@@ -129,7 +129,7 @@ The result of this optimization gives us the estimated state for the
129129previous :math: `N` steps in time, including the "current" time
130130:math: `x[N]`. The basic idea is thus to compute the state estimate that is
131131most consistent with our model and penalize the noise and disturbances
132- according to how likely the are (based on the given stochastic system
132+ according to how likely they are (based on the given stochastic system
133133model for each).
134134
135135Given a solution to this fixed-horizon optimal estimation problem, we can
@@ -344,7 +344,7 @@ following code::
344344
345345We consider an optimal control problem that consists of "changing lanes" by
346346moving from the point x = 0 m, y = -2 m, :math: `\theta ` = 0 to the point x =
347- 100 m, y = 2 m, :math: `\theta ` = 0) over a period of 10 seconds and with a
347+ 100 m, y = 2 m, :math: `\theta ` = 0) over a period of 10 seconds and
348348with a starting and ending velocity of 10 m/s::
349349
350350 x0 = np.array([0., -2., 0.]); u0 = np.array([10., 0.])
@@ -360,7 +360,7 @@ penalizes the state and input using quadratic cost functions::
360360 traj_cost = obc.quadratic_cost(vehicle, Q, R, x0=xf, u0=uf)
361361 term_cost = obc.quadratic_cost(vehicle, P, 0, x0=xf)
362362
363- We also constraint the maximum turning rate to 0.1 radians (about 6 degees )
363+ We also constrain the maximum turning rate to 0.1 radians (about 6 degrees )
364364and constrain the velocity to be in the range of 9 m/s to 11 m/s::
365365
366366 constraints = [ obc.input_range_constraint(vehicle, [8, -0.1], [12, 0.1]) ]
@@ -431,7 +431,7 @@ solutions do not seem close to optimal, here are a few things to try:
431431 good solutions with a small number of free variables (the example above
432432 uses 3 time points for 2 inputs, so a total of 6 optimization variables).
433433 Note that you can "resample" the optimal trajectory by running a
434- simulation of the sytem and using the `t_eval ` keyword in
434+ simulation of the system and using the `t_eval ` keyword in
435435 `input_output_response ` (as done above).
436436
437437* Use a smooth basis: as an alternative to parameterizing the optimal
@@ -445,14 +445,14 @@ solutions do not seem close to optimal, here are a few things to try:
445445 and `minimize_kwargs ` keywords in :func: `~control.solve_ocp `, you can
446446 choose the SciPy optimization function that you use and set many
447447 parameters. See :func: `scipy.optimize.minimize ` for more information on
448- the optimzers that are available and the options and keywords that they
448+ the optimizers that are available and the options and keywords that they
449449 accept.
450450
451451* Walk before you run: try setting up a simpler version of the optimization,
452452 remove constraints or simplifying the cost to get a simple version of the
453453 problem working and then add complexity. Sometimes this can help you find
454454 the right set of options or identify situations in which you are being too
455- aggressive in what your are trying to get the system to do.
455+ aggressive in what you are trying to get the system to do.
456456
457457See :ref: `steering-optimal ` for some examples of different problem
458458formulations.
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