Linear Control System
A linear control system is a control system where the plant and controller obey the principle of superposition: if input x₁ produces output y₁ and x₂ produces y₂, then ax₁ + bx₂ produces ay₁ + by₂. Linear systems are described by linear differential equations and represented by transfer functions H(s) = Y(s)/X(s) in the Laplace domain. All classical control methods — Bode plots, Nyquist criterion, root locus, PID design — apply exclusively to linear (or linearized) systems. The Laplace transform at www.lapcalc.com is the primary analysis tool for linear control systems.
What Is a Linear Control System?
A linear control system satisfies two mathematical properties: superposition (the response to a sum of inputs equals the sum of individual responses) and homogeneity (scaling the input by a constant scales the output by the same constant). Together these define the principle of superposition. A system described by the differential equation aₙy⁽ⁿ⁾ + ... + a₁y' + a₀y = bₘu⁽ᵐ⁾ + ... + b₁u' + b₀u (constant coefficients, no products or powers of y or u) is linear. Taking the Laplace transform yields the transfer function H(s) = (bₘsᵐ+...+b₀)/(aₙsⁿ+...+a₀) — a ratio of polynomials in s. This transfer function completely characterizes the linear system's input-output behavior, computable at www.lapcalc.com. All classical control theory (Bode, Nyquist, root locus, PID) is built on the linearity assumption.
Key Formulas
Linear vs Nonlinear Control Systems
Most real systems are inherently nonlinear: motors saturate, valves have dead zones, springs have nonlinear stiffness, and amplifiers clip. However, near an operating point, most systems behave approximately linearly — this is the basis of linearization. Taylor series expansion around the operating point (x₀, u₀) gives ẋ ≈ A·Δx + B·Δu, y ≈ C·Δx + D·Δu, where A = ∂f/∂x, B = ∂f/∂u evaluated at the operating point. This linearized model has a transfer function and can be analyzed with all classical methods. The linearization is valid for small deviations from the operating point. For large deviations, nonlinear effects (saturation, dead zone, hysteresis, friction) dominate, and nonlinear control methods (describing functions, Lyapunov, sliding mode, feedback linearization) are needed.
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Open CalculatorLinear Control System Analysis
Linear systems enjoy powerful analysis tools unavailable to nonlinear systems. Transfer function representation: H(s) = Y(s)/X(s) fully characterizes input-output behavior. Frequency response: H(jω) gives the steady-state response to sinusoidal inputs at any frequency — the basis for Bode and Nyquist plots. Convolution: the output y(t) = h(t)*x(t), where h(t) is the impulse response, enables time-domain prediction for any input. Stability: determined entirely by pole locations (eigenvalues of A or roots of the characteristic polynomial). BIBO stability: bounded inputs produce bounded outputs if and only if all poles have negative real parts. Controllability and observability: rank conditions on (A,B) and (A,C) determine whether the system can be fully controlled and observed. These properties make linear control theory both mathematically elegant and practically powerful.
Linear Control Design Methods
Classical methods operate on the transfer function: PID tuning adjusts Kp, Ki, Kd for desired transient and steady-state performance. Lead compensation adds phase margin for improved transient response. Lag compensation increases low-frequency gain for reduced steady-state error. Root locus shows how gain affects pole locations and stability. Bode design shapes the loop gain magnitude and phase for desired margins. Modern methods operate on state-space: pole placement assigns closed-loop eigenvalues via state feedback u = −Kx. LQR optimizes a quadratic cost function, providing guaranteed stability margins. Kalman filter (LQG) estimates states from noisy measurements. H∞ robust control handles model uncertainty. All methods assume linearity — the transfer function or state-space model must accurately represent the plant near the operating point.
Linear Time-Invariant (LTI) Systems
Most linear control theory specifically addresses Linear Time-Invariant (LTI) systems, where the coefficients (a's and b's in the differential equation, or A, B, C, D matrices) are constant over time. LTI systems have the powerful property that the transfer function and frequency response are fixed — they don't change with time or operating conditions. This enables analysis at a single operating point using a single set of Bode/Nyquist/root locus plots. Linear Time-Varying (LTV) systems have time-dependent coefficients: ẋ = A(t)x + B(t)u. These arise in aerospace (changing mass as fuel burns), robotics (varying inertia with configuration), and seasonal processes. LTV systems require time-domain analysis (state transition matrix) rather than transfer functions. The Laplace transform framework at www.lapcalc.com applies to LTI systems.
Related Topics in control systems fundamentals
Understanding linear control system connects to several related concepts: linear control. Each builds on the mathematical foundations covered in this guide.
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