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Sample and hold systems
Sample and Hold Systems
The continuous-time plant, P (s), is preceeding by a zero-order hold and followed by a sampler.
@@ T P (s) (^) ZOH C(z)
−
u 6
y(k) y(t) u(t) u(k) r(k)
Performance and stability is specified in terms of the digital domain signals, r(k), y(k), u(k), etc.
- Analog/Digital (A/D) board: sampler.
- Digital/Analog (D/A) board: zero-order hold.
Other options are possible but the above are by far the most common.
Modeling sampled systems
Modeling P (z)
P ( s ) C ( s )
P ( z )^ C ( z )
Approximation of C ( s ) with C ( z )
Model P ( s ), and sample/hold as P ( z )
Continuous-time design
Discrete-time design
- Develop a model of the discrete-time behavior of the plant.
- Allows digital designs to be performed directly.
- Evaluating the stability of the discrete-time system (C(z) and P (z) in feedback).
Roy Smith: ECE 147b 5 : 1
Sample and hold systems
Zero-order hold equivalence
This is a reasonable model of a typical digital to analog (D/A) converter.
At the sample-time, t = kT , the discrete input, u(k), is put on the output, u(t). This value is held constant for the entire sample period. So,
u(t) = u(k), for kT ≤ t < kT + T.
0 T 2T 3T 4T 5T 6T
Time: t
u ( k ) u ( t )
Sample and hold systems
Sample and Hold Systems
Model the system from the ZOH block to the sampler:
^ @@ T P (s) ZOH
y(k) y(t) u(t) u(k)
P (s) is an LTI system =⇒ the system from u(k) to y(k) is LSI.
It has an equivalent Z-transform, P (z).
f P (z) f
y(k) u(k)
Zero-order hold equivalence: The closed-loop combination of P (z) and C(z) exactly models P (s) in closed-loop at the sample times.
Roy Smith: ECE 147b 5 : 3
Sampling in closed-loop
Stability/Performance evaluation
P (z) is the zero-order hold equivalent of P (s).
P (z) C(z) (^)
−
v 6
y(k) r(k)
Remaining issues
- This analysis considers the system response at the sample times.
- Hidden oscillations
- Intersample behavior
- How should one modify C(z) if the discrete-time response is not satisfactory?
Sampling in closed-loop
Design closed-loop (continuous-time)
P (s) C(s) (^)
−
v 6
y(s) r(s)
Implemented closed-loop. Note C(z) approximates C(s).
@@ P (s) (^) ZOH C(z) T
−
v 6
y(k) y(t) u(t) u(k) r(k)
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Sampling in closed-loop
The example continued
Increasing Kp has the following effects:
- Decreasing the rise-time,
- Reducing the settling-time
- Reducing the steady-state tracking error.
- Increasing the controller output amplitude (more gain ⇒ more $).
In reality too much gain will eventually destabilize the continuous-time system (why?).
Digital implementation
ZOH equivalent for P (s):
P (z) = (1 − z−^1 )Z
P (s) s
= (1 − z−^1 )Z
a s(s + a)
1 − e−aT z − e−aT^
P (z) has a (stable) pole at z = e−aT^.
Approximation for C(s):
C(s) = Kp so C(z) = Kp
Sampling in closed-loop
An example
Consider a proportional controller: C(s) = C(z) = Kp.
And a simple plant: P (s) = a s + a
, a > 0.
Root locus
P (s)Kp has one pole and no zeros.
The closed-loop, with C(s) = Kp, is theoretically stable for all − 1 < Kp ≤ ∞.
Real
Imaginary s- plane
-a K > p (^0) K < p 0
K = p -
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Sample and hold
Precompensate by including a delay in the continuous design
P (s) e−sT / 2
y(t) u(t)
The additional delay approximates the phase lag that the ZOH will introduce in the digital implementation.
If C(s) is designed to work with P (s)e−sT /^2 , then it will probably work reasonably well for a ZOH implementation of P (s).
Sample and hold
Effect of a sample and hold
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.
-1.
-0.
-0.
-0.
-0.
0
Time [seconds]
Input sinusoid
Samples
ZOH output ZOH output 1st harmonic
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Delay approximations
Rational approximations to e−sT /^2 :
Pad´e Approximations P (s):
First order lag:
1 + sT / 2
First order Pad´e: 1 − sT / 4 1 + sT / 4
N th order Pad´e: e−θs^ ≈
1 − 2 θn s
)n ( 1 + 2 θn s
)n
We typically use a first order Pad´e approximation which adds one pole and one zero to the plant for our design of C(s).
If the plant dynamics are close to the Nyquist frequency we may choose to use a second order Pad´e approximation for greater accuracy.
Sample and hold
Precompensate by including a delay in the continuous design
10 -1^10 0 10 1 10 2 10
10 1
10 2
10 3
Frequency [rad/sec]
Magnitude
10 0 10 1 10 2 10 3
-15 0
0
50 Frequency [rad/sec]
Phase (degrees)
P(j ω )
P(j ω )
P(j ω )
ZOH equivalent of P(s)
ZOH equivalent of P(s)
Nyquist frequency
Nyquist frequency
e -j ω T/^2
e -j ω T/^2 P(j ω )
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