System modelling

This is a brief overview of the key concepts in system modelling, which we will use for control law design.

State, input, dynamics

A model has three pieces. The state \(x \in \mathbb{R}^n\) is a set of variables that describe the system at a given time and allow you to predict its future. The input \(u \in \mathbb{R}^m\) is what you are allowed to command - a force, a torque, a thrust, a wheel speed. The dynamics \(f\) tie them together:

$$\dot{x} = f(x, u).$$

The defining property of a state is sufficiency: if you know \(x(t_0)\) and the whole input signal \(u(\cdot)\) on \([t_0, t]\), you can compute \(x(t)\) without knowing anything else about what happened before \(t_0\).

Continuous time and discrete time

Physics is written in continuous time,

$$\dot{x} = f(x,u),$$

but controllers run on computers at a fixed rate, so the model you actually implement is a difference equation

$$x_{k+1} = F(x_k, u_k),$$

where the input is held constant between the sampling time \(\Delta t\). Both descriptions of the same robot coexist: you analyse in continuous time and implement in discrete time, or you discretise once and do everything discretely.

Three ways to get \(F\) from \(f\), in increasing order of cost and accuracy:

• Forward Euler. \(x_{k+1} = x_k + \Delta t\, f(x_k, u_k)\). 1st order prediction, common.
• Runge-Kutta 4. Four evaluations per step. 4th order prediction, more accurate.
• Exact discretization (linear systems only). For \(\dot x = Ax + Bu\) with \(u\) held constant over each interval, $$A_d = e^{A\Delta t}, \qquad B_d = \left(\int_0^{\Delta t} e^{A\tau}\,d\tau\right) B,$$ which is exact, not an approximation.

Linear and nonlinear systems

A system is linear when it can be written as

$$\dot{x} = A(t) x(t) + B(t) u, \qquad y = C(t) x + D(t) u.$$

inear systems are commonly divided into linear time-invariant (LTI) and linear time-varying (LTV) systems.

LTI systems

A linear system is time-invariant when its system matrices are constant:

$$\dot{x} = A x + B u, \qquad y = C x + D u.$$

The matrices \(A,B,C,D\) do not explicitly depend on time. Consequently, the same relationship between the state and input applies at every time. LTI systems have particularly convenient analytical properties and are the standard setting for many classical control methods, including LQR.

LTV systems

A linear system is time-varying when its system matrices depend explicitly on time:

$$\dot{x}=A(t)x+B(t)u, \qquad y=C(t)x+D(t)u.$$

The system is still linear in \(x\) and \(u\), but its dynamics change with time. For example, a robot following a time-varying reference trajectory may be described by an LTV model obtained by linearizing nonlinear dynamics along that trajectory.

The distinction can therefore be summarized as

$$ \begin{array}{lll} \text{LTI:} & \dot{x}=Ax+Bu, & A,B\text{ are constant}, \\ \text{LTV:} & \dot{x}=A(t)x+B(t)u, & A(t),B(t)\text{ vary with time}. \end{array} $$

Both are linear systems; the difference is whether their system matrices are constant or time-dependent.

Nonlinear systems

A general continuous-time nonlinear system is written as

$$\dot{x}=f(x,u),\qquad y=h(x,u),$$

where \(f\) and \(h\) are generally nonlinear functions of the state \(x\) and control input \(u\). Unlike linear systems, the dynamics may contain terms such as \(x^2\), \(xu\), \(\sin(x)\), or other nonlinear functions.

The distinction matters because linearity buys an enormous amount:

• Closed-form solutions, transfer functions, frequency response.
• Stability analysis is straightforward and well-developed for linear systems.
• Controllability and observability reduced to rank tests on a single matrix.
• Optimal control that solves in closed form (LQR) and convex MPC.

None of that survives nonlinearity intact. Most real robots are nonlinear: rotations, gravity, friction, contact, and aerodynamics all break superposition.

The control-affine systems

Fortunately, a huge fraction of robotic systems sit in a convenient middle class, control-affine:

$$\dot{x} = f(x) + g(x)\,u.$$

The state may enter as nonlinearly as it likes, but the input enters linearly. Here \(f(x)\) is the drift (what the system does with \(u = 0\)) and \(g(x)\) is the control-input matrix.

Linearization

Nonlinear systems are often well approximated by a linear one near an operating point \((x^\star, u^\star)\). Writing \(\delta x = x - x^\star\) and \(\delta u = u - u^\star\), a first-order Taylor expansion gives

$$\delta\dot{x} = A\,\delta x + B\,\delta u, \qquad A = \left.\frac{\partial f}{\partial x}\right|_{(x^\star,u^\star)}, \qquad B = \left.\frac{\partial f}{\partial u}\right|_{(x^\star,u^\star)}.$$

If \((x^\star, u^\star)\) is an equilibrium - \(f(x^\star,u^\star) = 0\) - then Lyapunov's indirect method applies: if every eigenvalue of \(A\) has strictly negative real part, the nonlinear system is locally asymptotically stable there. If any has strictly positive real part, it is unstable. Eigenvalues exactly on the imaginary axis tell you nothing and you have to go back to the nonlinear model.

Linearizing around a trajectory rather than a point gives a time-varying linear system \(\delta\dot x = A(t)\,\delta x + B(t)\,\delta u\), which is the basis of trajectory-tracking LQR and of iLQR/DDP.

Fully actuated and underactuated systems

A system is fully actuated when the dimension of input equals the number of degrees of freedom. It is underactuated when the dimension of input is less than the number of degrees of freedom.

Importantly, underactuated does not mean uncontrollable.An underactuated system can still be controllable; it simply cannot independently command every degree of freedom at every instant.

Common underactuated systems: quadrotors (6 DOF, 4 inputs - to translate sideways a quadrotor must first tilt), cart-poles, Acrobots, legged robots during flight phases, fixed-wing aircraft, and any car with nonholonomic constraints.

Modelling uncertainty

Everything above is the nominal model. Real systems differ from it, and how you write the difference determines what kind of guarantee you can get.

Bounded (set-based) uncertainty

$$\dot{x} = f(x,u) + d, \qquad d \in \mathcal{D},\ \ \|d\| \le \bar{d}.$$

The disturbance is unknown but confined to a known set. This supports worst-case guarantees - robust control, tube MPC, input-to-state stability, robust CBFs - at the cost of conservatism, since you design against the worst member of \(\mathcal D\) whether or not it ever occurs.

Stochastic uncertainty

$$dx = f(x,u)\,dt + \sigma(x,u)\,dW_t.$$

The disturbance is a random process with known statistics. Guarantees become probabilistic: expected cost, chance constraints \(\Pr[x \in \mathcal{X}] \ge 1-\epsilon\), etc. Less conservative, but a probabilistic guarantee.

Outputs and estimation

You rarely measure the full state. The measurement model is

$$y = h(x,u) + \nu, \qquad \text{or linearly} \qquad y = Cx + Du + \nu,$$

with \(\nu\) sensor noise. A GPS gives position but not velocity; an encoder gives joint angle but not joint rate; a camera gives pixels.

Observability asks whether \(x(0)\) can be reconstructed from \(y(\cdot)\) and \(u(\cdot)\) over a finite window. For LTI systems it is the rank condition

$$\operatorname{rank} \begin{bmatrix} C \\ CA \\ \vdots \\ CA^{n-1}\end{bmatrix} = n,$$

the dual of controllability. If a system is observable you can build an observer - a Luenberger observer, a Kalman filter, a moving-horizon estimator - and feed its estimate \(\hat x\) to a controller designed as if the state were measured. For LTI systems the separation principle says designing the controller and observer independently still yields a stable closed loop; for nonlinear systems it generally does not, and the coupling has to be handled explicitly.

Classifying a model

Every model can be placed against the same checklist, and the placement is what tells you which controllers are candidates.