PID control is one of the most widely used algorithms in engineering control systems. It is known for its mature technology, ease of use without the need to establish a mathematical model, flexible parameter settings, strong adaptability, and robust performance. Due to its effectiveness in control, it has been extensively applied in various fields. However, the tuning of PID parameters significantly affects the system's performance, and there is limited literature on the tuning methods specifically for incremental PID.
There are two main types of PID control: position-based PID and incremental PID. Position-based PID calculates the output based on the entire history of error values, which can lead to larger accumulated errors. In contrast, incremental PID computes the control action based only on the current and previous two error values, making it more efficient and less prone to large computational burdens. Additionally, if a microprocessor fails, the impact on the system is smaller because the actuator retains its last position due to its memory function. This makes incremental PID more reliable and widely used in practical applications.
**Incremental PID Control**
Incremental PID is a fundamental form of digital PID control. Instead of using the absolute value of the control signal, it calculates the change (increment) in the control output between consecutive sampling intervals. This approach reduces the need for extensive computation and storage, as it avoids accumulating the integral term over time.
The main advantages of incremental PID include:
1. It does not require accumulation of past error values. The control increment Δu(k) depends only on the last three sampled error values, allowing for better control performance through weighted calculations.
2. The computer outputs only the change in the control signal at each step, minimizing the impact of failures and reducing disturbances in the production process.
3. Manual-to-automatic switching causes minimal disturbance, enabling smooth transitions without causing sudden changes in the system.
However, since incremental PID requires storing the previous control value, it is typically not suitable for systems without memory capabilities, where positional PID would be preferred.
**Incremental PID Parameter Adjustment Formula**
The incremental PID control algorithm derives the control output at time n-1 from the position-based PID formula:
Assuming:
$$ u(n-1) = K_p e(n-1) + K_i \sum_{k=0}^{n-1} e(k) + K_d [e(n-1) - e(n-2)] $$
Then, by calculating the difference between the current and previous control outputs, we obtain the incremental form:
$$ \Delta u(n) = K_p [e(n) - e(n-1)] + K_i e(n) + K_d [e(n) - 2e(n-1) + e(n-2)] $$
This simplified equation allows for efficient computation and is commonly used in real-time control systems.
**Incremental PID Control Algorithm**
When the actuator requires only the change in the control signal (such as driving a stepper motor), the incremental PID algorithm becomes essential. The algorithm is derived from the standard PID formula and provides an efficient way to compute the control increment based on the most recent error measurements.
The output at the k-1th sampling instant can be calculated using:
$$ u(k-1) = K_p e(k-1) + K_i \sum_{i=0}^{k-1} e(i) + K_d [e(k-1) - e(k-2)] $$
By subtracting the previous control output from the current one, we get the incremental change:
$$ \Delta u(k) = K_p [e(k) - e(k-1)] + K_i e(k) + K_d [e(k) - 2e(k-1) + e(k-2)] $$
This approach reduces computational complexity and is widely adopted in industrial control systems.
**Incremental PID C51 Program**
Here is a sample implementation of the incremental PID algorithm in C for a microcontroller:
```c
typedef struct {
int SetPoint; // Target value
long SumError; // Error accumulation
double Proportion; // Proportional constant
double Integral; // Integral constant
double Derivative; // Derivative constant
int LastError; // Error[k-1]
int PrevError; // Error[k-2]
} PID;
static PID sPID;
static PID *sptr = &sPID;
void IncPIDInit(void) {
sptr->SumError = 0;
sptr->LastError = 0;
sptr->PrevError = 0;
sptr->Proportion = 0;
sptr->Integral = 0;
sptr->Derivative = 0;
sptr->SetPoint = 0;
}
int IncPIDCalc(int NextPoint) {
int iError, iIncpid;
iError = sptr->SetPoint - NextPoint;
iIncpid = sptr->Proportion * iError
- sptr->Integral * sptr->LastError
+ sptr->Derivative * sptr->PrevError;
sptr->PrevError = sptr->LastError;
sptr->LastError = iError;
return iIncpid;
}
```
**Combining Formulas to Explore General PID Control Law and Common Tuning Methods**
**(1) Discussing the Three Parameters of PID Control Law**
1. **Proportional (Kp):** Reacts immediately to the current deviation, helping to reduce it quickly.
2. **Integral (Ki):** Accumulates past errors to eliminate steady-state error and improve accuracy.
3. **Derivative (Kd):** Predicts future trends based on the rate of change of error, improving stability before the error occurs.
4. **PID Control:** Combines all three actions to achieve optimal control, eliminating residual error and enhancing system stability.
**(2) General Methods for PID Parameter Tuning**
1. **Experimental Testing Method:** Follows the "first proportional, then integral, finally derivative" approach, often used by beginners.
2. **Theoretical Calculation Method:** Based on the system’s mathematical model to calculate controller parameters.
3. **Empirical Tuning Method:** Often involves the critical proportional method, which is easy to implement and doesn't rely on complex models.
For new users, starting with equal coefficients for Kp, Ki, and Kd can simplify the process and help achieve desired results more easily.
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