Control Input Constraint Handling Using GA Optimized Predictive PID Controller Mohammad H. Moradi, Hemen Showkati Electrical Engineering Department, Bu Ali Sina University, Hamedan, Iran [email protected], [email protected] Abstract- In this paper a new algorithm is proposed to optimize the gains of Predictive PID controller using genetic algorithms. A Predictive PID controller is developed which can handle control input constraints including amplitude and rate of change limits. This is done by considering a weighted value of control signal used to drive the system into objective function. Predictive PID gains are calculated to meet specific characteristics of desired output. Predictive PID controller could be implemented on existing conventional PID controller hardware currently in use in industrial applications. The performance of the proposed scheme is compared with Generalized Predictive Controller (GPC) showing similar performance.

I.

INTRODUCTION

PID controllers are used extensively in industrial applications and process control. Although in the last decades new techniques and methods have been proposed, because of economic reasons and implementation issues of these new algorithms, PID controllers are still the most common controllers in industry. Their relatively robust performance and simple structure makes these controllers a feasible and satisfactory scheme to control typical processes and systems. The three parameters of PID controllers must be tuned to the process to give satisfactory performance. Over the years numerous methods have been suggested to tune PID gains. Several good methods could be found in [1]-[10]. Model-based Predictive control (MPC) is one of wellknown and effective control strategies in industry. For example in petrochemical industries, the GPC method [5] which is a famous MPC scheme has become one of the most popular Control methods to be implemented. The name Model predictive control arises from the manner in which the control law is computed. At the present time k, the behavior of the process over a horizon N is considered. Using a model the process response to change in the manipulated variable is predicted. The moves of the manipulated variables are selected such that the predicted response has certain desired characteristics. Only the first computed change in manipulated variable is implemented at time k+1 and the computation is repeated with the horizon moved by the time interval. This control strategy has been successfully implemented in a wide range of systems in industrial applications showing good performance and high degree of robust. In a typical plant-wide control system, PID controllers regulate most of the loops and changing these to a full MPC

controller requires a full upgrade of the process control hardware and software. Implementing model-based control may also require capital investment to support new hardware and software products and also resources to train personnel in the operational behavior of the new advanced controllers. Hence, MPC is often applied at a second level of control hierarchy to regulate the set points [3]. On the other hand, control constraints are major problem in design and implementation of control strategies. Among the earliest approaches to address the problem of input constraints in linear systems is integrator anti-windup, reaching back in the late 60s [8]. This method essentially consists of “switching off” the integrator action when the input signal saturates to avoid excessive built up in the control input. Ref [2] uses an observer inside the controller structure in order to adjust the states of the anti-windup compensator. Ref [7] surveys the field and proposes a model based H-infinity optimal strategy. In a different approach, the problem of input saturation and windup protection has been treated within a linear matrix inequality (LMI) framework [17]. Alternatively, actuator saturation can be posed as a constraint within the framework of model predictive control (MPC), and [6] present direct links between MPC designs and conventional anti-windup schemes. Increase in use of complex processes and growing need for robust controllers introduces a new field in academia, which is advanced techniques effective for controlling complex systems with simple structures like PID controllers. [10]. Many researchers have attempted to restrict these advanced methods to retrieve a PID structure, suitable to practical implementation in industrial applications. In [16] an Internal Model Control (IMC) based controller design for first order process model is proposed and [4] extended IMC-PID controller to cover the second order process model. The limitation of these methods is that tuning rules are derived for the delay free system. In [13] a PID control law mathematically equivalent to GPC was proposed. This was done by equating the discrete PID control law with linear form of GPC. Although there is no restriction on choice of GPC tuning parameters, the process model order is restricted to maximum of two. In [19] a least square algorithm was used to compute the closest equivalent PID controller to an IMC design, but still ineffective for time-delay and unstable systems. In [12] a model-based PID controller was used by utilizing multiple Distributed Control System (DCS) PID blocks to implement a model based predictive control

strategy. In our previous work [11] we have presented a predictive PID algorithm which has similar features to MPC. Corresponding to a prediction horizon of size M, a bank of M parallel conventional PID controllers is defined. Three term PID gains were calculated using optimal control signal matching with GPC. An important feature of this scheme was related to hardware implementation and current operator’s practice. If there exists a PID controller regulating the system, changing to a MPC strategy implies the removal of all the components currently in use. Such changes involve the approval of the operator and training in the use of new tuning rules of new control algorithm. The approach of this paper is based on existing PID controller configuration and requires only little extra additional hardware, because for most of typical industrial applications the calculations can be done off-line and be tested on a computer away from the actual system and then downloaded to the appropriate storage device. And more ever there are no changes from the point of view of operator’s training; the operator can continue to manage the plant in the same manner used before. In [15] we have proposed a genetic algorithm approach to predictive PID controller design and tuning in which the optimal values of Predictive PID gains were obtained using Genetic Algorithms. Using output signal matching to GPC, Predictive PID gains were found. As an extension of our previous works we will use an Integral of Absolute value of Error (IAE) index to our optimization problem and also consider the control signal in our objective function order to handle control amplitude constraints. In this approach no approximations will be used to achieve a straightforward answer and the values of Predictive PID gains are derived directly by genetic algorithm. By assigning an appropriate value to parameter α which is the weighting factor control signal in optimization objective function, control signal constraints including amplitude and rate of change could be handled in an optimal way. In this way the rate of change in output of the system could also be directly managed. The configuration of this paper is as follows: In section 2 Predictive PID structure will discussed. In section 3 the optimal values of Predictive PID gains will be calculated using genetic algorithms. In section 4 constraint handling capability of proposed method will be discussed. In section 5 the performance of proposed method will be studied. Finally conclusions close the paper. II. PREDICTIVE PID STRUCTURE A. Conventional PID structure PID controllers are used in industrial applications in different forms and configurations depending on process characteristics. The textbook discrete PID configuration has the following form: t ⎡ ⎤ u (t ) = ⎢k p e(t ) + ki ∑ e(t ) + k d [e(t ) − e(t − 1)]⎥ i =1 ⎣ ⎦

(1)

Or the controller could be written as:

q0 + q1 z −1 + q2 z −2 e(t ) (2) 1 − z −1 Where k p , ki and k d are the proportional, integral and u (t ) =

derivative gains respectively, and

q0 = k p + k i + k d ,

q1 = − k p − 2k d , q2 = k d . Taking the difference on both sides of the above equation, one can obtain the velocity form of PID controller:

Δu (t ) = u (t ) − u (t − 1) {k p [e(t ) − e(t − 1)] + ki e(t )

(3)

+ k d [e(t ) − 2e(t − 1) + e(t − 2)]} This can be written in matrix form as:

Δu(t ) = Ke(t ) = K[r (t ) − y (t )]

[

K = kp

ki

(4)

⎡0 − 1 1⎤ kd ⎢⎢0 0 1⎥⎥ ⎢⎣1 − 2 1⎥⎦

y (t ) = [ y (t − 2)

]

y (t )]

T

y (t − 1)

e(t ) = [e(t − 2) e(t − 1) e(t )]

T

r (t ) = [r (t − 2) r (t − 1) r (t )]

T

Compared to the popular MPC methods, the conventional PID controllers lack several features: 1. There is no mechanism to incorporate the future set point information if they are known. 2. The majority of the model-based tuning methods to tune PID controllers assume a second order model as an approximation of the process. 3. The control input constraints may be dealt with indirectly using integral wind-up methods. B. Predictive PID Structure Consider a control signal which is calculated by adding the output of M PID controllers, where the i-th PID operates on the error at future time (t+i). M can be also considered as the prediction horizon of the controller. Such a control signal is defined by [14]: M

M

i =0

i=0

Δu(t ) = K ∑ e(t + i ) = K ∑ [r (t + i ) − y (t + i )]

(5)

Where K is a vector of gains as defined in equation 4. r(t) is the reference signal, and

y (t + i ) = [ y (t + i − 2)

y (t + i − 1)

y (t + i )]

T

e(t + i ) = [e(t + i − 2) e(t + i − 1) e(t + i )]

T

r (t + i ) = [r (t + i − 2) r (t + i − 1) r (t + i )]

T

It has been proven that the controller of equation (5) is equivalent to M parallel PID controllers of the same three term gains, with the i-th PID controller operating on the error e at time (t+i) [11].

To implement the proposed controller the values of the error over the horizon {t: 1,…, M} are required. These values may be predicted using a model of the plant. We assume a state space model of process as follows: x k +1 = Ax k + BΔu k (6)

y k = Cx k + DΔu k In which X denotes state vector, y denotes process outputs,

Δu denotes control input and A, B, C, D are the matrices

defining the state-space model. Ordinarily for real processes D =0. Using equation (6) one can write the relations recursively for M-step ahead prediction [18]:

0 "⎤ ⎡ Δu k ⎤ ⎡ x k +1 ⎤ ⎡ A ⎤ ⎡ B ⎢ x ⎥ ⎢ A2 ⎥ ⎢ AB B "⎥⎥ ⎢⎢ Δu k +1 ⎥⎥ ⎢ k +2 ⎥ = ⎢ ⎥ x + ⎢ (7) ⎢ # ⎥ ⎢ # ⎥ k ⎢ # # # ⎥⎢ # ⎥ ⎥ ⎢ ⎥ ⎢ M⎥ ⎥⎢ ⎢ M −1 M −2 "⎦ ⎣Δu k + M −1 ⎦ ⎣A B A ⎣x k + M ⎦ ⎣ A ⎦ 0 "⎤ ⎡ Δu k ⎤ ⎡ yk +1 ⎤ ⎡ CA ⎤ ⎡ B ⎢ y ⎥ ⎢ CA 2 ⎥ ⎢ CAB CB "⎥⎥ ⎢⎢ Δu k +1 ⎥⎥ ⎢ k +2 ⎥ = ⎢ ⎥xk + ⎢ ⎢ # ⎥ ⎢ # ⎥ ⎢ # # # ⎥⎢ # ⎥ ⎢ ⎥ ⎢ M⎥ ⎥ ⎢ M −1 ⎥⎢ M −2 "⎦ ⎣ Δu k + M −1 ⎦ ⎣CA B CA ⎣ yk + M ⎦ ⎣CA ⎦ Assuming control horizon

N u =1, Δuk can be written in

terms of Predictive PID gains as follows: M

M

i=0

i =0

Δu k (t ) = K ∑ e(t + i ) = K ∑ [r (t + i ) − y (t + i )] Where K, y and r are as defined in (4). The block diagram of Predictive PID structure is shown in figure (1). III. OPTIMAL PREDICTIVE PID GAINS A. Genetic algorithms Genetic Algorithms have been introduced in early 70s by Holland [5]. In last decades GAs have attracted lots of attention to themselves. These algorithms simulate the biological theories of “survival of the fittest” and “natural selection”. Genetic algorithms provide a good search tool for a wide range of science and engineering optimizations. Instead of starting from just one point to find optimal values, GAs have a set of initial values called “initial population”. Because of this, GAs have a high degree of robustness and high probability to find the global optimum. GAs work with the coded version of problem instead of optimization problem itself, i.e. the optimization parameters are encoded in a string of binary or real digits called “chromosome”. This property makes GAs appropriate for controller parameter tuning. GAs work with 3 major operators: selection, crossover, and mutation. The selection operator selects two chromosomes based on their fitness value. Fitness value is assigned to every chromosome based on their properness to solve the problem (e.g. a quadratic cost function). There are several methods to select “parents” to produce “offspring”. Proportional and

Ranking are two well-known methods. In order to produce the next generation, GAs use another operator called crossover. The crossover operator uses parent’s chromosomes to produce the offspring. From a chosen point of chromosome the value of genes are swapped. This is called one-point crossover. Multi-point crossovers are also available. To help the search algorithm to escape local optimums and to cover wider range of search zone, GAs use mutation operator. According to the assigned value of mutation rate, this operator changes the value of digits in chromosome. B. Optimal values of predictive PID gains In section II structure of predictive PID controller has been discussed. The controller gets M future errors as inputs and takes a control action according to those errors signals. The control action took here is a conventional PID action. The control signal can be written as a linear combination of each conventional PID’s control signal. Sets of three controller parameters are the same for each PID controller. These three parameters of predictive PID should be tuned to the process to achieve a desirable closed-loop performance. In this section parameters of predictive PID controller are found using Genetic Algorithms. Here no approximations will be used to obtain optimal values of Predictive PID gains. The objective function is defined as follows: n

n

i =1

i =1

F = ∑ r − yiout + α ∑ Δui

(8)

The objective function is defined to make the output of the process as close as possible to set point with reasonable amount of control effort. The weighting factor α could be tuned to meet desired output trajectory. The effect of variation of α will be studied in section (4). The values of future error signals are found using equation (7). These error signals are fed to predictive PID and appropriate control signal is produced. The output of the plant is used to calculate the objective of genetic algorithm. One advantage of this method is that no approximations will be used to simplify the equations in order to obtain a satisfactory and straightforward answer. The objective function can be defined as complex as needed to solve the optimization problem. Optimization of the predictive PID gains is done offline. That’s because the genetic algorithms are usually computationally expensive. But once the PID gains are obtained they could be used in process control without any extra computational burden. The three Predictive PID gains are tuned to the process away from the real process itself, and then will be simulated to examine its efficiency. Then Predictive PID gains are saved in an appropriate storage device. These gains are used in Predictive PID structure presented in section II, to generate appropriate control signal. In each run of genetic algorithm we simulate the process for a step test signal. In our optimization we have minimized IAE index and also considering control signal needed to

control the system for a double step means a step in t=1 and another in t=50. In this way the result could be better generalized for other responses. For example when we minimize the cost function for a single step response, and then obtain optimal gains from the optimization, the obtained gains work well in a single step response. But maybe for another test signals the response will not be satisfactory (multiple steps or ramp signal). So in our optimization we have used two steps to optimize the response. The results show that the optimal gains obtained here could be used in a general form for any other test signals like multiple steps or ramp signal. In first run of GA, the first population is produced in which the individuals inside the population are coded versions of Predictive PID gains. These individuals are used to simulate the process for a double step test signal. The value of the fitness function is then calculated for each individual. According to the value of the fitness function, the next generation of the Predictive PID gains is produced. In first generations of Predictive PID gains, the mutation operator plays an important role. Because the value of the fitness function in first generations is very high; sometimes it approaches infinity. This is so because the first population individuals are generated randomly. By mutating the individuals a better population can be achieved. After several generations the population starts to converge. The order of the fitness function for the best individual of the population becomes as less as order of two or three. So adaptive mutation will be best suited mutation function for the purpose of our optimization. Because mutations in middle stage of the optimization process make the fitness function diverge. So an adaptive mutation function in which has a high mutation rate when fitness function order is high and low mutation rate when the population starts to converge. The overall procedure is shown in figure 2. IV. CONSTRAINT HANDLING All real world control systems must deal with constraints. The constraints acting on a process can originate from amplitude limits on control signal, slew rate limits of the actuators and limits on output signals. As a result of constraints, the actual plant input will be different from the output of the controller. When this happens, the controller output does not drive the plant as expected. Constraints on control signals, which are the input of the process, or manipulated variable are present in most of control systems. Most commonly these constraints are from the saturation characteristics of the valves with finite range of adjustment, flow rates with maximum values due to fixed pipe diameter, or control surfaces with limited deflection angles. Input constraints also appear in the in the form of rate constraints: valves and other actuators with limited slew rates. These constraints, especially of the saturation type, are also often active when a process is running at its most profitable condition. They may limit production rate, for example In a

semi-batch reactor, the reactant flow rate and the cooling water flow rate can both be adjusted between minimum and maximum limits. The economic objective is to finish the batch as quickly as possible. In practice, that is equivalent to keeping the reactant flow as large as possible. One of the advantages of MPC over conventional PID is the ease with which constraints can be handled. To handle control signal constraints we set the weighting factor α to an appropriate value. In this way the amount of control signal and its amplitude will be taken into account in optimization process. The Genetic algorithm will find the optimal solution of the problem in a way to balance the IAE error and the control signal effort used to drive the system. As mentioned earlier the objective of optimization is to make the output of the process as close as possible to the set point with a reasonable amount of control effort. The weighting factor α plays an important rule here. Variation of α from zero to 10 has been studied and settling time and total amount of control signal has been calculated (table 1). As the parameter α varies from 0 to 10, settling time also increases. when α =0, the response reaches the set point value within prediction horizon i.e. step response rises M seconds before changing the set point and reaches it in time instance that we change the set point. But in most of real industrial processes the amplitude of control signal is very important and because of actuator constraints the amplitude of control signal is limited to certain values. By increasing the value of α amplitude and rate of change in control signal decreases. By setting α to an appropriate value the constraints of system could be dealt with easily. Also the settling time of the system could be tuned be means of α . Figure (3) shows an example of variation of parameter α with other parameters of the system fixed. For the system with α =1 the response is quick but the amount of control signal and maximum signal amplitude is more than that of system with α =10. V. SIMULATION RESULTS Genetic algorithm optimized predictive PID controller has been designed and implemented for several benchmark systems. Results presented here are of a second order unstable system. The optimal values of Predictive PID gains were obtain through the procedure discussed in section III and the weighting factor α is set to 5. The results are compared with well known MPC method, Generalized Predictive Controller (GPC). Results reveal that this control scheme is effective and gives satisfactory performance compared with GPC. One advantage of proposed method over GPC is that this method could handle the constraints more easily without the need of QP problem solving. The control signal of Predictive PID controller is smoother than that of GPC so in practice it is so much easier to generate the Predictive PID control signal. Figure (4) shows that for a similar performance less control effort is needed.

VI. CONCLUSIONS In this paper a new method to optimize the Predictive PID gains was proposed. The optimal value of this predictive type PID controller was calculated using Genetic Algorithms. A constraint handling method in optimization problem was introduced which can effectively handle the constraints without solution of QP problem. By setting the parameters of objective function of optimization problem the constraints on control input could be handled. In this way the rate of change in output of the process could be directly managed. Simulation results reveal that this method is effective and gives similar performance in comparison with MPC schemes. This method can be applied to systems of any order without any approximations.

TABLE 1: VARIATION OF AND ITS EFFECT ON CONTROL OBJECTTIVES

α=

Settling Time(s)

0 0.1 0.25 0.5 1 2 5 10

4 4 4 5 7 12 16 20

Total Control Effort 1.6350 0.9568 0.8360 0.6723 0.5139 0.4609 0.3536 0.2836

Max Control Signal Amplitude 0.815 0.447 0.415 0.308 0.255 0.227 0.166 0.098

Min Control Signal Amplitude -0.658 -0.215 -0.187 -0.118 -0.065 -0.051 -0.030 -0.019

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Prediction d step Prediction d+1 step r(k)

Set-point Prediction

Δ u(k)

w(k) +

e(k)

+

PID

SYSTEM

Prediction d+M step

Figure 1: Block diagram of Predictive PID structure Generate Initial Population

Evaluate the Solution

Meet the Stopping Criterion

Yes

No Apply GA Operators

Produce New Population

Select the Optimal Gains from the Population

Figure 2: overall algorithm of genetic algorithm optimization

y(k)

Figure 3: step response of different choices of weighting factor.

Figure 4: comparison between step response of proposed method (red) and GPC method (blue).