Electronics Guide

A Series RLC Circuit

A voltage source u drives a resistance R, an inductance L, and a capacitance C in series. Take the inductor current i and the capacitor voltage v_C as states, so that x = [i; v_C]. Kirchhoff's voltage law and the capacitor's current law give

L di/dt = u − Ri − v_C, and C dv_C/dt = i

Dividing by L and by C gives

A = [−R/L, −1/L; 1/C, 0], B = [1/L; 0]

As an output, the capacitor voltage uses the row [0, 1] with D = 0, and the inductor voltage, u − Ri − v_C, uses [−R, −1] with D = 1, because a step in the source voltage appears across the inductor at once. With R = 10 Ω, L = 10 mH, and C = 100 µF, A = [−1000, −100; 10000, 0] and B = [100; 0] in SI units. Transient Response of RC, RL, and RLC Circuits derives this circuit's behavior from its second-order differential equation instead.

A DC Motor

A permanent-magnet DC motor turning an inertia combines mechanical and electrical states. Take the shaft angle θ, the speed ω, and the armature current i as states, and the applied voltage v and a load torque T_L as inputs. Newton's law for the rotor and Kirchhoff's law for the armature give

dθ/dt = ω, J dω/dt = K_ti − bω − T_L, and L_a di/dt = v − R_ai − K_eω

where J is the inertia, b the viscous friction coefficient, K_t the torque constant, K_e the back-EMF constant (numerically equal to K_t in SI units), and R_a and L_a the armature resistance and inductance. With x = [θ; ω; i] and u = [v; T_L],

A = [0, 1, 0; 0, −b/J, K_t/J; 0, −K_e/L_a, −R_a/L_a], B = [0, 0; 0, −1/J; 1/L_a, 0]

The load torque enters through the second column of B. The examples below use the motor from the speed-loop example in Time-Domain Response of Control Systems: K_t = K_e = 0.05 N·m/A, R_a = 1 Ω, negligible friction, and J = 1.25 × 10⁻⁴ kg·m², which give a speed gain of 20 (rad/s)/V and a 50 ms mechanical time constant, plus an assumed armature inductance of 1 mH. Then

A = [0, 1, 0; 0, 0, 400; 0, −50, −1000], B = [0, 0; 0, −8000; 1000, 0]

Modes and Eigenvectors

If A has n linearly independent eigenvectors v_i, with eigenvalues λ_i, collect them as the columns of a matrix V. Then A = VΛV⁻¹, where Λ is the diagonal matrix of eigenvalues, and

e^At = Ve^ΛtV⁻¹

where e^Λt is diagonal with entries e^λ_it. The free response is therefore a sum of modes, each keeping the shape of its eigenvector while it grows or decays as e^λ_it. Complex eigenvalues of a real matrix come in conjugate pairs, which combine into decaying or growing sinusoids, and a repeated eigenvalue without a full set of eigenvectors adds terms such as te^λt.

A Closed Form for Two States

A real 2 × 2 matrix with complex eigenvalues σ ± jω_d has a compact exponential. The Cayley–Hamilton theorem, which says that every square matrix satisfies its own characteristic equation, gives (A − σI)² = −ω_d²I, and the series collapses to

e^At = e^σt[cos(ω_dt) I + (sin(ω_dt)/ω_d)(A − σI)]

For the RLC circuit, σ = −500 s⁻¹ and ω_d = 866 rad/s, so A − σI = [−500, −100; 10000, 500]. With the capacitor charged to 10 V, no current, and a short circuit in place of the source, x(0) = [0; 10] and u = 0. Then, with t in seconds,

i(t) = −1.155e^−500t sin(866t) A

v_C(t) = 10e^−500t[cos(866t) + 0.577 sin(866t)] V

After 1 ms the current is −0.534 A, flowing against its reference direction as the capacitor discharges, and the capacitor holds 6.60 V; after 2 ms it holds 1.51 V.

For larger systems, summing the series can lose accuracy badly to cancellation, as Cleve Moler and Charles Van Loan explain in "Nineteen Dubious Ways to Compute the Exponential of a Matrix" (SIAM Review, 1978, revisited in 2003). MATLAB's expm and SciPy's scipy.linalg.expm use scaling and squaring with Padé approximants instead.

Internal Stability and BIBO Stability

A transfer function is bounded-input, bounded-output (BIBO) stable when all its poles lie in the open left half-plane. Because a cancellation can remove an eigenvalue from G(s), BIBO stability does not guarantee that the internal states stay bounded. Consider

A = [0, 1; 2, −1], B = [0; 1], C = [−1, 1], D = 0

The eigenvalues of A are +1 and −2, yet G(s) = (s − 1)/[(s − 1)(s + 2)] = 1/(s + 2). The unstable mode's eigenvector is [1; 1], which C maps to zero, so from x(0) = [0.001; 0.001] the output stays at zero while both states grow as e^t, reaching 22 after 10 s. In hardware such a state grows until an amplifier saturates or a component fails, and only the eigenvalues of A reveal it.

Lyapunov's Method

For the RLC circuit with its source shorted, the stored energy E = Li²/2 + Cv_C²/2 changes at the rate dE/dt = −Ri², so it never increases, and because the current cannot stay at zero unless the capacitor voltage is also zero, the energy drains away completely. Aleksandr Lyapunov generalized this argument in his 1892 doctoral thesis on the stability of motion. A symmetric matrix P is positive definite when x^TPx > 0 for every nonzero x, and positive semidefinite when x^TPx ≥ 0. A is Hurwitz if and only if, for any symmetric positive definite Q, the Lyapunov equation A^TP + PA = −Q has a symmetric positive definite solution P, and x^TPx is then an energy function that decreases along every trajectory except the equilibrium.

Controllability

A system is controllable if some input can drive the state from any initial value to any target value in finite time. A linear time-invariant system is controllable if and only if its controllability matrix,

M_c = [B, AB, A²B, …, Aⁿ⁻¹B]

has rank n. Its blocks sit side by side, so M_c has n rows and nm columns, and for a single input the test reduces to a nonzero determinant. Higher powers of A add nothing, because the Cayley–Hamilton theorem expresses Aⁿ through lower powers.

An equivalent test, named for Vasile Popov, Vitold Belevitch, and Malo Hautus, examines one eigenvalue at a time: the system is controllable if and only if [λI − A, B] has rank n for every eigenvalue λ of A. An eigenvalue that fails belongs to an uncontrollable mode, which no input can influence and no state feedback can move. A system whose uncontrollable modes are all stable is called stabilizable.

Two identical RC low-pass sections driven from one source are uncontrollable. With the capacitor voltages as states and a common time constant τ, A = [−1/τ, 0; 0, −1/τ] and B = [1/τ; 1/τ], so M_c = [1/τ, −1/τ²; 1/τ, −1/τ²], of rank 1. The difference v₁ − v₂ obeys d(v₁ − v₂)/dt = −(v₁ − v₂)/τ whatever the input, so no input can open or close the gap, but the gap decays: the pair is uncontrollable yet stabilizable. With time constants of 10 ms and 11 ms the pair becomes controllable, but barely: setting the capacitors to +0.5 V and −0.5 V within 50 ms takes 220 times the input energy that charges both to 1 V. The controllability Gramian, which sets that minimum energy, measures what the yes-or-no rank test cannot.

Observability

A system is observable if the output over a finite interval, together with the known input, determines the initial state and hence every later state. It is observable if and only if the observability matrix, which stacks C, CA, and so on as rows,

M_o = [C; CA; CA²; …; CAⁿ⁻¹]

has rank n. The eigenvalue form of the test requires [λI − A; C] to have rank n for every eigenvalue λ, and a system whose unobservable modes are all stable is called detectable. By duality, (A, C) is observable exactly when (A^T, C^T) is controllable, so every controllability result has an observability twin.

The choice of sensor matters. A tachometer on the motor measures speed, so C = [0, 1, 0] and

M_o = [0, 1, 0; 0, 0, 400; 0, −20000, −400000]

Its first column is zero and its rank is 2, so the angle is unobservable. No state equation has the angle on its right side, so speed and current evolve the same way wherever the shaft points. An encoder, with C = [1, 0, 0], gives M_o = [1, 0, 0; 0, 1, 0; 0, 0, 400], of rank 3: the angle's rate of change gives the speed, and the speed's rate of change reveals the current.

Controllable Canonical Form

For the strictly proper third-order transfer function

G(s) = (b₂s² + b₁s + b₀)/(s³ + a₂s² + a₁s + a₀)

the controllable canonical form, also called the phase-variable form, chains integrators and feeds the denominator coefficients back:

A = [0, 1, 0; 0, 0, 1; −a₀, −a₁, −a₂], B = [0; 0; 1], C = [b₀, b₁, b₂], D = 0

The pattern extends to any order, with ones above the diagonal, the negated denominator coefficients in the last row, and the numerator coefficients in C. The realization is always controllable, and it is observable unless the numerator and denominator share a factor. If they have equal degree, polynomial division first splits off a constant, which becomes D. Texts and tools order the states differently; SciPy's tf2ss, for example, places the negated coefficients in the first row.

Observable Canonical Form

Transposing the controllable form, with A_o = A^T, B_o = C^T, and C_o = B^T, gives the observable canonical form:

A_o = [0, 0, −a₀; 1, 0, −a₁; 0, 1, −a₂], B_o = [b₀; b₁; b₂], C_o = [0, 0, 1]

A scalar transfer function equals its own transpose, so both forms realize the same G(s). The observable form is always observable, and it is controllable unless a common factor exists. By duality, state-feedback calculations are simplest in controllable form and observer calculations in observable form.

Modal Form and Numerical Conditioning

When A has distinct eigenvalues, choosing T = V, the eigenvector matrix, makes T⁻¹AT diagonal, so each state is one mode. A zero row in T⁻¹B then marks an uncontrollable mode, and a zero column in CT an unobservable one. Companion forms, by contrast, are numerically fragile, because small errors in the coefficients of a high-order polynomial can move its roots a long way, so numerical work favors modal or balanced realizations. IIR Filter Structures meets the same problem when quantized coefficients perturb the poles of high-order direct-form filters.

Ackermann's Formula

In 1972 Jürgen Ackermann published a formula that performs the canonical-form calculation without changing coordinates:

K = [0, 0, …, 0, 1] M_c⁻¹ α_c(A)

where α_c(A) = Aⁿ + α_n−1Aⁿ⁻¹ + … + α₀I is the desired polynomial evaluated with the matrix A in place of s. The system must have a single input, so that M_c is square, and must be controllable, so that M_c is invertible. The formula follows from the Cayley–Hamilton theorem and serves discrete-time systems unchanged, but it is numerically fragile, because inverting M_c loses accuracy as the order rises or controllability weakens.

With several inputs, many gain matrices place the same poles, and the spare freedom can buy robustness. MATLAB's place uses the 1985 algorithm of Kautsky, Nichols, and Van Dooren to reduce the sensitivity of the closed-loop poles to perturbations, and SciPy's scipy.signal.place_poles offers that method alongside its default Tits–Yang algorithm.

The Reference Gain

State feedback alone drives x to zero. For a single output y = Cx to settle at a constant reference r, the steady state must satisfy 0 = (A − BK)x_ss + Bk_rr and Cx_ss = r, which with D = 0 gives

k_r = −1/[C(A − BK)⁻¹B]

This gain exists when the plant has no zero at s = 0. It depends on an exact model, and a constant disturbance still leaves an error, which integral action removes.

An Observer for the Servo

Suppose the servo has only its encoder, so y = θ. Observer poles with ζ = 0.7 and ω_n = 200 rad/s, four times faster than the controller, give α_o(s) = s² + 280s + 40,000. Here M_o = [1, 0; 0, 1], so L is the last column of α_o(A) = [40000, 260; 0, 34800]: L = [260; 34800], in s⁻¹ and s⁻². The eigenvalues of A − LC are −140 ± j142.8, and an error in the estimated speed decays to 2 percent of its initial value in about 30 ms.

The Separation Principle

Now feed back the estimate instead of the state: u = −Kx̂ + k_rr. In the coordinates [x; e], the closed loop is

d[x; e]/dt = [A − BK, BK; 0, A − LC][x; e] + [Bk_r; 0]r

The matrix is block triangular, so its eigenvalues are those of A − BK together with those of A − LC. Controller and observer can be designed separately, and joining them moves neither set of poles: for the servo, the combined model's eigenvalues are −35 ± j35.7 and −140 ± j142.8. Because the reference enters the plant and the observer identically, it cannot excite the estimation error, so with a correct initial estimate the step response matches the full-state design.

The principle rests on assumptions that practice can break:

• A linear plant: Saturation, dead zones, and friction couple the two designs.
• An accurate model: If the observer's A, B, or C differs from the plant's, the error equation picks up terms in x and u, and the eigenvalue sets no longer separate.
• The same input: When the amplifier clips, the observer needs the clipped value, not the commanded one.
• Nominal stability only: Separation guarantees pole locations, not robustness. John Doyle's 1978 paper "Guaranteed Margins for LQG Regulators" answered its implied question in a three-word abstract: "There are none."

A common rule of thumb places observer poles two to six times faster than the controller poles: a faster observer needs a larger L and passes more sensor noise into the estimate, and a slower one lets estimation transients show in the response. For a linear system with Gaussian noise of known statistics, the Kalman filter, published in 1960, chooses the gain that minimizes the mean-square estimation error, and pairing it with the linear quadratic regulator gives linear quadratic Gaussian (LQG) control. The State Observers section of Control Algorithms describes the Luenberger, extended Kalman filter, sliding-mode, and disturbance observers that motor drives use to estimate speed, position, and load torque.

Integral Action for the Servo

For the servo, A_a = [0, 1, 0; 0, −20, 0; 1, 0, 0] and B_a = [0; 400; 0]. Keeping the dominant pair at −35 ± j35.7 and adding a pole at −50 s⁻¹ gives the target (s² + 70s + 2,500)(s + 50) = s³ + 120s² + 6,000s + 125,000. Matching the closed-loop polynomial s³ + (20 + 400k₂)s² + 400k₁s + 400k_I gives k₁ = 15 V/rad, k₂ = 0.25 V/(rad/s), and k_I = 312.5 V/(rad·s). With the reference entering only through the integrator, a 1 rad step overshoots by 1.5 percent, settles within 2 percent in 96 ms, and needs at most 2.1 V. The same load torque now deflects the shaft by at most 0.054 rad, 40 ms after it arrives, and the error then returns to zero.

In practice the integrator winds up while the actuator saturates, so controllers clamp x_I or stop integrating while the drive is limited.

LQR for the Servo

Penalize only the angle of the two-state servo, with an acceptable error of 1 rad, and limit the drive to the amplifier's 24 V: Q = [1, 0; 0, 0] rad⁻² and R = 1/576 V⁻². The Riccati solution gives K = [24, 0.3] and closed-loop poles at −70 ± j68.6, a damping ratio of 0.71. With k_r = 24, a 1 rad step overshoots by 4.0 percent and settles within 2 percent in 61 ms, and the drive starts at exactly the 24 V that the weights encoded. For this plant k₁ = √(Q₁₁/R), the largest allowed voltage divided by the largest allowed error.

Loosening the acceptable error to 10 rad gives k₁ = 2.4 V/rad, a natural frequency of 31 rad/s, and a damping ratio of 0.78. Tightening it to 0.1 rad gives 240 V/rad, 310 rad/s, and 0.71, but with the armature inductance restored, that fast design's damping ratio falls to 0.57.

Guaranteed Margins

LQR gains come with guaranteed margins when every state is measured and fed back, R is diagonal if there are several inputs, and the loop is broken at the plant input. The loop then tolerates any increase in gain, any reduction by less than a factor of 2 (6 dB), and phase shifts of up to 60 degrees in each input channel. For a single input the guarantee follows from the return-difference inequality |1 + L(jω)| ≥ 1, which keeps the Nyquist plot of the loop gain outside the unit circle centered on −1; the three servo designs have 66 to 75 degrees of phase margin. The guarantee does not survive an observer, as Doyle's paper showed, and discrete-time LQR carries weaker guarantees.

Design in Discrete Time

The controllability and observability tests, Ackermann's formula, and observer design carry over with A_d and B_d in place of A and B and with target poles in the z-plane. Mapping the servo's target poles through z = e^sT gives 0.96499 ± j0.03447, and Ackermann's formula gives K_d = [6.096, 0.1237], close to the continuous gains because 1 kHz sampling is far faster than the closed-loop dynamics.

Sampling can destroy controllability or observability, but only when two eigenvalues of A share a real part and their imaginary parts differ by a whole multiple of 2π/T. An undamped 50 Hz oscillator sampled exactly twice per cycle, at T = 10 ms, has A_d = −I, so its controllability matrix [B_d, A_dB_d] = [B_d, −B_d] has rank 1; at T = 9 ms the rank returns to 2.

State-Space Averaging of Switching Converters

A switching converter is not time-invariant, because its circuit changes each time a switch turns on or off. R. D. Middlebrook and Slobodan Ćuk showed how to average it into a tractable model at the 1976 IEEE Power Electronics Specialists Conference. In continuous conduction the converter alternates between two linear circuits, with matrices A₁ and B₁ for the fraction d of each period and A₂ and B₂ for the rest, and weighting each by the time it applies gives

dx/dt = [d × A₁ + (1 − d) × A₂]x + [d × B₁ + (1 − d) × B₂]v_in

which holds when the switching frequency lies well above the converter's natural frequencies, so that the ripple is small. Because d multiplies the state, the model is nonlinear. At a steady duty cycle D (not the feedthrough matrix), with A and B the averaged matrices, the operating point is X = −A⁻¹BV_in, and linearizing about it gives a duty-cycle input matrix of (A₁ − A₂)X + (B₁ − B₂)V_in.

For an ideal boost converter with inductor current and capacitor voltage as states, load resistance R, and output capacitance C, A₁ = [0, 0; 0, −1/(RC)] with the switch on, A₂ = [0, −1/L; 1/C, −1/(RC)] with it off, and B₁ = B₂ = [1/L; 0]. With D′ = 1 − D, the output is V = V_in/D′, the inductor current is I_L = V/(D′R), and the duty-cycle-to-output transfer function is

G_vd(s) = [V/(D′RC)](D′²R/L − s) / [s² + s/(RC) + D′²/(LC)]

The numerator vanishes at s = +D′²R/L, a right-half-plane zero, which is why a boost converter's output first dips when the duty cycle rises. With V_in = 12 V, D = 0.5, R = 10 Ω, L = 100 µH, and C = 220 µF, the output is 24 V, the resonant poles lie near 540 Hz, and the zero sits at 25,000 rad/s, about 3.98 kHz, so the voltage loop must cross over well below it. Switching Converter Control describes the loops built on such models. Averaged models miss effects near the switching frequency, such as the subharmonic oscillation of peak current-mode control.

Motor Drives

Motor drives use state-space models at several levels. In the rotor's d–q reference frame, a permanent-magnet synchronous motor's stator currents form a linear parameter-varying model with cross-coupling terms proportional to the electrical speed, which field-oriented control cancels by decoupling feedforward. Speed and position loops can use state feedback with integral action, as in the servo example, and a load-torque observer adds the load as a constant state: augmented this way, the servo model passes the observability test with the encoder as its only sensor. Sensorless drives estimate position and speed from measured currents and voltages, and model predictive control optimizes a cost over a short horizon at each sample using a discrete-time state model. Control Algorithms covers both.