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Control theory models

Move from differential equations to feedback, transient response, frequency response and state-space reasoning, with twenty solved examples.

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Matter pathway: atom → solid → liquid → gas → plasma. Quantum mechanics and quantum field theory provide foundations across the pathway; they are not additional phases. This is a connected modeling route, not a universal heating curve. Actual phases depend on pressure, composition, and kinetics.

1. Dynamics and transfer functions

Definitions & inputs. u input,y output,τ time constant,K static gain,s Laplace variable.

  1. A first-order balance describes one dominant storage mode.

    τy˙+y=Ku\tau\dot y+y=Ku

  2. Laplace transformation turns differentiation into multiplication by s.

    G(s)=Y(s)/U(s)=K/(τs+1)G(s)=Y(s)/U(s)=K/(\tau s+1)

  3. Invert the unit-step response with zero initial output.

    y(t)=K(1−e−t/τ)y(t)=K(1-e^{-t/\tau})

Interpretation. Initial-state responses must be added when the system does not start at rest.

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2. Negative feedback

Definitions & inputs. r reference,e error,C controller,G plant,T closed-loop transfer.

  1. Substitute the controller and plant relations into the feedback equation.

    e=r−y,y=GC(r−y)e=r-y,\quad y=GC(r-y)

  2. Solve for output and error; T+S=1.

    T=GC/(1+GC),S=1/(1+GC)T=GC/(1+GC),\quad S=1/(1+GC)

  3. Proportional, integral and derivative actions shape the loop.

    C(s)=Kp+Ki/s+KdsC(s)=K_p+K_i/s+K_ds

Interpretation. Ideal differentiation amplifies high-frequency noise and normally needs a filter.

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3. Poles and transient response

Definitions & inputs. ζ damping ratio,ωn natural frequency,ωd damped frequency.

  1. Normalize a second-order characteristic polynomial.

    G(s)=ωn2/(s2+2ζωns+ωn2)G(s)=\omega_n^2/(s^2+2\zeta\omega_ns+\omega_n^2)

  2. Complex poles set the oscillation rate and first response peak.

    ωd=ωn1−ζ2,tp=π/ωd\omega_d=\omega_n\sqrt{1-\zeta^2},\quad t_p=\pi/\omega_d

  3. Exponential decay gives overshoot and a common approximate two-percent settling estimate.

    Mp=e−πζ/1−ζ2,ts≃4/(ζωn)M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}},\quad t_s\simeq4/(\zeta\omega_n)

Interpretation. Extra poles, zeros, delays and nonlinearities can invalidate these estimates.

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4. Frequency and state space

Definitions & inputs. x state,A dynamics matrix,B input matrix,C output matrix,Ts sample period.

  1. Separate internal memory from measured output.

    x˙=Ax+Bu,y=Cx\dot x=Ax+Bu,\quad y=Cx

  2. Integrate the held-input state equation exactly.

    xk+1=eATsxk+(∫0TseAτB dτ)ukx_{k+1}=e^{AT_s}x_k+\left(\int_0^{T_s}e^{A\tau}B\,d\tau\right)u_k

  3. Measure phase distance from negative feedback becoming positive at gain crossover.

    PM=180∘+∠L(jωc),∣L(jωc)∣=1PM=180^\circ+\angle L(j\omega_c),\quad |L(j\omega_c)|=1

Interpretation. Margin interpretation needs the complete loop and treatment of unstable open-loop poles.

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Graphical worked example

Open-loop first-order model G(s)=2/(0.5s+1), zero initial state and unit input step. X axis: Time (s). Y axis: Unit-step output (normalized units).
Open-loop first-order model G(s)=2/(0.5s+1), zero initial state and unit input step. Related worked calculation · Download SVG · Plot data

Twenty worked examples

Open a problem to see its defined inputs, assumptions, equation, numerical substitution, result, and interpretation. Values are illustrative analytical exercises.

Example 01. First-order pole

Definitions & inputs. τ=.5 s,K=2.

  1. Choose the governing model and isolate the requested quantity.

    p=−1/τp=-1/\tau

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    p=−1/0.5p=-1/0.5

  3. Evaluate the expression; the result uses the units shown.

    Result=−2 s−1\mathrm{Result}=-2\ {\rm s}^{-1}

Interpretation. A negative real pole gives exponential decay.

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Example 02. Step after one time constant

Definitions & inputs. K=2,unit step,t=τ.

  1. Choose the governing model and isolate the requested quantity.

    y=K(1−e−1)y=K(1-e^{-1})

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    y=2(1−e−1)y=2(1-e^{-1})

  3. Evaluate the expression; the result uses the units shown.

    Result=1.264241 \mathrm{Result}=1.264241\ {}

Interpretation. Final output is 2 in compatible output units.

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Example 03. First-order 95% rise

Definitions & inputs. τ=.5 s,target y/K=.95.

  1. Choose the governing model and isolate the requested quantity.

    t=−τln⁡(1−0.95)t=-\tau\ln(1-0.95)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    t=−0.5ln⁡0.05t=-0.5\ln0.05

  3. Evaluate the expression; the result uses the units shown.

    Result=1.497866 s\mathrm{Result}=1.497866\ {\rm s}

Interpretation. This is time to 95%, not the 10–90% rise-time definition.

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Example 04. Closed-loop static gain

Definitions & inputs. Plant static gain 2,proportional gain 3,unity negative feedback.

  1. Choose the governing model and isolate the requested quantity.

    T(0)=KpG0/(1+KpG0)T(0)=K_pG_0/(1+K_pG_0)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    T(0)=6/7T(0)=6/7

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8571429 \mathrm{Result}=0.8571429\ {}

Interpretation. Finite proportional gain leaves steady tracking error.

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Example 05. Unit-step error

Definitions & inputs. Same static loop gain 6,stable type-0 loop.

  1. Choose the governing model and isolate the requested quantity.

    e∞=1/(1+L0)e_\infty=1/(1+L_0)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    e∞=1/7e_\infty=1/7

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1428571 \mathrm{Result}=0.1428571\ {}

Interpretation. The final-value theorem requires a stable closed-loop response.

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Example 06. Sensitivity magnitude at DC

Definitions & inputs. Real DC loop gain 9.

  1. Choose the governing model and isolate the requested quantity.

    S0=1/(1+L0)S_0=1/(1+L_0)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    S0=1/10S_0=1/10

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1 \mathrm{Result}=0.1\ {}

Interpretation. Sensitivity reduction at DC may be offset at other frequencies.

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Example 07. Closed-loop time constant

Definitions & inputs. G=2/(.5s+1),Kp=3,unity feedback.

  1. Choose the governing model and isolate the requested quantity.

    τcl=τ/(1+KKp)\tau_{cl}=\tau/(1+KK_p)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    τcl=0.5/7\tau_{cl}=0.5/7

  3. Evaluate the expression; the result uses the units shown.

    Result=0.07142857 s\mathrm{Result}=0.07142857\ {\rm s}

Interpretation. The output gain also changes; speed alone is not the whole response.

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Example 08. Damped natural frequency

Definitions & inputs. ωn=10 rad/s,ζ=.6.

  1. Choose the governing model and isolate the requested quantity.

    ωd=ωn1−ζ2\omega_d=\omega_n\sqrt{1-\zeta^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ωd=101−0.36\omega_d=10\sqrt{1-0.36}

  3. Evaluate the expression; the result uses the units shown.

    Result=8 rad s−1\mathrm{Result}=8\ {\rm rad\,s}^{-1}

Interpretation. This is the oscillatory part of the pole pair.

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Example 09. Second-order peak time

Definitions & inputs. ωd=8 rad/s.

  1. Choose the governing model and isolate the requested quantity.

    tp=π/ωdt_p=\pi/\omega_d

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    tp=π/8t_p=\pi/8

  3. Evaluate the expression; the result uses the units shown.

    Result=0.3926991 s\mathrm{Result}=0.3926991\ {\rm s}

Interpretation. Standard zero-free second-order unit-step response is assumed.

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Example 10. Second-order overshoot

Definitions & inputs. ζ=.6.

  1. Choose the governing model and isolate the requested quantity.

    Mp=e−πζ/1−ζ2M_p=e^{-\pi\zeta/\sqrt{1-\zeta^2}}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Mp=e−0.75πM_p=e^{-0.75\pi}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.09478022 \mathrm{Result}=0.09478022\ {}

Interpretation. Multiply by 100 to obtain percent overshoot.

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Example 11. Approximate settling time

Definitions & inputs. ζ=.6,ωn=10 rad/s,two-percent criterion.

  1. Choose the governing model and isolate the requested quantity.

    ts≃4/(ζωn)t_s\simeq4/(\zeta\omega_n)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ts≃4/6t_s\simeq4/6

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6666667 s\mathrm{Result}=0.6666667\ {\rm s}

Interpretation. This is a common envelope estimate, not an exact last crossing.

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Example 12. First-order gain at corner

Definitions & inputs. G=1/(τs+1),ωτ=1.

  1. Choose the governing model and isolate the requested quantity.

    ∣G∣=1/1+(ωτ)2|G|=1/\sqrt{1+(\omega\tau)^2}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ∣G∣=1/2|G|=1/\sqrt2

  3. Evaluate the expression; the result uses the units shown.

    Result=0.7071068 \mathrm{Result}=0.7071068\ {}

Interpretation. The magnitude is approximately −3.01 dB.

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Example 13. First-order phase at corner

Definitions & inputs. ωτ=1.

  1. Choose the governing model and isolate the requested quantity.

    ϕ=−arctan⁡(ωτ)\phi=-\arctan(\omega\tau)

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    ϕ=−arctan⁡1\phi=-\arctan1

  3. Evaluate the expression; the result uses the units shown.

    Result=−45 deg\mathrm{Result}=-45\ {\rm deg}

Interpretation. The phase approaches −90° only at much higher frequency.

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Example 14. Phase margin

Definitions & inputs. Loop phase −135° at a unit-magnitude crossover.

  1. Choose the governing model and isolate the requested quantity.

    PM=180∘+ϕcPM=180^\circ+\phi_c

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    PM=180−135PM=180-135

  3. Evaluate the expression; the result uses the units shown.

    Result=45 deg\mathrm{Result}=45\ {\rm deg}

Interpretation. Other crossovers and unstable open-loop poles must also be considered.

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Example 15. Delay-induced phase

Definitions & inputs. Added delay .02 s at ω=10 rad/s.

  1. Choose the governing model and isolate the requested quantity.

    Δϕ=−ωTd\Delta\phi=-\omega T_d

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Δϕ=−10(0.02)\Delta\phi=-10(0.02)

  3. Evaluate the expression; the result uses the units shown.

    Result=−0.2 rad\mathrm{Result}=-0.2\ {\rm rad}

Interpretation. Pure delay does not change magnitude but consumes phase margin.

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Example 16. Integrator accumulation

Definitions & inputs. Ki=2 s⁻¹,constant error .1 for 3 s,initial integral zero.

  1. Choose the governing model and isolate the requested quantity.

    uI=Ki∫e dtu_I=K_i\int e\,dt

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    uI=2(0.1)(3)u_I=2(0.1)(3)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.6 \mathrm{Result}=0.6\ {}

Interpretation. Saturation needs anti-windup treatment.

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Example 17. Derivative command

Definitions & inputs. Kd=.5 s,error slope .2/s.

  1. Choose the governing model and isolate the requested quantity.

    uD=Kde˙u_D=K_d\dot e

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    uD=0.5(0.2)u_D=0.5(0.2)

  3. Evaluate the expression; the result uses the units shown.

    Result=0.1 \mathrm{Result}=0.1\ {}

Interpretation. Real derivative action is filtered to limit noise amplification.

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Example 18. Exact sampled decay

Definitions & inputs. Continuous dx/dt=−2x,Ts=.1 s.

  1. Choose the governing model and isolate the requested quantity.

    Ad=e−2TsA_d=e^{-2T_s}

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ad=e−0.2A_d=e^{-0.2}

  3. Evaluate the expression; the result uses the units shown.

    Result=0.8187308 \mathrm{Result}=0.8187308\ {}

Interpretation. Repeated multiplication reproduces exact sample values for the unforced model.

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Example 19. Euler stability limit

Definitions & inputs. dx/dt=−2x,forward Euler.

  1. Choose the governing model and isolate the requested quantity.

    ∣1−2Ts∣<1⇒0<Ts<1|1-2T_s|<1\Rightarrow0<T_s<1

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    Ts,upper=2/2T_{s,upper}=2/2

  3. Evaluate the expression; the result uses the units shown.

    Result=1 s\mathrm{Result}=1\ {\rm s}

Interpretation. The upper endpoint itself is not asymptotically stable; accuracy usually needs much smaller steps.

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Example 20. Scalar state-feedback pole

Definitions & inputs. dx/dt=−x+u,u=−3x.

  1. Choose the governing model and isolate the requested quantity.

    p=A−BKp=A-BK

  2. Insert the stated inputs in consistent units or the explicitly defined normalized units.

    p=−1−1(3)p=-1-1(3)

  3. Evaluate the expression; the result uses the units shown.

    Result=−4 s−1\mathrm{Result}=-4\ {\rm s}^{-1}

Interpretation. State measurement and actuator authority are assumed.

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Symbols and units

Each derivation and problem defines its own symbols and inputs. Symbols may be reused with different meanings in other subjects. Keep units consistent, retain sufficient precision during calculation, and apply the stated validity limits.