M2-T4 — Trim and Linearization

Learning objectives

After this lesson, you should be able to:

• Find a trim of a nonlinear model by solving the steady-state equations numerically.
• Linearize the model about a trim to state-space form with a Jacobian.
• Read the poles of the linear model and relate them to the behaviour of the vehicle.
• Estimate the range of the linear model's validity, and name the couplings it leaves out.

The intuition

Hopper's nonlinear model, from the last two lessons, has seven states and two inputs. Its lateral acceleration is $(T/m)\sin(\theta + \delta)$, which is not a linear function of anything. The methods of the next modules (poles, frequency response, linear-quadratic regulators) need a model of the form $\dot{\mathbf{x}} = \mathbf{A}\mathbf{x} + \mathbf{B}\mathbf{u}$. The way to get one is to look at the vehicle close up, near one steady condition, where the curved relations are nearly straight.

Trim: a condition the vehicle can hold

A trim is a steady operating condition: the inputs are constant and the accelerations that matter are constant too. Every vehicle has its own.

• A multirotor in hover: zero velocity, zero tilt, rotor thrusts that add up to the weight.
• An aircraft in level flight at a given airspeed: an angle of attack, an elevator angle and a throttle setting at which lift equals weight and thrust equals drag.
• A satellite spinning at a constant rate about a principal axis.
• Hopper in hover (zero tilt, thrust equal to weight) or in a steady acceleration (a constant lateral acceleration, with the thrust tilted to give it).

Finding a trim is solving equations: set the derivatives that must vanish to zero, and solve for the inputs and the states that remain. For hover the answer is obvious. For a steady acceleration it is not: the thrust, the pitch and the gimbal angle have to be found together, because the aerodynamic moment and the gimbal moment have to cancel at the same time as the forces balance. A numerical solver (Newton's method, with a numerical Jacobian) does it for any model.

Linearize: the straight line that touches the curve

Around a trim, small deviations from it obey equations that are linear in the deviations. The figure shows the idea for one term: the straight line $g\theta$ touches the curve $g\sin\theta$ at the origin and stays close for a while. The slope of that line is a partial derivative, and the collection of such slopes for every state and every input is a pair of matrices, $\mathbf{A}$ and $\mathbf{B}$.

What the linear model tells you

The poles of $\mathbf{A}$, its eigenvalues, say how the vehicle behaves near the trim: which modes grow, which decay, which neither. For Hopper in hover they are 0, 0, +0.7 and −0.7 per second: two integrators (position and velocity) and the unstable pair of the pitch axis. The same four numbers were already in the model; the linearization exposes them and makes them computable for a model whose structure is not obvious by inspection.

What it leaves out

A linear model is true at one point and an approximation near it. Two things go wrong as the vehicle moves away. The curvature of each term shows up, as in the figure. And couplings that vanished at the trim come back: at hover, the lateral and vertical motions are independent to first order, but a Hopper tilted by 40° has lost 2.3 m/s² of vertical acceleration that the lateral model knows nothing of.

Trim finds the condition. Linearization gives the model around it. The range of validity, which the linear model cannot tell you about itself, is something you compute separately.

Definitions

Nonlinear model

$\dot{\mathbf{x}} = \mathbf{f}(\mathbf{x}, \mathbf{u})$ with state $\mathbf{x}$ and input $\mathbf{u}$.

Equilibrium or trim $(\mathbf{x}_0, \mathbf{u}_0)$

A state and input at which the part of the motion that must be steady has no acceleration. For a true equilibrium, $\mathbf{f}(\mathbf{x}_0, \mathbf{u}_0) = 0$. For a steady acceleration the equations are solved for the states and inputs that make the other derivatives vanish.

Deviations $\delta\mathbf{x}, \delta\mathbf{u}$

The difference from the trim: $\delta\mathbf{x} = \mathbf{x} - \mathbf{x}_0$, $\delta\mathbf{u} = \mathbf{u} - \mathbf{u}_0$.

Jacobian

The matrix of partial derivatives of a vector function: its entry $(i, j)$ is how output $i$ changes with input $j$.

Pole

An eigenvalue of $\mathbf{A}$. A mode with eigenvalue $\lambda$ behaves as $e^{\lambda t}$ (M1-T3: One Block Diagram, Four Vehicles explains why).

The linearization is the first-order expansion of the model about the trim:

For a steady lateral acceleration $a$, the trim of Hopper has no vertical or pitch acceleration. Its three unknowns, the thrust $T$, the pitch $\theta$ and the gimbal $\delta$, satisfy three equations:

Linearizing the lateral channel, with states $(x, v_x, \theta, q)$ and input $\delta$, gives:

Derivation: the Jacobian of Hopper, its poles, and how far it can be trusted

The linearization follows from one expansion. Then the entries of $\mathbf{A}$ and $\mathbf{B}$ for Hopper can be written down by hand, and the model's range of validity estimated.

1. $$\mathbf{f}(\mathbf{x}_0 + \delta\mathbf{x},\,\mathbf{u}_0 + \delta\mathbf{u}) = \mathbf{f}(\mathbf{x}_0, \mathbf{u}_0) + \mathbf{A}\,\delta\mathbf{x} + \mathbf{B}\,\delta\mathbf{u} + \text{(second order)}$$

   Expand each component of $\mathbf{f}$ in a Taylor series about the trim. The first-order coefficients are the partial derivatives, collected into the Jacobians $\mathbf{A}$ and $\mathbf{B}$.

2. $$\delta\dot{\mathbf{x}} = \dot{\mathbf{x}} - \dot{\mathbf{x}}_0 \approx \mathbf{A}\,\delta\mathbf{x} + \mathbf{B}\,\delta\mathbf{u}$$

   At a trim the derivative of the part of the state that must stay steady is zero (or constant), so it drops out of the difference. Neglecting the second-order terms leaves the linear model.

The entries for Hopper

1. $$\frac{\partial}{\partial\theta}\!\left[\frac{T}{m}\sin(\theta + \delta)\right] = \frac{T}{m}\cos(\theta_0 + \delta_0) = g$$

   Differentiate the lateral acceleration with respect to pitch. At the trim the vertical balance of the trim equations says $(T/m)\cos(\theta_0 + \delta_0) = g$, so the entry is exactly $g$ at every steady acceleration, not just in hover. The same holds for the derivative with respect to $\delta$.

2. $$\frac{\partial}{\partial\theta}\!\left[\lambda^2\theta + \frac{T l}{J}\sin\delta\right] = \lambda^2, \qquad \frac{\partial}{\partial\delta}\!\left[\cdots\right] = \frac{T\,l}{J}\cos\delta_0$$

   Differentiate the pitch acceleration. The aerodynamic term is already linear. The gimbal term has slope $b = T l\cos\delta_0/J$, the authority at the trim.

3. $$\det(s\mathbf{I} - \mathbf{A}) = s^2\,(s^2 - \lambda^2)$$

   Expand the determinant. The chain $x, v_x$ is two integrators driven by the pitch, and the pitch pair is $\ddot\theta = \lambda^2\theta$. The characteristic polynomial factors into $s^2$ and $s^2 - \lambda^2$.

4. $$s = 0,\ 0,\ +\lambda,\ -\lambda$$

   The poles: two integrators and a divergent and a convergent pitch mode. They do not depend on the trim, and they do not depend on the pitch authority. Only $\mathbf{B}$ changes with the thrust.

How far can it be trusted?

1. $$\sin\theta = \theta - \frac{\theta^3}{6} + \cdots \;\Rightarrow\; \frac{g\theta - g\sin\theta}{g\sin\theta} \approx \frac{\theta^2}{6}$$

   The first neglected term of the series for the sine. The relative error of the linear lateral acceleration grows as the square of the tilt.

2. $$\frac{\theta^2}{6} = 0.05 \Rightarrow \theta = 0.55\ \text{rad} = 31^\circ, \qquad \frac{\theta^2}{6} = 0.10 \Rightarrow \theta = 0.77\ \text{rad} = 44^\circ$$

   Choose an acceptable error and solve for the tilt. Both numbers match the figure (the 10% point, at 43°, is a little lower because the next term of the series matters there).

3. $$\Delta a_z = g\,(\cos\theta - 1) \approx -\frac{g\,\theta^2}{2}$$

   The vertical acceleration that the lateral model leaves out when thrust is held at the hover value. It is a second-order effect that is missing from the linear model of the lateral channel.

The linearization is exact only at the trim. Near it the error is second order in the distance, and the question is never "is the model linear?" but "how far from the trim will the vehicle go?".

A worked example

Find the trim of Hopper for a lateral acceleration of 2 m/s² at liftoff mass, linearize its lateral channel about hover and about that trim, and measure where the hover model stops being accurate.

1. Trim. Newton's method on the three equations of the definitions, with a numerical Jacobian, starting from the weight and small angles, finds a thrust of 1201.4 N (1.0206 times the weight), a pitch of 11.81° and a gimbal angle of −0.289°. The thrust direction $\theta + \delta$ is 11.52°, which is $\arctan(2/9.81)$: the direction of the vector sum of weight and the required acceleration.
2. Linearize. A central-difference Jacobian of the nonlinear model gives the matrices at both operating points. At hover $\mathbf{A}$ has the entries $g = 9.81$ and $\lambda^2 = 0.49$ and $\mathbf{B}$ has $(g, b) = (9.81, 19.62)$. At the trim $\mathbf{A}$ is identical, as the derivation says, and $b$ is 20.02.
3. Poles. The eigenvalues are −0.7, 0, 0 and +0.7 per second at both operating points.
4. Validity. The linear lateral acceleration $g\theta$ is 0.5% too high at 10° of tilt, 2.1% at 20°, 4.7% at 30°, 8.6% at 40° and 20.9% at 60°. The vertical acceleration that it leaves out is −0.15 m/s² at 10° and −2.3 m/s² at 40°, nearly a quarter of gravity.
5. Read it. For a hop whose tilt stays below about 20° the hover model is good to 2% and the missing vertical coupling is below 0.6 m/s². For manoeuvres beyond 30° one linear model is not enough, and the options are several linearizations (gain scheduling, module 5) or a nonlinear controller.

import math
from gnckit import HOPPER, hopper_deriv, jacobian, newton, mat_eigvals

P = HOPPER
g, m, J, l, lam = P['g'], P['m0'], P['J'], P['l'], P['lam']

def f(state, T, delta):
    return hopper_deriv(list(state) + [m], T, delta)[:6]

# trim for a steady lateral acceleration a: no vertical acceleration, no pitch acceleration
def residual(u, a=2.0):
    T, th, delta = u
    d = hopper_deriv([0.0, 10.0, th, 0.0, 0.0, 0.0, m], T, delta)
    return [d[3] - a, d[4], d[5]]
T, th, dl = newton(residual, [m * g, 0.1, -0.01])
print(f'trim at a = 2 m/s^2: thrust {T:.1f} N ({T / (m * g):.4f} x weight), pitch {math.degrees(th):.2f} deg, gimbal {math.degrees(dl):.3f} deg')

lat_idx = (0, 3, 2, 5)
def lateral_AB(T, th, dl):
    x0 = [0.0, 10.0, th, 0.0, 0.0, 0.0]
    A = jacobian(lambda s: f(s, T, dl), x0)
    h = 1e-6
    dfdu = [(a - b) / (2 * h) for a, b in zip(f(x0, T, dl + h), f(x0, T, dl - h))]
    return [[A[i][j] for j in lat_idx] for i in lat_idx], [dfdu[i] for i in lat_idx]
for name, args in (('hover', (m * g, 0.0, 0.0)), ('a = 2 trim', (T, th, dl))):
    A, B = lateral_AB(*args)
    ev = sorted(mat_eigvals(A), key=lambda c: complex(c).real)
    print(f'{name:10s} A[1][2] = {A[1][2]:.3f}  A[3][2] = {A[3][2]:.3f}  B = ({B[1]:.3f}, {B[3]:.3f})  poles {[round(complex(e).real, 3) for e in ev]}')

print('how far the linear model is from the nonlinear one (hover linearization):')
for deg in (10, 20, 30, 40, 60):
    t = math.radians(deg)
    a_nl = g * math.sin(t); a_lin = g * t
    az = g * (math.cos(t) - 1.0)
    print(f'  theta {deg:2d} deg: lateral accel {a_nl:5.2f} vs linear {a_lin:5.2f} m/s^2 ({100 * (a_lin / a_nl - 1):4.1f}% high), vertical accel the linear model leaves out {az:6.2f} m/s^2')

trim at a = 2 m/s^2: thrust 1201.4 N (1.0206 x weight), pitch 11.81 deg, gimbal -0.289 deg
hover      A[1][2] = 9.810  A[3][2] = 0.490  B = (9.810, 19.620)  poles [-0.7, 0.0, 0.0, 0.7]
a = 2 trim A[1][2] = 9.810  A[3][2] = 0.490  B = (9.810, 20.023)  poles [-0.7, 0.0, 0.0, 0.7]
how far the linear model is from the nonlinear one (hover linearization):
  theta 10 deg: lateral accel  1.70 vs linear  1.71 m/s^2 ( 0.5% high), vertical accel the linear model leaves out  -0.15 m/s^2
  theta 20 deg: lateral accel  3.36 vs linear  3.42 m/s^2 ( 2.1% high), vertical accel the linear model leaves out  -0.59 m/s^2
  theta 30 deg: lateral accel  4.90 vs linear  5.14 m/s^2 ( 4.7% high), vertical accel the linear model leaves out  -1.31 m/s^2
  theta 40 deg: lateral accel  6.31 vs linear  6.85 m/s^2 ( 8.6% high), vertical accel the linear model leaves out  -2.30 m/s^2
  theta 60 deg: lateral accel  8.50 vs linear 10.27 m/s^2 (20.9% high), vertical accel the linear model leaves out  -4.90 m/s^2

In robotics

Linearization is the first step of almost every control design, and the choice of trim shapes the result.

• Aircraft. The trim is a level flight at a chosen airspeed and altitude, found by solving for the angle of attack, elevator and throttle. Linearizing there gives the longitudinal and lateral models whose modes (short period, phugoid, roll, spiral, Dutch roll) module 10 studies. A different airspeed is a different trim and a different model, so an aircraft has a family of them.
• Multirotors. The hover trim and its linear model are the basis of the attitude and position controllers of UAV M5-T2: Quadrotor dynamic modelling. The lateral channel of Hopper has the same structure as the translational channel of a quadrotor: two integrators fed by the tilt.
• Tools. Simulation environments linearize a model with a numerical Jacobian exactly as in the worked example: Simulink's linmod, python-control's linearize and the linearization functions of robotics toolboxes. The step size of the numerical derivative matters: too large and it measures curvature, too small and it measures rounding error.
• Gain scheduling. A family of linearizations over a changing parameter, such as the mass or the dynamic pressure, gives a family of controllers (module 5). Hopper's $\mathbf{B}$ depends on the thrust, so its pitch authority is 19.6 at liftoff and 11.8 at burnout.
• Stability of the nonlinear system. If every pole of $\mathbf{A}$ is in the left half-plane, the nonlinear system is locally stable at the trim, and if one is in the right half-plane, it is unstable. Poles at zero decide nothing. This is Lyapunov's indirect method (module 5). It says nothing about how large the region of stability is.

The Jacobian is the same object that appears in kinematics and in estimation: Linear Algebra M8-T4: Jacobian Matrices derives it for a robot arm, and the extended Kalman filter of module 7 linearizes its measurement model in the same way.

Common misconceptions

Key takeaways

1. A trim is a steady operating condition that a vehicle can hold: hover, level flight at an airspeed, a steady acceleration. It is found by solving the steady-state equations, numerically if need be.
2. Linearizing about a trim gives $\delta\dot{\mathbf{x}} = \mathbf{A}\,\delta\mathbf{x} + \mathbf{B}\,\delta\mathbf{u}$ with the Jacobians of the model as $\mathbf{A}$ and $\mathbf{B}$.
3. Hopper's lateral channel has poles 0, 0, +0.7 and −0.7 per second at every steady acceleration; only the input matrix changes with thrust and mass.
4. The error of the linear model is second order in the distance from the trim: for Hopper about $\theta^2/6$ in lateral acceleration.
5. The linear model also drops couplings: at 40° of tilt, 2.3 m/s² of vertical acceleration is missing.
6. Compute the range over which the linear model holds before designing on it, and use several linearizations when the vehicle will go farther.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• K. J. Åström and R. M. Murray, Feedback Systems, 2nd edition, Princeton University Press, 2021 , the section on linearization. Free to read as the authors' PDF; the book is copyrighted.
• B. L. Stevens, F. L. Lewis and E. N. Johnson, Aircraft Control and Simulation, 3rd edition, Wiley, 2016, the chapter on aircraft trim and linearization. The standard treatment, with the numerical methods. Paid book.
• J. How, MIT 16.333 Aircraft Stability and Control (OpenCourseWare) , the lectures on trim and linearized equations. Free; licence CC BY-NC-SA 4.0.
• R. Tedrake, Underactuated Robotics , the chapter on linearization and stability of nonlinear systems, with runnable notebooks. Free to read; no licence statement was found.

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What’s next

Related content

Nothing below is required to finish this lesson.

• Before
  M2-T3: Forces, Moments and Actuator Models

  Model gravity, thrust and aerodynamic forces with coefficients, and model an actuator by its lag, rate limit and saturation.

• Lab
  M2-L1: Trim and Linearize Hopper

  Find the gimbal angle and thrust that hold Hopper in hover with an off-centre mass, linearize about that point, and show where the linear model stops matching the simulator.

Other courses on this site treat neighbouring topics from their own angle:

• UAV M1-T4: UAV coordinate frames
• UAV M1-T5: Six-degree-of-freedom UAV dynamics
• UAV M5-T1: UAV equations of motion
• UAV M5-T2: Quadrotor dynamic modelling
• Linear Algebra M7-T3: Three-Dimensional Rotation Matrices
• Linear Algebra M7-T8: Euler Angles and Quaternions
• Linear Algebra M8-T4: Jacobian Matrices
• Navigation M1-T3: Robot Motion Models for Navigation
• Robot Learning M7-T2: System Identification

This course leaves the following to the pages that already teach them:

• Rotation matrices, Euler angles and quaternions from first principles
• The quadrotor model, motor mixing and thrust generation
• Unicycle and car-like kinematic models