1. System, State, Input, and Output
Control theory starts by separating what evolves internally from what can be manipulated and what can be measured. The state x is the smallest set of variables that, together with future inputs, determines future system evolution. The input u is what the controller can change. The output y is what sensors or the task expose. This distinction is fundamental because the controller may need to regulate states that are not directly measured.
Key points
- State is not the same as sensor readings; unmeasured state can still be essential to predict dynamics.
- A model can be continuous-time or discrete-time, linear or nonlinear, deterministic or stochastic.
- Feedback means the control action depends on measured or estimated current state, so disturbances can be corrected online.
2. State-Space Representation in Continuous and Discrete Time
A linear time-invariant (LTI) continuous-time model is ẋ=Ax+Bu, y=Cx+Du. Matrix A describes internal dynamics, B how inputs affect states, C which state combinations are measured, and D any direct input-to-output path. The discrete counterpart is x_{k+1}=Ax_k+Bu_k, y_k=Cx_k+Du_k. Most digital controllers ultimately operate on a discrete model.
Key points
- The eigenvalues of A determine natural modes when u=0.
- Different state-space realizations can represent the same input-output transfer function.
- State coordinates may be changed with an invertible transformation without changing physical input-output behavior.
3. Equilibrium Points and Linearization
Real systems are often nonlinear: ẋ=f(x,u). An equilibrium (x*,u*) satisfies f(x*,u*)=0, meaning the state remains constant under that input. Around an equilibrium, small perturbations δx=x−x* and δu=u−u* can often be approximated by a linear model whose A and B are Jacobians. Linear control theory then describes local behavior near the operating point.
Key points
- Linearization is local; large excursions may invalidate the approximation.
- For trajectory tracking, time-varying linearization along a nominal trajectory leads to linear time-varying models.
- Always define the operating point before discussing local stability of a nonlinear system.
4. Natural Response, Eigenvalues, and Modes
For the autonomous linear system ẋ=Ax, the solution is x(t)=e^{At}x(0). Eigenvalues of A determine whether modes grow, decay, or oscillate. In continuous time, a mode e^{λt} decays when Re(λ)<0. In discrete time, λ^k decays when |λ|<1. Complex-conjugate eigenvalues create oscillatory modes, while their real part or magnitude determines damping.
Key points
- Eigenvectors describe modal directions; eigenvalues describe modal time behavior.
- Fast negative poles decay quickly; poles near the imaginary axis decay slowly.
- A stable eigenvalue set does not by itself quantify transient amplification in highly non-normal systems.
5. Stability and Lyapunov Analysis
Stability asks what happens to trajectories that start near an equilibrium. Lyapunov stability means they remain near; asymptotic stability additionally requires convergence to the equilibrium. A Lyapunov function V(x) acts like a generalized energy: if V is positive away from equilibrium and decreases along trajectories, the equilibrium is stable. For LTI systems, a quadratic Lyapunov function V=x^TPx leads to a matrix inequality/equation.
Key points
- For continuous-time LTI systems, A is Hurwitz iff for every Q≻0 there exists P≻0 solving A^TP+PA=−Q.
- Lyapunov methods generalize beyond linear systems and do not require an explicit trajectory solution.
- Stability is distinct from performance: a stable controller can still be too slow, oscillatory, or energy-intensive.
6. Controllability
Controllability asks whether the input has enough authority to move the state through all independent state directions. For an n-dimensional LTI system, construct the controllability matrix 𝒞=[B,AB,…,A^{n−1}B]. Full rank n means every state can be reached from any initial state in finite time in the ideal linear model. If a mode is uncontrollable, no state-feedback gain can arbitrarily move that mode's eigenvalue.
Key points
- Controllability is a structural property of (A,B), independent of a particular feedback gain.
- Poor controllability can be numerical as well as binary: a theoretically controllable mode may require enormous input energy.
- The controllability Gramian quantifies how difficult different state directions are to reach.
7. Observability
Observability is the dual question: can the internal state be reconstructed from measured outputs over time? For an n-dimensional LTI system, the observability matrix 𝒪=[C;CA;…;CA^{n−1}] has rank n exactly when all state directions can be inferred. An unobservable unstable mode is especially problematic because it can grow without being detectable from the measurements used by the controller.
Key points
- Observability depends on both the dynamics A and sensor placement C.
- Detectability is weaker than observability: unobservable modes are allowed as long as they are already stable.
- State estimation quality depends on noise and conditioning, not only on binary rank tests.
8. State Feedback and Pole Placement
With state feedback u=−Kx, the closed-loop dynamics become ẋ=(A−BK)x. If (A,B) is controllable, K can be chosen to place the closed-loop poles at desired locations. Pole placement directly shapes decay rate and oscillation, but it does not explicitly optimize control effort or robustness; very aggressive pole locations may demand unrealistic inputs.
Key points
- Pole placement specifies dynamics first and solves for K; LQR specifies a cost and lets the optimal poles emerge.
- For multi-input systems there can be many gains realizing the same poles.
- Actuator saturation breaks the linear closed-loop model and can cause windup or performance degradation.
9. LQR: Optimal State Feedback
The linear quadratic regulator (LQR) chooses state feedback by minimizing a quadratic infinite-horizon cost. Q penalizes undesirable state deviation and R penalizes control effort. Solving the algebraic Riccati equation gives P, then K=R^{-1}B^TP for continuous time. Under stabilizability/detectability conditions, the resulting feedback stabilizes the system and provides a principled trade-off instead of manually placing every pole.
Key points
- Increasing a diagonal Q weight generally makes the corresponding state more expensive, but cross-coupling means the effect on poles is not one-to-one.
- Increasing R discourages control effort and usually slows the response.
- LQR assumes the full state is available; with estimated state it becomes LQG when combined with a Kalman filter under standard assumptions.
10. Observers and Kalman Filtering
When x is not measured directly, a state observer runs a copy of the model and corrects it using the output residual y−C x̂. A Luenberger observer uses a fixed gain L. The Kalman filter derives a time-varying or steady-state gain from process-noise covariance Q_w and measurement-noise covariance R_v, producing the minimum-variance linear estimate under Gaussian linear assumptions.
Key points
- Observer pole placement is the dual of controller pole placement; observability enables arbitrary observer pole assignment.
- A larger assumed process noise makes the Kalman filter trust measurements more; larger measurement noise makes it trust the model more.
- For nonlinear systems, EKF linearizes the model and UKF propagates sigma points; particle filters handle more general distributions at higher cost.
11. Separation Principle, Reference Tracking, and Integral Action
For suitable linear systems, controller and observer can be designed separately: if A−BK and A−LC are both stable, the observer-based feedback is stable. This is the separation principle. Regulation to zero is not the same as tracking a nonzero reference. Reference feedforward, state augmentation, and integral action are common ways to remove steady-state tracking error, especially under constant disturbances or model mismatch.
Key points
- Integral action adds accumulated tracking error as a state and forces constant steady-state error toward zero when the augmented system is controllable.
- Feedforward uses a model to generate the nominal input required by a reference; feedback corrects deviations.
- This feedforward + feedback structure also appears inside tracking MPC.
12. A Practical Control-Design Workflow
A reliable workflow is: define states/inputs/outputs and operating point; derive or identify the model; discretize if needed; verify stability, controllability, and observability; choose feedback design such as pole placement, LQR, or MPC; design an observer if states are missing; simulate with constraints/noise/model mismatch; only then deploy and retune. Each theoretical test answers a different question and should not be skipped simply because a numerical controller appears to work in one simulation.
Key points
- Ask 'is it stabilizable?' before asking 'which K is best?'.
- Ask 'is the state observable/detectable?' before assuming a state-feedback law can be implemented.
- Use LQR when quadratic performance and unconstrained linear feedback are adequate; use MPC when explicit constraints or previewed references are central.
What you should remember
After this note, you should be able to look at (A,B,C), explain stability/controllability/observability, design basic state feedback and an observer, and understand how LQR, Kalman filtering, and MPC fit into one control-theory picture.