Controllability

A linear time-invariant (LTI) system is controllable if, for any initial state x₀ and any desired final state x_f, there exists a finite time T and an admissible control input u(t) that steers the system from x₀ to x_f. The primary test is the Kalman rank condition: the system's controllability matrix C = [B, AB, A²B, ..., Aⁿ⁻¹B] must have full row rank (rank = n, the state dimension). This ensures the control input can influence all modes of the system.

The set of all states reachable from the origin is the controllable subspace, which is precisely the column space of the controllability matrix C. If the system is not fully controllable, the state space decomposes into a controllable and an uncontrollable part (via the Kalman decomposition). Only states within the controllable subspace can be influenced by the control input u(t). For unstable systems, uncontrollable modes lead to instability that cannot be corrected by feedback.

Stabilizability is a weaker, more practical property than full controllability. A system is stabilizable if all its uncontrollable modes are stable (i.e., have negative real parts). This means that while you may not be able to steer the system to an arbitrary state, you can design a feedback controller to drive all unstable states to zero, ensuring the overall system is stable. It is the essential condition for designing a stabilizing Linear Quadratic Regulator (LQR).

Distinct from state controllability, output controllability concerns whether the control input can steer the system's output y(t), not its internal state. A system is output controllable if the matrix [CB, CAB, CA²B, ..., CAⁿ⁻¹B] has full row rank equal to the output dimension. A system can be output controllable even if it is not state controllable, as long as the uncontrollable states do not affect the output.

For nonlinear systems, controllability is analyzed locally using Lie brackets and the concept of differential controllability. The system is locally accessible if the accessibility distribution, built from Lie brackets of the system vector fields, has full rank. A stronger property, small-time local controllability (STLC), holds if the system can reach nearby states in arbitrarily small time. This is foundational for planning for nonholonomic systems like cars and drones.

In robotics, controllability analysis determines feasible motions.

• Underactuated Systems (e.g., quadrotors, walking robots): Have fewer control inputs than degrees of freedom. They are often not fully controllable in all configurations but may be small-time locally controllable, enabling complex maneuvers through dynamic coupling.
• Nonholonomic Systems (e.g., wheeled robots): Subject to constraints like "no sideways slip." They are not controllable via linearization but are often STLC, requiring nonlinear planning (e.g., using RRTs).
• Kinematic vs. Dynamic Controllability: A robot arm may be kinematically controllable (any joint configuration is reachable) but require dynamic analysis to see if it can be moved there given torque limits and inertia.

A classic example of a fully controllable linear system.

• System Model: A point mass (the cart) with position x and velocity v. A single force F is applied.
• State-Space: States are [x, v]. The control input is F.
• Why Controllable? The applied force directly influences acceleration, which integrates to velocity and then position. With a well-designed controller, you can drive the cart from any starting [x, v] to any desired [x_desired, v_desired] in finite time.

A controllable but constrained nonlinear system. It is controllable but not stabilizable by continuous, time-invariant feedback (Brockett's condition).

• System Model: States are [x, y, θ] (position and heading). Control inputs are forward velocity v and steering angle φ.
• The Constraint: The car cannot move directly sideways. This is a nonholonomic constraint: ẋ sinθ - ẏ cosθ = 0.
• Why Controllable? Despite the constraint, complex maneuvers (parallel parking) allow the car to reach any [x, y, θ]. The system is nonlinearly controllable. The Linearized model around a straight path, however, may lose controllability in the lateral direction.

An underactuated system where controllability depends on the configuration. It has fewer independent control inputs than degrees of freedom.

• System Model: Two linked rods. States are the angles and angular velocities of both joints [θ1, θ2, ω1, ω2]. Often, only the base joint has a torque actuator.
• Controllability Analysis: The linearized system around its unstable upright equilibrium is controllable with a single actuator—you can swing it up. However, linearized around some hanging configurations, it may be uncontrollable, as the torque cannot independently influence all state directions.
• Key Insight: Controllability is a local property evaluated at a specific operating point.

An example of a system transitioning from controllable to uncontrollable due to an actuator failure.

• Nominal System: A quadcopter has four independent rotor thrusts. Its 6-DOF pose [x, y, z, roll, pitch, yaw] is fully controllable.
• Failure Mode: One rotor seizes. The system loses independent control authority over yaw and a translational direction.
• Result: The system becomes uncontrollable in the full state-space. It may only be controllable within a degraded subspace (e.g., it can still translate in some directions but cannot achieve arbitrary orientation). This is critical for fault-tolerant control design.

A canonical fully controllable linear system used as a benchmark in control theory.

• System Model: ẋ1 = x2, ẋ2 = x3, ..., ẋ_n = u. The control input u is the nth derivative of the first state x1.
• State-Space: x = [x1, x2, ..., x_n]^T. The system matrix A is a nilpotent Jordan block, and the control matrix B is [0, 0, ..., 1]^T.
• Why Controllable? The controllability matrix [B, AB, A^2B, ...] is full rank (it's the identity matrix). This structure is the basis for feedback linearization techniques, where nonlinear systems are transformed into this controllable form.

An example of natural uncontrollability due to a conserved quantity or symmetry in the dynamics.

• System Model: Consider a satellite tumbling in space with no external torques. Its total angular momentum is a conserved quantity.
• The Limitation: If the satellite only has internal reaction wheels, it can change its orientation but cannot alter its total angular momentum vector. The component of the state corresponding to total angular momentum is uncontrollable.
• Engineering Implication: You cannot steer the system to an arbitrary angular momentum state; you can only maneuver within the subspace defined by the initial conserved momentum. This requires control moment gyroscopes or thrusters to break the symmetry and achieve full controllability.

Observability is the dual property to controllability. It determines whether the internal state of a dynamical system can be inferred from knowledge of its external outputs over a finite time. A system must be observable for a controller to make informed decisions based on sensor measurements.

• Key Relationship: The Kalman rank condition provides a test for both controllability and observability using the system's A (state) and C (output) matrices.
• Practical Implication: An unobservable system has hidden internal modes that cannot be monitored, potentially leading to instability even if controllable.

Stabilizability is a weaker condition than full controllability. A system is stabilizable if all its unstable modes are controllable. This means it is possible to design a feedback controller such that the closed-loop system is stable, even if some stable modes cannot be directly influenced.

• Contrast with Controllability: Full controllability requires steering all states. Stabilizability only requires controlling the states that could cause instability.
• Engineering Relevance: For many practical systems (e.g., aircraft, power grids), achieving stability is the primary goal, making stabilizability a critical, often sufficient, property.

Reachability is closely related to controllability and concerns the set of all states that can be reached from a given initial state using admissible controls within a finite time. While controllability asks if any state can be reached from any other state, reachability analyzes the specific set of achievable states from a particular starting point.

• Reachable Set: The collection of all states reachable from x₀. For linear systems, the reachable set is a subspace spanned by the controllability matrix.
• Application: Central to verification and safety analysis, e.g., ensuring a robot's trajectory stays within a safe set of states.

The Linear Quadratic Regulator (LQR) is an optimal feedback control law for a linear system with quadratic cost functions. Its design and stability guarantees are fundamentally predicated on the system's controllability.

• Prerequisite: The system must be controllable (or at least stabilizable) for the standard infinite-horizon LQR solution to exist.
• Mechanism: LQR computes a gain matrix K that minimizes a cost function J = ∫ (xᵀQx + uᵀRu) dt, providing optimal state regulation.
• Result: The LQR controller guarantees closed-loop stability for controllable systems.

The Controllability Gramian is a matrix that quantifies the degree of controllability of a system. For a linear time-invariant system, it is defined as W_c = ∫₀^∞ e^(Aτ) B Bᵀ e^(Aᵀτ) dτ. It provides more nuanced information than the binary Kalman rank test.

• Interpretation: The Gramian's eigenvalues indicate how much control energy is required to move the system in the direction of the corresponding eigenvectors. Small eigenvalues mean a state direction is poorly controllable.
• Usage: Used in model order reduction to truncate weakly controllable (and observable) states, and in minimum-energy control calculations.

A nonholonomic constraint is a non-integrable kinematic restriction on a system's velocities. It is a primary reason why many robotic systems (like cars or differential-drive robots) are not controllable in the linear sense but may be nonlinearly controllable.

• Example: A car cannot move directly sideways. This constraint is expressed as ẏ cosθ - ẋ sinθ = 0.
• Effect on Controllability: While it restricts instantaneous velocity, a car can still reach any position/orientation (x, y, θ) through maneuvers like parallel parking. This is analyzed via Lie algebra rank condition for nonlinear systems.