Control Systems, List of Equations

The following is a list of the important equations from the text, arranged by subject. For more information about these equations, including the meaning of each variable and symbol, the uses of these functions, or the derivations of these equations, see the relevant pages in the main text.

Fundamental Equations

[Euler's Formula]

{e}^{j ω} = cos ⁡ ( ω ) + j sin ⁡ ( ω )

[Convolution]

( a ∗ b ) ( t ) = {∫}_{− ∞}^{∞} a ( τ ) b ( t − τ ) d τ

[Convolution Theorem]

ℒ [ f ( t ) ∗ g ( t ) ] = F ( s ) G ( s )

ℒ [ f ( t ) g ( t ) ] = F ( s ) ∗ G ( s )

[Characteristic Equation]

| A − λ I | = 0

A v = λ v

w A = λ w

[Decibels]

d B = 20 log ⁡ ( C )

Basic Inputs

[Unit Step Function]

u ( t ) = \{ \begin{matrix}0 , & t < 0 \\ 1 , & t ≥ 0\end{matrix}

[Unit Ramp Function]

r ( t ) = t u ( t )

[Unit Parabolic Function]

p ( t ) = \frac{1}{2} {t}^{2} u ( t )

Error Constants

[Position Error Constant]

{K}_{p} = {lim}_{s → 0} G ( s )

{K}_{p} = {lim}_{z → 1} G ( z )

[Velocity Error Constant]

{K}_{v} = {lim}_{s → 0} s G ( s )

{K}_{v} = {lim}_{z → 1} ( z − 1 ) G ( z )

[Acceleration Error Constant]

{K}_{a} = {lim}_{s → 0} {s}^{2} G ( s )

{K}_{a} = {lim}_{z → 1} ( z − 1 {)}^{2} G ( z )

System Descriptions

[General System Description]

y ( t ) = {∫}_{− ∞}^{∞} g ( t , r ) x ( r ) d r

[Convolution Description]

y ( t ) = x ( t ) ∗ h ( t ) = {∫}_{− ∞}^{∞} x ( τ ) h ( t − τ ) d τ

[Transfer Function Description]

Y ( s ) = H ( s ) X ( s )

Y ( z ) = H ( z ) X ( z )

[State-Space Equations]

{x}^{′} ( t ) = A x ( t ) + B u ( t )

y ( t ) = C x ( t ) + D u ( t )

[Transfer Matrix]

C [ s I − A {]}^{− 1} B + D = 𝐇 ( s )

C [ z I − A {]}^{− 1} B + D = 𝐇 ( z )

[Transfer Matrix Description]

𝐘 ( s ) = 𝐇 ( s ) 𝐔 ( s )

𝐘 ( z ) = 𝐇 ( z ) 𝐔 ( z )

[Mason's Rule]

M = \frac{{y}_{o u t}}{{y}_{i n}} = {∑}_{k = 1}^{N} \frac{{M}_{k} Δ {\text{ }}_{k}}{Δ \text{ }}

Feedback Loops

[Closed-Loop Transfer Function]

{H}_{c l} ( s ) = \frac{K G p ( s )}{1 + K G p ( s ) G b ( s )}

[Open-Loop Transfer Function]

{H}_{o l} ( s ) = K G p ( s ) G b ( s )

[Characteristic Equation]

F ( s ) = 1 + {H}_{o l}

Transforms

[Laplace Transform]

F ( s ) = ℒ [ f ( t ) ] = {∫}_{0}^{∞} f ( t ) {e}^{− s t} d t

[Inverse Laplace Transform]

f ( t ) = {ℒ}^{− 1} \{ F ( s ) \} = \frac{1}{2 π} {∫}_{c − i ∞}^{c + i ∞} {e}^{s t} F ( s ) \ d s

[Fourier Transform]

F ( j ω ) = ℱ [ f ( t ) ] = {∫}_{0}^{∞} f ( t ) {e}^{− j ω t} d t

[Inverse Fourier Transform]

f ( t ) = {ℱ}^{− 1} \{ F ( j ω ) \} = \frac{1}{2 π} {∫}_{− ∞}^{∞} F ( j ω ) {e}^{− j ω t} d ω

[Star Transform]

{F}^{∗} ( s ) = {ℒ}^{∗} [ f ( t ) ] = {∑}_{i = 0}^{∞} f ( i T ) {e}^{− s i T}

[Z Transform]

X ( z ) = 𝒵 \{ x [ n ] \} = {∑}_{i = − ∞}^{∞} x [ n ] {z}^{− n}

[Inverse Z Transform]

x [ n ] = {Z}^{− 1} \{ X ( z ) \} = \frac{1}{2 π j} {∮}_{C} X ( z ) {z}^{n − 1} d z \text{ }

[Modified Z Transform]

X ( z , m ) = 𝒵 ( x [ n ] , m ) = {∑}_{n = − ∞}^{∞} x [ n + m − 1 ] {z}^{− n}

Transform Theorems

[Final Value Theorem]

x ( ∞ ) = {lim}_{s → 0} s X ( s )

x [ ∞ ] = {lim}_{z → 1} ( z − 1 ) X ( z )

[Initial Value Theorem]

x ( 0 ) = {lim}_{s → ∞} s X ( s )

State-Space Methods

[General State Equation Solution]

x ( t ) = {e}^{A t − {t}_{0}} x ( {t}_{0} ) + {∫}_{{t}_{0}}^{t} {e}^{A ( t − τ )} B u ( τ ) d τ

x [ n ] = {A}^{n} x [ 0 ] + {∑}_{m = 0}^{n − 1} {A}^{n − 1 − m} B u [ n ]

[General Output Equation Solution]

y ( t ) = C {e}^{A t − {t}_{0}} x ( {t}_{0} ) + C {∫}_{{t}_{0}}^{t} {e}^{A ( t − τ )} B u ( τ ) d τ + D u ( t )

y [ n ] = C {A}^{n} x [ 0 ] + {∑}_{m = 0}^{n − 1} C {A}^{n − 1 − m} B u [ n ] + D u [ n ]

[Time-Variant General Solution]

x ( t ) = ϕ ( t , {t}_{0} ) x ( {t}_{0} ) + {∫}_{{t}_{0}}^{t} ϕ ( τ , {t}_{0} ) B ( τ ) u ( τ ) d τ

x [ n ] = ϕ [ n , {n}_{0} ] x [ {t}_{0} ] + {∑}_{m = {n}_{0}}^{n} ϕ [ n , m + 1 ] B [ m ] u [ m ]

[Impulse Response Matrix]

G ( t , τ ) = \{ \begin{matrix}C ( τ ) ϕ ( t , τ ) B ( τ ) & \text{ if } t ≥ τ \\ 0 & \text{ if } t < τ\end{matrix}

G [ n ] = \{ \begin{matrix}C {A}^{k − 1} N & \text{ if } k > 0 \\ 0 & \text{ if } k ≤ 0\end{matrix}

Root Locus

[The Magnitude Equation]

{}1 + K G ( s ) H ( s ) = 0

{}1 + K \overset{‾}{G H} ( z ) = 0

[The Angle Equation]

∠ K G ( s ) H ( s ) = 18 {0}^{∘}

∠ K \overset{‾}{G H} ( z ) = 18 {0}^{∘}

[Number of Asymptotes]

{N}_{a} = P − Z

[Angle of Asymptotes]

{ϕ}_{k} = ( 2 k + 1 ) \frac{π}{P − Z}

[Origin of Asymptotes]

{σ}_{0} = \frac{{∑}_{P} − {∑}_{Z}}{P − Z}

[Breakaway Point Locations]

\frac{G ( s ) H ( s )}{d s} = 0 or \frac{\overset{‾}{G H} ( z )}{d z} = 0

Lyapunov Stability

[Lyapunov Equation]

M A + {A}^{T} M = − N

Controllers and Compensators

[PID]

D ( s ) = {K}_{p} + \frac{{K}_{i}}{s} + {K}_{d} s

D ( z ) = {K}_{p} + {K}_{i} \frac{T}{2} [ \frac{z + 1}{z − 1} ] + {K}_{d} [ \frac{z − 1}{T z} ]

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