Control Systems: A Complete Revision Guide

Part I — Foundations: Modelling and Transfer Functions

What Is a Control System?

A control system is an arrangement of physical components connected to command, direct or regulate another system so as to achieve a desired objective. The controller computes a corrective action from the difference between what is wanted and what is measured.

Open-Loop versus Closed-Loop Systems

An open-loop system applies a control action without measuring the result; a closed-loop (feedback) system senses the output and compares it with the reference, correcting itself continuously.

Further Classification of Control Systems

Effects of Feedback and Sensitivity

The sensitivity of the closed-loop transfer function \(T\) with respect to a parameter \(\alpha\) measures the fractional change in \(T\) for a fractional change in \(\alpha\):

For \(T=\dfrac{G}{1+GH}\) this gives the two standard results

Disturbance Rejection and Noise Suppression

Writing the loop gain as \(L=G_cG_pH\), the closed-loop transfer functions to the various inputs are

• Reference tracking: \(\dfrac{Y}{R}=\dfrac{G_cG_p}{1+L}\;(=T)\).
• Disturbance: \(\dfrac{Y}{D}=\dfrac{G_p}{1+L}\).
• Sensor noise: \(\dfrac{Y}{N}=\dfrac{-L}{1+L}=-T\).
• Error: \(\dfrac{E}{R}=\dfrac{1}{1+L}\;(=S)\).

Control Components

Armature-Controlled DC Servomotor

The governing electrical and mechanical equations are

Neglecting the armature inductance \((L_a\approx0)\) gives the familiar speed-type transfer function

with motor gain \(K_m=\dfrac{K_t}{R_aB+K_tK_b}\) and mechanical time constant \(\tau_m=\dfrac{R_aJ}{R_aB+K_tK_b}\). A field-controlled DC motor has an extra field pole, \(\dfrac{\theta(s)}{E_f(s)}=\dfrac{K_m}{s(1+s\tau_f)(1+s\tau_m)}\), and is slower.

Gear Trains and Reflected Inertia

For an input gear with \(N_1\) teeth driving an output gear with \(N_2\) teeth,

with power conserved, \(\omega_1T_1=\omega_2T_2\). Inertia and damping on the output shaft, referred to the input shaft, scale by the square of the gear ratio:

Gears match a high-speed, low-torque motor to a low-speed, high-torque load and reduce reflected inertia. The ratio giving maximum load acceleration is \(n_{\mathrm{opt}}=\sqrt{J_L/J_m}\).

Mathematical Modelling

Translational Mechanical Systems

Applying Newton's second law to a mass–spring–damper gives

Rotational, Electrical Elements and Analogies

For a series RLC circuit, \(L\ddot q+R\dot q+\dfrac{q}{C}=e(t)\), so \(\dfrac{I(s)}{E(s)}=\dfrac{sC}{s^2LC+sRC+1}\). The dual force–current analogy maps \(f\leftrightarrow i\), \(M\leftrightarrow C\), \(B\leftrightarrow1/R\), \(K\leftrightarrow1/L\).

Laplace Transform Essentials

The Laplace transform converts differential equations into algebraic equations:

Key properties used constantly in control work include linearity, the differentiation rules \(\mathcal{L}\{\dot f\}=sF(s)-f(0)\) and \(\mathcal{L}\{\ddot f\}=s^2F(s)-sf(0)-\dot f(0)\), integration \(\mathcal{L}\{\int_0^t f\}=F(s)/s\), the time-shift \(\mathcal{L}\{f(t-a)u(t-a)\}=e^{-as}F(s)\), the frequency-shift \(\mathcal{L}\{e^{-at}f(t)\}=F(s+a)\), and convolution \(\mathcal{L}\{f*g\}=F(s)G(s)\).

Transfer Functions

The transfer function of an LTI system, with all initial conditions zero, is

A transfer function applies to LTI systems only, is independent of the input, and its poles dictate stability and dominant dynamics while its zeros shape the transient response.

Block Diagram Algebra

Signal Flow Graphs and Mason's Gain Formula

In a signal flow graph each node is a variable and each branch is a directed edge with a gain. A forward path runs from input to output without repeating a node; a loop is a closed path repeating no node; non-touching loops share no node.

Worked Example

The forward paths are \(P_1=G_1G_2G_3\) and \(P_2=G_1G_4\). The loops are \(L_1=-G_2H_1\), \(L_2=-G_1G_2G_3H_2\) and \(L_3=-G_1G_4H_2\); none are non-touching, so \(\Delta=1-(L_1+L_2+L_3)\) and \(\Delta_1=\Delta_2=1\). Hence

Part II — Time-Domain Analysis

Standard Test Signals

First-Order Systems

The standard first-order transfer function is \(G(s)=\dfrac{Y(s)}{R(s)}=\dfrac{K}{1+s\tau}\), where \(\tau\) is the time constant. Its unit-step response (for \(K=1\)) is

The response reaches \(63.2\%\) at \(t=\tau\), \(86.5\%\) at \(2\tau\), \(95.0\%\) at \(3\tau\), \(98.2\%\) at \(4\tau\) and \(99.3\%\) at \(5\tau\). The settling time is \(t_s\approx4\tau\) (2%) and the rise time \(t_r\approx2.2\tau\) (10–90%). The single pole sits at \(s=-1/\tau\) (real and negative, hence stable): a smaller \(\tau\) gives a faster response.

Second-Order Systems — Standard Form

Here \(\omega_n\) is the undamped natural frequency (rad/s) and \(\zeta\) the dimensionless damping ratio. The characteristic equation \(s^2+2\zeta\omega_ns+\omega_n^2=0\) has poles \(s_{1,2}=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1}\).

The damped natural frequency is \(\omega_d=\omega_n\sqrt{1-\zeta^2}\) and the damping coefficient \(\sigma=\zeta\omega_n\).

Second-Order Step Response — All Cases

For the underdamped case \((0\lt\zeta\lt1)\),

For the critically damped case \((\zeta=1)\), \(y(t)=1-e^{-\omega_nt}(1+\omega_nt)\). For the overdamped case \((\zeta\gt1)\), with \(s_{1,2}=-\zeta\omega_n\pm\omega_n\sqrt{\zeta^2-1}\),

Time-Domain Specifications (Underdamped)

• Delay time (to 50%): \(t_d\approx\dfrac{1+0.7\zeta}{\omega_n}\).
• Rise time (0–100%): \(\boxed{t_r=\dfrac{\pi-\phi}{\omega_d}}\), with \(\phi=\cos^{-1}\zeta\).
• Peak time: \(\boxed{t_p=\dfrac{\pi}{\omega_d}=\dfrac{\pi}{\omega_n\sqrt{1-\zeta^2}}}\).
• Peak overshoot: \(\boxed{M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}}\times100\%}\).
• Settling time: \(t_s=\dfrac{4}{\zeta\omega_n}\) (2%) or \(\dfrac{3}{\zeta\omega_n}\) (5%).
• Oscillations to settle: \(N=\dfrac{t_s\,\omega_d}{2\pi}\).

Specifications — Quick Reference

Typical designs use \(\zeta\in[0.4,\,0.8]\); the value \(\zeta=0.707\) is often optimal, giving a fast response with about \(5\%\) overshoot. The damping ratio can be recovered from a measured overshoot using \(\zeta=\dfrac{-\ln(M_p)}{\sqrt{\pi^2+\ln^2(M_p)}}\).

Effect of Additional Poles and Zeros; Dominant Pole

A right-half-plane zero makes the system non-minimum phase, causing initial undershoot and limiting the achievable bandwidth.

Steady-State Error and Static Error Constants

For a unity-feedback system, the final-value theorem gives

The static error constants are

Steady-State Error — Worked Example

1. One pole at the origin makes this a Type-1 system.
2. Error constants: \(K_p=\lim_{s\to0}\dfrac{10}{s(s+2)(s+5)}=\infty\) and \(K_v=\lim_{s\to0}\dfrac{10}{(s+2)(s+5)}=\dfrac{10}{2\cdot5}=1\).
3. Superposition: the step part (amplitude 2) gives \(e_{ss_1}=\dfrac{2}{1+K_p}=0\); the ramp part (slope 3) gives \(e_{ss_2}=\dfrac{3}{K_v}=3\).

Part III — Stability Analysis

Concept of Stability

A system is bounded-input bounded-output (BIBO) stable if every bounded input produces a bounded output. For an LTI system this is equivalent to absolute integrability of the impulse response and to all poles lying in the left half-plane:

Routh–Hurwitz Stability Criterion

The Routh–Hurwitz criterion determines stability without computing the roots. For a characteristic equation \(a_ns^n+a_{n-1}s^{n-1}+\cdots+a_1s+a_0=0\), a necessary condition is that all coefficients are present and of the same sign. The sufficient condition uses the Routh array, whose first two rows hold the coefficients and whose later entries are formed by

Special Cases

• First element of a row is zero: replace the zero with a small \(\epsilon\gt0\), continue, then let \(\epsilon\to0^+\).
• An entire row is zero: this signals symmetric roots. Form the auxiliary equation \(A(s)=0\) from the row above, differentiate to get \(dA/ds\), and use its coefficients to replace the zero row; the roots of \(A(s)=0\) are also roots of the characteristic equation. For \(s^5+2s^4+24s^3+48s^2-25s-50=0\) the \(s^3\) row vanishes; the \(s^4\) row gives \(A(s)=2s^4+48s^2-50\) and \(dA/ds=8s^3+96s\).

Routh–Hurwitz — Worked Example

The Routh array is

No sign changes in the first column requires \(\dfrac{6-K}{3}\gt0\Rightarrow K\lt6\) and \(K\gt0\).

Root Locus — Introduction

The root locus is the trajectory of the closed-loop poles in the \(s\)-plane as the open-loop gain \(K\) varies from \(0\) to \(\infty\). For \(1+KG(s)H(s)=0\) with \(G(s)H(s)=\dfrac{\prod(s-z_i)}{\prod(s-p_j)}\), a point lies on the locus when it satisfies the

• Angle condition: \(\boxed{\angle G(s)H(s)=(2k+1)\cdot180^\circ}\), expanded as \(\sum_{i=1}^m\angle(s-z_i)-\sum_{j=1}^n\angle(s-p_j)=(2k+1)180^\circ\);
• Magnitude condition (used to find \(K\)): \(|KG(s)H(s)|=1\Rightarrow K=\dfrac{1}{|G(s)H(s)|}\).

Rules for Root-Locus Construction

1. Symmetry: the locus is symmetric about the real axis.
2. Branches: there are \(n\) branches (number of open-loop poles, \(n\ge m\)).
3. Start/end: branches start at poles \((K=0)\) and end at zeros or at infinity \((K\to\infty)\); \(n-m\) branches go to infinity.
4. Real-axis segments: a point lies on the locus if the total number of real poles and zeros to its right is odd.
5. Asymptotes: \(n-m\) asymptotes at angles \(\theta_A=\dfrac{(2k+1)180^\circ}{n-m}\), meeting the real axis at the centroid \(\sigma_A=\dfrac{\sum\text{poles}-\sum\text{zeros}}{n-m}\).
6. Breakaway/break-in points: solutions of \(\dfrac{dK}{ds}=0\) that lie on the locus.
7. \(j\omega\)-axis crossing: found from the Routh array; the auxiliary equation gives the crossing frequency and critical \(K\).
8. Angle of departure (from a pole): \(\theta_d=180^\circ-\sum\angle(\text{other poles})+\sum\angle(\text{zeros})\).
9. Angle of arrival (at a zero): \(\theta_a=180^\circ+\sum\angle(\text{poles})-\sum\angle(\text{other zeros})\).

Root Locus — Worked Example

The poles are at \(0,-2,-4\) \((n=3)\) with no finite zeros \((m=0)\), so there are three branches, all tending to infinity. The real-axis segments are \([-2,0]\) and \((-\infty,-4]\). The asymptotes are at \(\pm60^\circ\) and \(180^\circ\), with centroid \(\sigma_A=\dfrac{0-2-4}{3}=-2\). The breakaway point comes from \(K=-s(s+2)(s+4)\), \(\dfrac{dK}{ds}=-(3s^2+12s+8)=0\), giving \(s=-0.845\) (on the locus) or \(s=-3.155\) (rejected).

For the \(j\omega\)-axis crossing, the characteristic equation \(s^3+6s^2+8s+K=0\) gives the Routh array

so at \(K=48\) the auxiliary equation \(6s^2+48=0\) gives \(s=\pm j2\sqrt{2}\).

Part IV — Frequency-Domain Analysis

Frequency Response — Basics

The frequency response is the steady-state response of an LTI system to a sinusoid of varying frequency, obtained by setting \(s=j\omega\) in \(G(s)\): \(G(j\omega)=|G(j\omega)|\,\angle G(j\omega)\). For an input \(r(t)=A\sin\omega t\) the steady-state output is \(y_{ss}(t)=A\,|G(j\omega)|\,\sin\!\bigl(\omega t+\angle G(j\omega)\bigr)\) — the same frequency, scaled in amplitude and shifted in phase.

For a standard second-order system, \(\omega_r=\omega_n\sqrt{1-2\zeta^2}\), \(M_r=\dfrac{1}{2\zeta\sqrt{1-\zeta^2}}\), \(\omega_b=\omega_n\sqrt{1-2\zeta^2+\sqrt{2-4\zeta^2+4\zeta^4}}\).

Bode Plot — Concept

A Bode plot consists of a magnitude plot of \(20\log_{10}|G(j\omega)|\) (dB) against \(\log_{10}\omega\) and a phase plot of \(\angle G(j\omega)\) (degrees) against \(\log_{10}\omega\). The logarithmic scale turns multiplication of factors into addition, so each factor contributes an asymptotic straight line.

Bode Plot — Sketching Procedure

1. Convert \(G(s)\) to time-constant (Bode) form.
2. Identify the corner frequencies: \(\omega_c=1/\tau\) for first-order factors and \(\omega_c=\omega_n\) for second-order factors.
3. At low frequency the initial slope is \(-20N\) dB/dec, where \(N\) is the number of poles at the origin.
4. The initial line passes through \(20\log K\) at \(\omega=1\) (Type 0) or through \(20\log K\) at \(\omega=K^{1/N}\) (Type \(N\)).
5. At each corner frequency, change the slope by \(\pm20\) (simple) or \(\pm40\) (quadratic) dB/dec.
6. Sketch the phase by summing the individual contributions.

Bode Plot — Example Sketch

This is a Type-1 system with corner frequencies \(\omega_{c1}=10\) and \(\omega_{c2}=50\) rad/s.

The slope is \(-20\) dB/dec until \(\omega=10\), then \(-40\) up to \(\omega=50\), then \(-60\) beyond. At \(\omega=1\) the magnitude is \(20\log100=40\) dB, and the gain crossover is \(\omega_{gc}\approx\sqrt{100\cdot10}\approx31.6\) rad/s.

Polar Plot

A polar (Nyquist) plot traces \(|G(j\omega)|\angle G(j\omega)\) in the complex plane as \(\omega\) goes from \(0\) to \(\infty\). One computes the start \((\omega=0)\), the end \((\omega\to\infty)\), and the real- and imaginary-axis crossings. For \(G(s)=\dfrac{1}{1+s\tau}\) the plot starts at \(1\angle0^\circ\), passes through \(\tfrac{1}{\sqrt2}\angle{-45^\circ}\) at \(\omega=1/\tau\), and ends at \(0\angle{-90^\circ}\), forming a semicircle in the lower half-plane.

Nyquist Stability Criterion

By Cauchy's argument principle, for a rational \(F(s)\) and a closed contour \(\Gamma_s\), \(N=P-Z\), where \(N\) is the number of counter-clockwise encirclements of the origin by \(F(s)\), \(P\) the poles of \(F(s)\) inside \(\Gamma_s\) and \(Z\) the zeros inside.

Nyquist Plot Construction

1. Compute \(GH(j\omega)\) at \(\omega=0^+\) and as \(\omega\to\infty\).
2. Find the real-axis crossings, where \(\mathrm{Im}\{GH\}=0\).
3. Find the imaginary-axis crossings, where \(\mathrm{Re}\{GH\}=0\).
4. Sketch for \(\omega\gt0\) and mirror about the real axis for \(\omega\lt0\).
5. Count the counter-clockwise encirclements \(N\) of \((-1,0)\).
6. Apply \(Z=P-N\); the closed loop is stable if and only if \(Z=0\).

Relative stability follows from the same plot: \(\text{GM}=\dfrac{1}{|GH(j\omega_{pc})|}\) (in dB, \(-20\log|GH|\) at \(\omega_{pc}\)) and \(\text{PM}=180^\circ+\angle GH(j\omega_{gc})\).

Correlation Between Time and Frequency Domains

Useful design relations: \(t_r\cdot\omega_b\approx1.5\) to \(2.5\); for \(\zeta=0.707\), \(\omega_b=\omega_n\) and \(M_r=1\) (no peaking); a finite \(M_r\) occurs only for \(\zeta\lt1/\sqrt2=0.707\). Higher bandwidth gives a faster response and better tracking but more noise susceptibility; a common target is \(M_r\approx1.1\)–\(1.5\) and PM \(\approx45^\circ\)–\(60^\circ\).

Constant-\(M\)/Constant-\(N\) Circles and the Nichols Chart

For unity feedback the closed-loop frequency response is \(M(\omega)e^{j\alpha(\omega)}=\dfrac{G(j\omega)}{1+G(j\omega)}\).

The Nichols chart plots open-loop magnitude (dB) against phase (deg) with constant-\(M\) and constant-\(N\) loci superimposed, which is more convenient than polar plots for compensator design.

Minimum-Phase versus Non-Minimum-Phase Systems

A minimum-phase system has all poles and zeros in the LHP; a non-minimum-phase system has at least one pole or zero in the RHP; an all-pass system has poles and zeros mirror-symmetric about the imaginary axis, so \(|G(j\omega)|=1\) for all \(\omega\). Among systems with the same magnitude response, the minimum-phase one has the smallest phase lag, obeys Bode's gain–phase relation and is invertible. Non-minimum-phase systems exhibit initial undershoot, extra phase lag and limited bandwidth.

Compensator Design in the Frequency Domain

Lead Compensator (Bode Design)

1. Adjust the gain \(K\) to meet the steady-state error specification.
2. Sketch the Bode plot of \(KG_p\) and find the existing phase margin \(\phi_0\).
3. Required lead: \(\phi_{\max}=\phi^*-\phi_0+\epsilon\), with \(\epsilon=5^\circ\)–\(10^\circ\).
4. Compute \(\alpha=\dfrac{1-\sin\phi_{\max}}{1+\sin\phi_{\max}}\).
5. Find the new gain crossover \(\omega'_{gc}\) where \(|KG_p|=-10\log(1/\alpha)\) dB.
6. Set \(\omega_m=\omega'_{gc}\), so \(\tau=\dfrac{1}{\omega_m\sqrt{\alpha}}\).
7. Use \(G_c(s)=\dfrac{1+s\tau}{1+s\alpha\tau}\) (with \(K_c=1/\alpha\)).

Lag Compensator (Bode Design)

1. Adjust \(K\) for the error specification.
2. Find the frequency where the phase of \(KG_p\) gives PM \(=\phi^*+\epsilon\); call it \(\omega'_{gc}\).
3. At \(\omega'_{gc}\), with \(|KG_p|_{\text{dB}}=A\), set \(\beta=10^{A/20}\).
4. Place the zero at \(\omega_z=\omega'_{gc}/10\), i.e. \(\tau=10/\omega'_{gc}\).
5. Use \(G_c(s)=\dfrac{1+s\tau}{1+s\beta\tau}\).

Part V — Advanced Topics

PID Controllers

The standard PID control law and its transfer function are

Lead, Lag and Lead–Lag Compensators

The general first-order compensator is \(G_c(s)=K_c\dfrac{s+z}{s+p}\) with \(z,p\gt0\).

For the lead compensator \(G_c(s)=\alpha\dfrac{1+s\tau}{1+s\alpha\tau}\) with \(\alpha\lt1\), the maximum phase lead is \(\phi_{\max}=\sin^{-1}\!\left(\dfrac{1-\alpha}{1+\alpha}\right)\), occurring at \(\omega_m=\dfrac{1}{\tau\sqrt{\alpha}}\), where \(|G_c(j\omega_m)|=\dfrac{1}{\sqrt{\alpha}}\). For the lag compensator \(G_c(s)=\beta\dfrac{1+s\tau}{1+s\beta\tau}\) with \(\beta\gt1\), the phase contribution is negative, so \(\omega_m\) is placed well below \(\omega_{gc}\) to avoid degrading the phase margin.

Time Delay and Padé Approximation

A pure transport lag, \(y(t)=u(t-T_d)\), has \(G_d(s)=e^{-sT_d}\) with \(|G_d(j\omega)|=1\) (all-pass) and \(\angle G_d(j\omega)=-\omega T_d\) (an unbounded, linear phase lag). Delays arise from transport, sensing and computation; they subtract phase and therefore reduce the phase margin, limiting bandwidth (rule of thumb \(\omega_{gc}T_d\lesssim1\)), and can cause oscillation or instability if ignored.

The Padé approximations replace the exponential with a rational function:

The first-order form introduces an RHP zero at \(2/T_d\), making it a non-minimum-phase approximation. At \(\omega_{gc}\) a delay subtracts \(\omega_{gc}T_d\) rad \(=57.3\,\omega_{gc}T_d\) degrees of phase; the Smith predictor compensates known delays explicitly.

State-Space Representation

The state-space model uses state variables capturing the minimum information needed to predict the system's future behaviour:

Here \(\mathbf x\in\mathbb{R}^n\) is the state, \(\mathbf u\in\mathbb{R}^r\) the input, \(\mathbf y\in\mathbb{R}^m\) the output, with \(\mathbf A_{n\times n}\), \(\mathbf B_{n\times r}\) and \(\mathbf C_{m\times n}\). The transfer function recovered from the model is

with characteristic polynomial \(|s\mathbf I-\mathbf A|=0\). The state equation solves to \(\mathbf x(t)=e^{\mathbf At}\mathbf x(0)+\int_0^te^{\mathbf A(t-\tau)}\mathbf B\mathbf u(\tau)\,d\tau\), where the state-transition matrix is \(\phi(t)=e^{\mathbf At}=\mathcal{L}^{-1}\{(s\mathbf I-\mathbf A)^{-1}\}\).

Canonical Forms

For a SISO system \(G(s)=\dfrac{b_2s^2+b_1s+b_0}{s^3+a_2s^2+a_1s+a_0}\), two common realisations are the controllable and observable canonical forms.

The diagonal (modal) form uses \(\mathbf A=\mathrm{diag}(p_1,\ldots,p_n)\) for distinct poles, decoupling the system into first-order modes; the Jordan form handles repeated eigenvalues with a block-diagonal structure. All forms describe the same system, related by similarity transformations \(\mathbf x'=\mathbf P\mathbf x\): \(\mathbf A'=\mathbf P\mathbf A\mathbf P^{-1}\), \(\mathbf B'=\mathbf P\mathbf B\), \(\mathbf C'=\mathbf C\mathbf P^{-1}\).

Controllability and Observability

By the duality principle, \((\mathbf A,\mathbf B,\mathbf C)\) is controllable if and only if \((\mathbf A^T,\mathbf C^T,\mathbf B^T)\) is observable. The PBH test gives an equivalent check: controllable if \(\mathrm{rank}[\,s\mathbf I-\mathbf A\ \ \mathbf B\,]=n\) for all \(s\) (checked at the eigenvalues of \(\mathbf A\)), and observable if \(\mathrm{rank}\!\begin{bmatrix}s\mathbf I-\mathbf A\\\mathbf C\end{bmatrix}=n\).

Controllability/Observability — Example

For controllability,

so the system is controllable. For observability,

so the system is observable.

State Feedback and Pole Placement

The state-feedback law \(\mathbf u=-\mathbf K\mathbf x+\mathbf r\) gives closed-loop dynamics \(\dot{\mathbf x}=(\mathbf A-\mathbf B\mathbf K)\mathbf x+\mathbf B\mathbf r\), whose eigenvalues are the roots of \(|s\mathbf I-(\mathbf A-\mathbf B\mathbf K)|=0\). For a SISO system, Ackermann's formula gives \(\mathbf K=[\,0\ 0\ \cdots\ 0\ 1\,]\,\mathbf Q_c^{-1}\,\phi_d(\mathbf A)\), where \(\phi_d(s)\) is the desired characteristic polynomial. By the pole-placement theorem, if \((\mathbf A,\mathbf B)\) is controllable, the closed-loop poles can be placed arbitrarily.

Digital Control and the Z-Transform

Sampling a signal gives \(x[k]=x(kT)\), and the Z-transform is

Discrete-Time Stability and the \(s\to z\) Mapping

Under \(z=e^{sT}\), the left half of the \(s\)-plane maps to the interior of the unit circle, the imaginary axis maps to the unit circle \(|z|=1\), and the right half-plane maps to the exterior.

Stability can be tested with Jury's test (the discrete analogue of Routh–Hurwitz) or by the bilinear transformation \(z=\dfrac{1+w}{1-w}\) followed by Routh–Hurwitz on the \(w\)-plane. A zero-order hold, \(G_{ZOH}(s)=\dfrac{1-e^{-sT}}{s}\), converts samples into a piecewise-constant signal.

Inverse Z-Transform and Jury's Test

The inverse Z-transform can be obtained by long division (\(X(z)=x[0]+x[1]z^{-1}+x[2]z^{-2}+\cdots\)), by partial fractions of \(X(z)/z\), or by the residue method \(x[k]=\sum\text{Residues of }X(z)z^{k-1}\). For \(F(z)=a_0+a_1z+\cdots+a_nz^n=0\), Jury's test requires \(F(1)\gt0\), \((-1)^nF(-1)\gt0\) and \(|a_0|\lt a_n\), together with the conditions \(|b_0|\gt|b_{n-1}|\), \(|c_0|\gt|c_{n-2}|\), … built from the Jury array whose entries are \(b_k=\begin{vmatrix}a_0&a_{n-k}\\a_n&a_k\end{vmatrix}\).

Discrete State-Space Model

The discrete state equations are \(\mathbf x[k+1]=\mathbf G\mathbf x[k]+\mathbf H\mathbf u[k]\) and \(\mathbf y[k]=\mathbf C\mathbf x[k]+\mathbf D\mathbf u[k]\). Discretising a continuous system with a zero-order hold and sampling period \(T\) gives

and the pulse transfer function is \(G(z)=\mathbf C(z\mathbf I-\mathbf G)^{-1}\mathbf H+\mathbf D\). Controllability and observability use the same forms with \(\mathbf G,\mathbf H\). A feature unique to discrete time is deadbeat control: choosing \(\mathbf K\) to place all closed-loop poles at \(z=0\), giving finite settling in at most \(n\) steps.

Nonlinear Systems — Describing Function

For a nonlinearity \(y=N(x)\) driven by \(x(t)=X\sin\omega t\), the describing function is the ratio of the fundamental Fourier component of the output to the input amplitude:

A limit cycle is predicted when \(1+N(X)G(j\omega)=0\), i.e. \(G(j\omega)=-1/N(X)\); plotting \(G(j\omega)\) and \(-1/N(X)\) together, their intersections indicate possible limit cycles.

Phase-Plane Analysis

For a second-order system \(\ddot x=f(x,\dot x)\), trajectories are plotted in the \((x_1,x_2)=(x,\dot x)\) plane, with singular (equilibrium) points where \(\dot x_1=\dot x_2=0\). Linearising at an equilibrium, \(\dot{\mathbf x}=\mathbf A\mathbf x\), the type follows from the eigenvalues.

A limit cycle is an isolated closed trajectory (for example, the Van der Pol oscillator); linear systems cannot have isolated limit cycles — only nonlinear systems can.

Common Nonlinearities and Lyapunov Stability

Typical Lyapunov candidates include the energy-based \(V=\tfrac12\dot x^2+\int_0^xf(\xi)\,d\xi\) and the quadratic \(V(\mathbf x)=\mathbf x^T\mathbf P\mathbf x\) with \(\mathbf P\gt0\); a radially unbounded \(V\) gives global stability.

Introduction to Optimal Control (LQR)

The linear–quadratic regulator finds the input minimising a quadratic cost. Given \(\dot{\mathbf x}=\mathbf A\mathbf x+\mathbf B\mathbf u\), minimise

The optimal control is \(\mathbf u^*(t)=-\mathbf K\mathbf x(t)\) with \(\mathbf K=\mathbf R^{-1}\mathbf B^T\mathbf P\), where \(\mathbf P\succ0\) uniquely solves the algebraic Riccati equation

Additional Worked Examples

Block-Diagram Reduction

Reducing the inner loop, \(G_{\mathrm{in}}=\dfrac{G_1}{1+G_1H_1}=\dfrac{10}{s+6}\). Cascading with \(G_2\), \(G_{\mathrm{fwd}}=\dfrac{10}{s+6}\cdot\dfrac{5}{s+2}=\dfrac{50}{(s+2)(s+6)}\). Closing the unity outer loop gives

This has \(\omega_n=\sqrt{62}\approx7.87\) and \(\zeta=8/(2\omega_n)\approx0.508\) — stable, underdamped, with \(M_p\approx15.4\%\).

Bode-Plot Analysis

Requiring \(\angle G(j\omega_{gc})=-180^\circ+45^\circ=-135^\circ\),

which solves to \(\omega_{gc}\approx1.2\) rad/s. Setting \(|G(j\omega_{gc})|=1\),

State-Space: Matrix Exponential

The eigenvalues come from \(|s\mathbf I-\mathbf A|=s^2+5s+6=(s+2)(s+3)\), so \(\lambda_1=-2,\ \lambda_2=-3\). The resolvent is

and inverting each entry (for example \(\dfrac{s+5}{(s+2)(s+3)}=\dfrac{3}{s+2}-\dfrac{2}{s+3}\)) gives

which correctly reduces to the identity at \(t=0\).

Summary, Exam Strategy and Common Pitfalls

Master Formula Sheet

Other essentials worth memorising: the Z-transform pairs \(\delta[k]\to1\), \(u[k]\to z/(z-1)\), \(a^k\to z/(z-a)\), with discrete stability requiring all poles \(|z|\lt1\) and the mappings \(z=e^{sT}\), \(s=\ln(z)/T\). The PID controller is \(G_c(s)=K_p+\dfrac{K_i}{s}+K_ds\). For a lead network \(\alpha\lt1\) with \(\phi_m=\sin^{-1}\dfrac{1-\alpha}{1+\alpha}\), and for a lag network \(\beta\gt1\). The linear Lyapunov equation is \(\mathbf A^T\mathbf P+\mathbf P\mathbf A=-\mathbf Q\).

GATE / ESE Exam Strategy

Common Pitfalls and Tricks

The Big Picture

References and Further Reading

Standard Textbooks

• K. Ogata, Modern Control Engineering, 5th ed., Pearson.
• N. S. Nise, Control Systems Engineering, 8th ed., Wiley.
• B. C. Kuo and F. Golnaraghi, Automatic Control Systems, 10th ed.
• I. J. Nagrath and M. Gopal, Control Systems Engineering, 6th ed., New Age.
• R. C. Dorf and R. H. Bishop, Modern Control Systems, 13th ed., Pearson.

Advanced and State-Space

• G. Franklin, J. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems.
• H. K. Khalil, Nonlinear Systems, 3rd ed.
• K. J. Åström and R. M. Murray, Feedback Systems.

For Competitive Exams (GATE / ESE / PSU)

• Made Easy, Control Systems (theory and practice sets).
• ACE Engineering Academy materials.
• Past 20 years' GATE and ESE question papers.