^3 is stable for x in (-1,1)
sdpvar x;
f = -x + x^3;
V = x^2;
[lambda,c] = polynomial(x,2);
constraint = [sos(-jacobian(V,x)*f + lambda*(V-1)), sos(lambda)];
sol = solvesos(constraint,[],[],c)
x
xdot
1-1
S-procedure
To write as a SOS program, replace positivity with SOS where lambda is a polynomial in x of a specified degree
Example: van der Pol Oscillator
Find region of attraction represented as a sum of squares
Polynomial representation of equations of motion
H(s,c) sines and cosinesH(s,c) qddot - ... = B us^2 + c^2 = 1
Apply as a constraint!
Examples from Dynamic Walking
● Fit return maps with linear model despite noisy sensors / animals
● Approximate inverse dynamics (w/ constraints)○ Find muscle activations to match data in
musculoskeletal model○ Linear optimal control○ Nonlinear stability analysis○ Region of attraction○ Nonlinear control design
Tools● Linear programming● Quadratic programming● Semidefinite programming
○ Sum of squares Optimization
Software● SeDuMi: Solver of semidefinite programs
○ SDP○ LP, QP, etc.
● YALMIP○ Matlab wrapper for many solvers
Referenceshttp://www.stanford.edu/~boyd/cvxbook/
Optimization
Problemsf(x) has local minimaslow convergence
Decision Variables
Scalar Objective
Convex programming
No local minima (only 1 global minimum)
Definition of convex functions
Examples: linear function, quadratic, etc.x
f(x)
Convex programming
Constraints from a convex set
Linear half-space also a convex setCombinations of linear constraints result in convex sets
x1
x2any straight line intersecting set always has only one contiguous region inside set, never crosses out and then back in
Convex Optimization, Stephen BoydConvex Analysis, Rockafellar
Example: Fit model to data
t
y
trip
noisy data of recovery from trip
Quadratic Objectives
Linear Constraints
Example: linear_guassian.m [sic]sdpvar Ahat % defines the unknown variableerr = ...objective = err'*err;constraints = [Ahat<=1, Ahat>=-1]; % defines allowable bounds on Ahat% yalmip is pretty smart and allows all sorts of convex constraintssolvesdp(constraints, objectives) % picks an appropriate solver and runs it% returns a structure showing how well it did
Very noisy data being fit with a first-order model
Dynamics
Approximate inverse dynamics as QP
inertial matrixCoriolis, gravity Applied
torqueq1
q2
q3
Example: approximate inverse dynamics
ground reaction forces
Quadratic program
Solve arm muscle forces
muscle model based on:http://www.springerlink.com/content/j750t73548076000/?MUD=MP
Solve arm muscle forcesload meas_arm_data; % previously provided data, including q_meas, qd_meas, etc.
n=length(t);
ahat = sdpvar(6,n); % define 6 x n matrix to be solved for
A = MuscleMapping; % matrix of moment arms of muscles
con =[];
for i=1:n
[H,C,B] = ArmDynamics(q_meas(:,i),qd_meas(:,i));
% ArmDynamics returns mass matrix, other terms numerically
con = [con; H*qdd_meas(:,i) + C - B*A*ahat(:,i) == 0];
end
obj = norm (ahat(:),2); % define quadratic objective
con = [con;ahat(:)>=0]; % with non-negativity constraints
solvesdp(con,obj) % puts soln in ahat
load true_arm_data;
plot(t,double(ahat(1,:)),t,a(1,:));
% note ahat is an object that can be cast into a
% double matrix
legend('ahat','a')
Example: min max problem
convert to
Turn into a convex program by introducing new variable cThere are many ways to convert particular optimization problems (that don't necessarily look convex) into convex programs
This often arises when we have a piecewise linear objective function. We can write it like:
Semidefinite programming
The set of semidefinite matrices is convex.Semidefinite matrix:
Convex combination of semidefinite matrices:
Stability
Another option: Lyapunov stability
Convert into an SDP by optimizing a positive semi-definite matrix S that defines a Lyapunov function.Hard to check eigenvalues of A, but easier to find an S
meaning that we need an S such that
suggestion: use ^{\top} instead of ^T for transpose
Could check stability using
Robust stabilitySuppose A is not known precisely, but that it has possible perturbations that can be expressed as a weighted sum of matrices.Now find an S (i.e. Lyapunov function) that yields a stable closed-loop system despite any weights/perturbations within defined allowable bounds.
Robust stability via Common-Lyapunov function
Polynomial dynamics
Approximate ODE's state-derivative function with a polynomial in xAlso make V a polynomial in xWrite polynomial as sum of squares
Example: Test stability of a simple system%% Example 1
%% prove that \dot{x} = -x^3 is globally stable
sdpvar x;
f = -x^3;
V = x^2;
constraint = sos(-jacobian(V,x)*f);
sol = solvesos(constraint)
%% Example 2
%% prove that \dot(X) = -x + x^3 is stable for x in (-1,1)
sdpvar x;
f = -x + x^3;
V = x^2;
[lambda,c] = polynomial(x,2);
constraint = [sos(-jacobian(V,x)*f + lambda*(V-1)), sos(lambda)];
sol = solvesos(constraint,[],[],c)
x
xdot
1-1
S-procedure
To write as a SOS program, replace positivity with SOS where lambda is a polynomial in x of a specified degree
Example: van der Pol Oscillator
Find region of attraction represented as a sum of squares
Polynomial representation of equations of motion
H(s,c) sines and cosinesH(s,c) qddot - ... = B us^2 + c^2 = 1
Apply as a constraint!
Examples from Dynamic Walking
● Fit return maps with linear model despite noisy sensors / animals
● Approximate inverse dynamics (w/ constraints)○ Find muscle activations to match data in
musculoskeletal model○ Linear optimal control○ Nonlinear stability analysis○ Region of attraction○ Nonlinear control design
Tools● Linear programming● Quadratic programming● Semidefinite programming
○ Sum of squares Optimization
Software● SeDuMi: Solver of semidefinite programs
○ SDP○ LP, QP, etc.
● YALMIP○ Matlab wrapper for many solvers
Referenceshttp://www.stanford.edu/~boyd/cvxbook/
Optimization
Problemsf(x) has local minimaslow convergence
Decision Variables
Scalar Objective
Convex programming
No local minima (only 1 global minimum)
Definition of convex functions
Examples: linear function, quadratic, etc.x
f(x)
Convex programming
Constraints from a convex set
Linear half-space also a convex setCombinations of linear constraints result in convex sets
x1
x2any straight line intersecting set always has only one contiguous region inside set, never crosses out and then back in
Convex Optimization, Stephen BoydConvex Analysis, Rockafellar
Example: Fit model to data
t
y
trip
noisy data of recovery from trip
Quadratic Objectives
Linear Constraints
Example: linear_guassian.m [sic]sdpvar Ahat % defines the unknown variableerr = ...objective = err'*err;constraints = [Ahat<=1, Ahat>=-1]; % defines allowable bounds on Ahat% yalmip is pretty smart and allows all sorts of convex constraintssolvesdp(constraints, objectives) % picks an appropriate solver and runs it% returns a structure showing how well it did
Very noisy data being fit with a first-order model
Dynamics
Approximate inverse dynamics as QP
inertial matrixCoriolis, gravity Applied
torqueq1
q2
q3
Example: approximate inverse dynamics
ground reaction forces
Quadratic program
Solve arm muscle forces
muscle model based on:http://www.springerlink.com/content/j750t73548076000/?MUD=MP
Solve arm muscle forcesload meas_arm_data; % previously provided data, including q_meas, qd_meas, etc.
n=length(t);
ahat = sdpvar(6,n); % define 6 x n matrix to be solved for
A = MuscleMapping; % matrix of moment arms of muscles
con =[];
for i=1:n
[H,C,B] = ArmDynamics(q_meas(:,i),qd_meas(:,i));
% ArmDynamics returns mass matrix, other terms numerically
con = [con; H*qdd_meas(:,i) + C - B*A*ahat(:,i) == 0];
end
obj = norm (ahat(:),2); % define quadratic objective
con = [con;ahat(:)>=0]; % with non-negativity constraints
solvesdp(con,obj) % puts soln in ahat
load true_arm_data;
plot(t,double(ahat(1,:)),t,a(1,:));
% note ahat is an object that can be cast into a
% double matrix
legend('ahat','a')
Example: min max problem
convert to
Turn into a convex program by introducing new variable cThere are many ways to convert particular optimization problems (that don't necessarily look convex) into convex programs
This often arises when we have a piecewise linear objective function. We can write it like:
Semidefinite programming
The set of semidefinite matrices is convex.Semidefinite matrix:
Convex combination of semidefinite matrices:
Stability
Another option: Lyapunov stability
Convert into an SDP by optimizing a positive semi-definite matrix S that defines a Lyapunov function.Hard to check eigenvalues of A, but easier to find an S
meaning that we need an S such that
suggestion: use ^{\top} instead of ^T for transpose
Could check stability using
Robust stabilitySuppose A is not known precisely, but that it has possible perturbations that can be expressed as a weighted sum of matrices.Now find an S (i.e. Lyapunov function) that yields a stable closed-loop system despite any weights/perturbations within defined allowable bounds.
Robust stability via Common-Lyapunov function
Polynomial dynamics
Approximate ODE's state-derivative function with a polynomial in xAlso make V a polynomial in xWrite polynomial as sum of squares
Example: Test stability of a simple system%% Example 1
%% prove that \dot{x} = -x^3 is globally stable
sdpvar x;
f = -x^3;
V = x^2;
constraint = sos(-jacobian(V,x)*f);
sol = solvesos(constraint)
%% Example 2
%% prove that \dot(X) = -x + x^3 is stable for x in (-1,1)
sdpvar x;
f = -x + x^3;
V = x^2;
[lambda,c] = polynomial(x,2);
constraint = [sos(-jacobian(V,x)*f + lambda*(V-1)), sos(lambda)];
sol = solvesos(constraint,[],[],c)
x
xdot
1-1
S-procedure
To write as a SOS program, replace positivity with SOS where lambda is a polynomial in x of a specified degree
Example: van der Pol Oscillator
Find region of attraction represented as a sum of squares
Polynomial representation of equations of motion
H(s,c) sines and cosinesH(s,c) qddot - ... = B us^2 + c^2 = 1
Apply as a constraint!