Modeling & Control

State-space, transfer-function, impulse/step and frequency-response models are different views of dynamics. Each preserves particular questions and hides others; choosing among them requires stating linearity, operating point, initial-state, causality and validity assumptions.

Classical automatic-control theory is a disciplined way to turn a physical objective into a verified feedback loop. It connects modeling, transfer functions, poles and zeros, time and frequency response, stability tests, compensation, nonlinear limits, sampling, and implementation without confusing any one view for the whole system.

A control model begins with conserved quantities, constitutive laws, interfaces, and units. Linearization then produces a local perturbation model around an equilibrium or trajectory; its Jacobians explain the nearby dynamics but do not make the original nonlinear plant globally linear.

Standard dynamic elements are reusable causal patterns—gain, integrator, differentiator, lag, second-order mode, zero, and delay. Their poles and zeros explain decay, growth, oscillation, phase, inverse response, and high-frequency behavior, while physical realizability determines which algebraic forms can exist alone.

A block diagram is a causal contract: arrows carry named signals, blocks implement transformations, and summing signs define the loop. Reduction rules can derive transfer functions, but they preserve meaning only when pickoff points, disturbances, initial conditions, and internal stability remain explicit.

Time-domain performance describes how a closed loop moves, peaks, settles, tracks, rejects disturbances, and uses the actuator. Rise time, overshoot, settling time, and integral criteria are meaningful only with explicit inputs, thresholds, initial/final values, horizons, constraints, and measurement conventions.

Frequency response measures how a stable linear system transforms sinusoidal amplitude and phase. Bode plots assemble pole, zero, gain, delay, and crossover effects across decades, while minimum-phase structure determines how much phase can be inferred from magnitude and exposes limits hidden by identical gain curves.

Classical stability criteria answer the same pole-location question from coefficients, argument change, or loop frequency response. Routh–Hurwitz counts right-half-plane roots without solving them; Mikhailov tracks a characteristic polynomial; Nyquist relates loop encirclements of −1 to closed-loop unstable poles.

D-decomposition maps a chosen root-domain boundary into controller or plant parameter space. The resulting curves or surfaces partition parameters into connected regions with invariant root counts, revealing every stabilizing low-order design rather than one tuned point.

A root locus is the set of closed-loop pole locations generated as a real gain varies. Its angle and magnitude conditions expose how open-loop poles and zeros constrain motion, while compensator poles and zeros reshape that geometry to meet damping, speed, and robustness goals.

Steady-state error is determined by the low-frequency structure of a stable closed loop. System type counts uncancelled open-loop integrators and predicts which polynomial references have zero finite error, while disturbance location, nonunity sensing, saturation, leakage, and hidden cancellations set the real boundary.

Lead compensation adds phase near crossover to increase damping or bandwidth; lag compensation raises low-frequency loop gain relative to crossover to improve accuracy. Their pole-zero placement is a controlled redistribution of gain and phase, constrained by delay, nonminimum-phase dynamics, noise, uncertainty, and actuator effort.

System identification builds a dynamic model from deliberately informative input/output data. The real work is experiment design, structure choice, independent validation, residual diagnosis and a validity envelope—not obtaining one parameter vector with a good in-sample fit.

Identification data are created by an experiment, not merely collected. Excitation, operating range, sampling, sensor dynamics, time alignment, preprocessing, feedback, and validation splits determine which dynamics can be learned and which artifacts will be mistaken for the plant.

Nonparametric identification estimates an impulse response or frequency-response function without first compressing the dynamics into a small pole-zero model. It is the diagnostic layer between raw experiments and parametric fitting, exposing delay, resonances, usable bandwidth, uncertainty and model violations.

ARX, ARMAX, output-error, Box–Jenkins and innovation state-space models differ in how they constrain plant and disturbance dynamics. Prediction-error estimation fits those assumptions; order criteria rank candidates, while independent simulation, prediction and residual tests decide whether the resulting model is adequate for its use.

Monte Carlo replaces an expectation or integral with random draws and reports the sampling error of the resulting estimator. Importance sampling, control variates, stratification, Markov-chain Monte Carlo, and Gibbs updates change how evidence is obtained; diagnostics must distinguish target invariance from finite-run mixing and effective information.

Feedback measures consequences and changes the input to reduce error or reject disturbances. It can create robustness and regulation, but it also closes a dynamic loop in which delay, noise, saturation and wrong sign can amplify the very deviation it was meant to suppress.

The three-term control law and everything the textbook equation leaves out — derivative on the measurement instead of the error, a filtered derivative, integral windup and the anti-windup schemes that fix it, Ziegler-Nichols with its real coefficients and its real weaknesses, and how to pick a sample rate.

A digital controller observes and acts at discrete instants while the plant evolves continuously. Sampling, anti-alias filtering, conversion, computation, communication, output holds, quantization, jitter, missed deadlines, and state update order are therefore closed-loop dynamics—not implementation details added after design.

A discrete LTI model is asymptotically stable when every pole lies strictly inside the unit circle. Exact sampling maps continuous poles by z=e^{sT}; approximate integration, delayed implementation, controller conversion, finite precision, and sampling zeros can move the actual closed loop across that boundary even when the continuous design was stable.

Relay controllers choose among discrete actuator states. Hysteresis, deadband, dwell time and pulse timing turn an ideal threshold into a physically usable law by trading tracking precision against chatter, wear, minimum on-time and plant ripple.

Sliding-mode control switches structure so trajectories reach and remain near a designed manifold whose reduced dynamics have desired properties. Its robustness is specific to matched uncertainty and ideal switching; real actuators introduce chattering, delay and bandwidth limits that must be designed explicitly.

Robustly tracking a class of references or rejecting a class of persistent disturbances requires the feedback controller to contain their generating dynamics. Integral action is the simplest case: it is an internal model of constants, not a generic promise of zero error.

Robust control asks whether stability and performance survive a stated set of plant errors and disturbances. Sensitivity functions expose the trade-offs; loop shaping allocates gain by frequency; H∞ synthesis turns weighted worst-case amplification into an optimization problem.

A phase portrait shows the geometry of a two-state autonomous nonlinear system without requiring a closed-form time solution. Equilibria, nullclines, local linearization, invariant manifolds, basins, and periodic orbits explain behavior that a single transfer function cannot represent.

Lyapunov analysis proves stability without solving every trajectory by finding a scalar function that decreases along the dynamics. Its sign conditions distinguish stability, asymptotic convergence, exponential decay, and invariance; sublevel sets can certify local regions of attraction, but a candidate function is only as strong as the region where its derivative condition is proved.

Lur'e analysis separates a linear dynamic block from a memoryless nonlinearity. Sector and passivity criteria can certify stability for every nonlinearity in a class, while describing functions approximate a particular nonlinearity by its fundamental harmonic to predict candidate oscillations—not prove their existence or stability.

A sampled-data model connects a continuous plant to input and output sequences through an explicit sampler and hold. The pulse transfer function predicts sample-to-sample behavior for that interface, while intersample motion, fractional delay, aliasing, and multirate timing remain part of the hybrid system.

Digital controllers can be obtained by discretizing a continuous design, designing directly in the z-plane, solving a polynomial model-matching equation, or using discrete state space. Each route must satisfy causality, internal stability, sampling-zero, delay, robustness, and actuator constraints; deadbeat response is a fragile finite-sample ideal, not free speed.

A digital transfer function becomes control only through a particular state update, arithmetic format, execution order, saturation policy, and timing contract. Direct, cascade, parallel, and state-space realizations are algebraically equivalent at infinite precision but can differ sharply in coefficient sensitivity, internal range, quantization noise, overflow, and limit cycles.

Modern control theory treats internal state as the object that evolves, can be influenced, and must often be estimated. This roadmap connects realizations, state propagation, structural tests, feedback synthesis, integral augmentation, and observers into one design workflow.

One input/output behavior has infinitely many state-coordinate descriptions. Similarity transformations preserve behavior while changing matrix structure; canonical and modal forms expose algebraic features, but minimality, physical meaning, and numerical conditioning determine whether a realization is useful.

The state transition matrix propagates an initial condition through linear dynamics; convolution with the same propagator accumulates input and disturbance effects. This chapter derives both parts, relates modes to eigenstructure, and connects exact propagation to sampled-data models.

Controllability asks which state directions inputs can move; observability asks which directions output history can distinguish. Rank tests express the exact theory, while PBH tests, Gramians, staircase forms, stabilizability, detectability, and scaling make the conclusions useful in real designs.

Full-state feedback changes the state matrix from A to A-BK, allowing controllable modes to be assigned. Pole placement makes that algebra explicit, but pole choice must also respect actuator limits, uncertainty, noise, neglected dynamics, reference tracking, and numerical conditioning.

A state observer runs a model beside the plant and corrects it with output prediction error. Observability or detectability determines whether its error can converge; the separation principle then allows linear state-feedback and observer poles to be designed independently under precise model assumptions.

How to turn a physical control objective into a mathematically well-posed dynamic optimization problem: dynamics, admissible trajectories, endpoint conditions, Bolza cost, units, existence, and the differences among time, fuel, and quadratic criteria.

A derivation-first account of the first variation, Euler–Lagrange equation, natural and moving-endpoint boundary conditions, conserved quantities, constraints, and the second-order tests that distinguish an extremal from a minimum.

The maximum principle as a necessary condition for constrained dynamic optimization: Hamiltonian convention, adjoint dynamics, pointwise maximization, transversality, normal and abnormal extremals, switching functions, singular arcs, and the bridge to HJB.

Full phase-plane synthesis for bounded minimum-time control: the double integrator's one-switch proof, parabolic switching curve, exact feedback and travel time, plus harmonic-oscillator arcs, repeated switches, and tracking-coordinate translations.

Why bounded L1-optimal control develops bang-off-bang arcs: thresholded Hamiltonian maximization, scalar and vector cases, lower bounds, dead zones, singular thresholds, free-time nonattainment, and μ-optimal approximations.

How Bellman's principle of optimality turns a control problem into a value-function PDE, how feedback follows from its minimizer, when verification is sufficient, why switching surfaces make the value nonsmooth, and how HJB connects to PMP.

A complete continuous- and discrete-time LQR derivation: quadratic HJB ansatz, optimal feedback, differential and algebraic Riccati equations, stabilizing-solution conditions, value function, weight scaling, and actuator-limit boundaries.

How to optimize a quadratic closed-loop criterion when the controller architecture is prescribed: stability domain, Parseval and residue evaluation, Lyapunov cost equations, scalar derivation, parameter gradients, and why deleting LQR feedback paths is not optimal redesign.