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Control engineering

Zero-order-hold discretisation of a first-order plant

Discrete-time model of a first-order plant when its input is held constant over each sample interval.

Ref: CTL-22
Rev: v1
Field: Control engineering
Result: A_d

A_{d} = e^{- a T_{s}} B_{d} = \frac{b}{a} (1 - e^{- a T_{s}})

If the input is held constant over each sample interval, the exact discrete-time model of ẋ = −ax + bu is x[k+1] = A_dx[k] + B_du[k] with A_d = e^−aT_s and B_d = (b/a)(1 − e^−aT_s). The continuous pole at −a maps to the discrete pole A_d, which lies inside the unit circle for any positive a and any sample interval. Here A_d is the output and B_d is shown among the derived results.

Discrete-time model of a first-order plant when its input is held constant over each sample interval. For the plant ẋ = −ax + bu the model is x[k+1] = A_dx[k] + B_du[k].

Calculator

Solve for

A_dDiscrete state coefficientComputed

aPlant decay rate a

1/s

0.510

bInput gain b

1/s

0.55

T_sSample interval

0.010.5

Input coefficient, B_d = (b/a)(1 − A_d)0.09063
Discrete DC gain, B_d / (1 − A_d)0.5
Plant time constant in samples, 1/(aT_s)5 samples

What drives the result

Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.

Discrete state coefficient () current value 0.8187

T_sSample intervalKey driver

0.3680.98

aPlant decay rate a

0.3680.951

bInput gain b

At the defaults x[k+1] = 0.8187 x[k] + 0.09063 u[k]. The discrete DC gain B_d/(1 − A_d) is 0.5, the same as the continuous DC gain b/a, as it should be.
The plant time constant is 5 samples long. Sampling as slowly as one time constant (T_s = 0.5 s) would give A_d = 0.3679.

Chart data as tables

Ranked variables

Variable	Typical range	Lowest result	Highest result	Swing ()	Local step	Change in result
T_s Sample interval	0.01 to 0.5 s	0.3679	0.9802	0.6123	+10%	−1.98 %
a Plant decay rate a	0.5 to 10 1/s	0.3679	0.9512	0.5833	+10%	−1.98 %
b Input gain b	0.5 to 5 1/s	0.8187	0.8187	0	+10%	0 %

Sweep of Sample interval

Sample interval (s)	Discrete state coefficient
0.01	0.9802
0.07125	0.8672
0.1325	0.7672
0.1938	0.6788
0.255	0.6005
0.3163	0.5313
0.3775	0.47
0.4387	0.4158
0.5	0.3679

Sweep one variable

Vary

Discrete state coefficient ()Hover, touch, or use the arrow keys on the chart

Variables and how each one contributes

A_d

Discrete state coefficient

Result ·

What the equation gives you. Everything else feeds it.

Plant decay rate a

1/s · typical 0.5 to 10

The decay rate of the plant in 1/s: its time constant is 1/a. A faster plant decays more in each sample, so A_d is smaller. Raising it by 10 % (to 2.2) moves the result by −1.98 %, from 0.8187 to 0.8025.

Input gain b

1/s · typical 0.5 to 5

The input gain in 1/s. No effect on A_d, which depends on a and T_s only. It scales B_d: 1 gives B_d = 0.09063 at the defaults.

T_s

Sample interval

s · typical 0.01 to 0.5

The sample interval in s. A longer interval lets more of the state decay between samples, so A_d falls. Raising it by 10 % (to 0.11) moves the result by −1.98 %, from 0.8187 to 0.8025.

Worked example with the values above

Exponent: aT_s = 2 × 0.1 = 0.2, so A_d = e^−0.2 = 0.8187.
Input coefficient: B_d = (b/a)(1 − A_d) = (1/2) × (1 − 0.8187) = 0.09063.
The model is x[k+1] = 0.8187 x[k] + 0.09063 u[k].

Assumptions and limits

A first-order plant with a constant input held over each sample (a zero-order hold) and a constant sample interval.
No delay between sampling and the update of the input.

Common mistakes

Using the forward-Euler approximation A_d ≈ 1 − aT_s for a long sample interval. It becomes unstable when aT_s exceeds 2, while the exact value stays stable.
Using the formula when the input is not held, for example a signal interpolated between samples. The result is then an approximation.

Statistics and uncertainty

A_d is an exponential of the product aT_s, so a relative error in a gives a relative error of aT_s times as much in A_d (and the opposite sign). At the defaults aT_s = 0.2, so a 10 % error in a changes A_d by about 2 %. Jitter in the sample interval has the same effect. With very slow sampling A_d is close to zero, and the model barely depends on a at all.

Where it is used

Turning a continuous model into the difference equation used in a digital controller or simulation.
Checking that a sample rate is fast enough compared with the plant's time constant.

Maths you need

Exponential decay: e^(−at) for a first-order plant
Difference equations: x[k+1] in terms of x[k] and u[k]

Related formulas

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