CN114114909B - Intermittent process 2D output feedback prediction control method based on particle swarm optimization - Google Patents
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Abstract
The invention discloses an intermittent process 2D output feedback prediction control method based on particle swarm optimization, which is characterized by comprising the following steps: step 1: establishing an uncertainty two-dimensional intermittent process state space model with a bounded norm; step 2: designing a two-dimensional iterative learning output feedback prediction controller; step 3: and designing a 2D output feedback robust predictive tracking controller optimized based on a particle swarm algorithm. The invention designs the controller with the extension information, which has better tracking performance and robustness and stronger resistance to system uncertainty and external interference. The output feedback prediction controller realizes the control target and meets the actual production requirement by considering the output and input constraints and the unmeasurable state in the actual production. The use of PSO algorithms to find better solutions near the legacy controller compensates for the effect that some parameters of the performance index function may not be optimal due to manual adjustment.
Description
Technical Field
The invention belongs to the technical field of information, and particularly relates to an intermittent process 2D output feedback prediction control method based on particle swarm optimization.
Background
For intermittent processes with repetitive characteristics, ILC (iterative learning) is a good choice. However, the use of ILC alone for batch processes results in poor convergence and stability due to environmental factors and variations in operating conditions, particularly chemical batch processes, which do not repeat well, i.e., batch processes have a degree of non-repeatability (or uncertainty). The MPC (model predictive control) is widely applied in the production process which cannot be accurately modeled, so that the combination design of the ILC and the MPC has important significance for batch production under a two-dimensional framework.
In an actual system, the state of the system is not easy to directly measure, or the state feedback cannot be realized physically due to the limitation of the measuring equipment in terms of economy and usability, so that the iterative learning robust prediction control based on the output feedback has more practical significance. Meanwhile, in order to prevent overshoot in the production process, a certain limit needs to be given to input and output, however, the research results for the problem are few at present. Since the essence of MPC optimization is to achieve optimization by manually adjusting the parameter variables of the performance index, this is not consistent with today's high-efficiency and high-precision control. The genetic algorithm is used as an intelligent search algorithm, and has strong advantages in the intermittent process optimization problem. Compared with a genetic algorithm, the particle swarm optimization algorithm has the characteristics of simplicity, fewer parameters, easiness in implementation, no need of coding, high convergence speed and the like. The control performance can be further improved if the controller parameters are optimized by using a particle swarm optimization algorithm.
Disclosure of Invention
Aiming at the defects in the prior art, the invention provides an intermittent process 2D output feedback prediction control method based on particle swarm optimization. The method designs the controller with the extended information, and the controller has better tracking performance and robustness and stronger resistance to system uncertainty and external interference.
In order to solve the problems in the prior art, the invention adopts the following technical scheme:
the intermittent process 2D output feedback prediction control method based on particle swarm optimization comprises the following steps:
step 1: establishing an uncertainty two-dimensional intermittent process state space model with a bounded norm;
step 2: designing a two-dimensional iterative learning output feedback prediction controller;
step 3: and designing a 2D output feedback robust predictive tracking controller optimized based on a particle swarm algorithm.
Further, the two-dimensional intermittent process state space model is as follows:
wherein T is the time, k is the batch, T p To terminate the moment, x (t, k) ∈R n ,y(t,k)∈R l ,u(t,k)∈R m ,w(t,k)∈R n Respectively represent the firstState, output, input and disturbance signals at time k of t batch, { A, B, C } are n×n, n×m, l×n dimensional matrices, respectively, ΔA (t, k), ΔB (t, k) represents uncertainty of system parameters.
Further, the step 2 includes the steps of:
step 2.1: designing a two-dimensional iterative learning prediction controller:
t＝0,1,2,...,T p ；l,m＝0,1,2,...,
where u (t+l|t, k+m|k) represents the predicted value of the input variable at time t of k lot,
r(t+l|t,k+m|k)∈R m representing an iterative update law to be designed at the moment of t of k batches, and r (t|t, k|k) =r (t, k), and u (t, 0) represents an initial value of iteration;
step 2.2: introducing expansion information to obtain an expanded two-dimensional closed-loop state space model;
wherein:
step 2.3: designing an output feedback prediction controller:
wherein x is c (t+l|t,k+m|k)∈R n+2l Is the internal state of the controller, { A ci ,B ci ,C ci ,D ci } i＝1,2 The controller parameters are (n+2l) × (n+2l), (n+2l) ×3l, m× (n+2l), and m×3l dimensions, respectively.
Further, the step 3 includes the following steps:
step 3.1: selecting a performance index function:
step 3.2: solving the controller:
step 3.3: and optimizing the controller parameters obtained based on the particle swarm algorithm.
Further, the performance index function in step 3.1 is:
wherein V is m (z (t+n|t, k+n|k)) is a terminal constraint;
when the interference is non-repetitive, at infinity, [ t, ] and [ k, ] times, [ t, ] are, a "worst case" performance indicator is defined as the kth batch at time t of the uncertainty system:
further, the constraint conditions of the performance index function are:
wherein Y is 1 ＝[D c1 G C c1 ],Y 2 ＝[D c2 G C c2 ]，Q 1 ,Q 2 R is a corresponding weight matrix, gamma is more than 0, R m ,y m The variables r (t+l|t, k+m|k) and y, respectively e Upper bound of (t+l|t, k+m|k) [ A B C]e.OMEGA.where Ω is an uncertainty set.
Further, the output feedback prediction controller parameters of the step 3.2 solving controller are designed as follows:
wherein the full order matrix M, N satisfies the condition XY+MN T =i, obtainable by singular value decomposition of matrix I-XY.
Further, the speed update formula and the position update formula for optimizing the obtained controller parameters based on the particle swarm algorithm in the step 3.3 are respectively:
v i,k+1 ＝wv i,k +c 1 ξ(p i,k -x i,k )+c 2 η(p g,k -x i,k )
x i,k+1 ＝x i,k +v i,k+1 (18)
wherein v is i,k Velocity vector v for the ith particle at the kth iteration i,0 For a given initial iteration speed of the ith particle, x i,k For the position of the ith particle at the kth iteration, w is the inertial weight, c 1 And c 2 For learning factors or acceleration factors, ζ and η are two [0,1]]Random numbers uniformly distributed on, p i,k For the optimal position of the ith particle at the kth iteration, p g ,k Is the global optimum at the kth iteration.
Further, the optimizing the controller parameters based on the particle swarm algorithm in the step 3.3 includes the following steps:
3.31: let x be the controller A c1 ,B c1 ,C c1 ,D c1 ,A c2 ,B c2 ,C c2 ,D c2 Vectors of all elements in the list;
3.32: an objective function J (x) to be optimized is determined.
3.33: initializing a particle swarm;
3.34: and solving a local optimal solution of x.
Further, the step 3.33 initializes the particle swarm, including: initializing a population size N, a position x of each particle i And velocity v i And with constraints for each particle positionIs the jth component of the position of the ith particle at the kth iteration, with the number of iterations k=300.
The invention has the advantages and beneficial effects that:
the invention designs a controller with extended information aiming at an intermittent process with uncertainty and interference by combining Iterative Learning Control (ILC) and predictive control (MPC) under the framework of a two-dimensional system theory, and the controller has better tracking performance and robustness and stronger resistance to system uncertainty and external interference. The output feedback controller realizes the control target and meets the actual production requirement by considering the output and input constraints and the unmeasurable state in the actual production. The use of PSO algorithms to find better solutions near the legacy controller compensates for the effect that some parameters of the performance index function may not be optimal due to manual adjustment.
Drawings
The invention is described in detail below with reference to the attached drawing figures:
FIG. 1 is a flow chart of an intermittent process 2D output feedback predictive control method based on particle swarm optimization;
FIG. 2 is a diagram of a 2D output feedback prediction control ILC framework in accordance with the present invention;
FIG. 3 is a graph showing a comparison of tracking performance of the system of the present invention with or without expansion information under repetitive disturbance;
FIG. 4 is a graph showing a comparison of tracking performance of whether a particle swarm optimization is adopted when the system has expansion information under repetitive disturbance;
FIG. 5 is a graph of the output trace of the system without expansion information under repetitive disturbance in the present invention;
FIG. 6 is a graph of output trace of the system with expansion information under repetitive disturbance in the present invention;
FIG. 7 is a graph of the output trace of the system of the present invention optimized using a particle swarm algorithm when there is information of the expansion under repetitive disturbance.
Detailed Description
The present invention will be described in further detail with reference to the following examples, but the scope of the present invention is not limited to the examples, and the claims should be construed. In addition, any modification or variation which can be easily realized by those skilled in the art without departing from the technical scheme of the present invention falls within the scope of the claims of the present invention.
As shown in fig. 1, the intermittent process 2D output feedback prediction control method based on particle swarm optimization of the invention comprises the following steps:
step 1: establishing an uncertainty two-dimensional intermittent process state space model with a bounded norm;
the two-dimensional intermittent process state space model is as follows:
wherein T is the time, k is the batch, T p To terminate the moment, x (t, k) ∈R n ,y(t,k)∈R l ,u(t,k)∈R m ,w(t,k)∈R n The states, outputs, inputs, perturbation signals, { A, B, C } are n, n m, l n dimensional matrices, respectively, at time k of the t-th batch, ΔA (t, k), ΔB (t, k) represents the uncertainty of the system parameters, and is assumed to have the following structural form:
ΔA(t,k)＝EΔ(t,k)F 1 ,ΔB(t,k)＝EΔ(t,k)F 2 wherein E, F 1 ,F 2 Respectively n×1,1×n,1×m-dimensional constant arrays, and Δ (t, k) is unknown parameter perturbation and satisfies Δ T (t, k) delta (t, k) is less than or equal to I, and I is a first-order identity matrix.
Step 2: based on the repeated characteristic and the two-dimensional characteristic of the intermittent process, a two-dimensional iterative learning output feedback prediction controller is designed by combining the prediction control and the output feedback, and the expansion information is introduced, so that a two-dimensional closed-loop state space model of the intermittent process is obtained;
the step 2 comprises the following steps:
step 2.1: using predictive control theory, let x (t+l|t, k+m|k), y (t+l|t, k+m|k), u (t+l|t, k+m|k) represent predicted values of the state variable, the output variable, and the input variable at the time of k batch t, respectively, for the uncertainty system model given in (1), the following two-dimensional iterative learning predictive controller is designed:
t＝0,1,2,...,T p ；l,m＝0,1,2,...,
wherein R (t+l|t, k+m|k) ∈R m Represents an iterative update law to be designed at time t of k batches, and r (t|t, k|k) =r (t, k), and u (t, 0) represents an initial value of the iteration.
Step 2.2: introducing expansion information, and combining the expansion information with the two-dimensional closed-loop state space model (2) to obtain an expanded two-dimensional closed-loop state space model;
defining an output tracking error of the system:
e(t+l|t,k+m|k)＝y(t+l|t,k+m|k)-y r (t), (3)
wherein y is r And (t) is the set output tracking value.
Defining an error function in the batch direction:
where σ may take input, output or state variables, and r (t+l|t, k+m|k) =u e (t+l|t,k+m|k)。
From the formulae (1), (2), (3), (4):
when (when)In the case of a repetitive disturbance, when +.>And is a non-repetitive disturbance.
Introducing new state variables
An extended equivalent 2D system is as follows:
wherein:
step 2.3: when only output is detectable in the process, the output feedback prediction controller is designed by combining the 2D output feedback prediction control ILC framework diagram of the figure 2:
wherein x is c (t+l|t,k+m|k)∈R n+2l Is the internal state { A } of the controller ci ,B ci ,C ci ,D ci } i＝1,2 The controller parameters are (n+2l) × (n+2l), (n+2l) ×3l, m× (n+2l), and m×3l dimensions, respectively. Order theSubstituting (10) into (9) to obtain the following 2D control system:
wherein the method comprises the steps of
Step 3: 2D output feedback robust predictive tracking controller design based on particle swarm optimization.
The step 3 comprises the following steps:
step 3.1: selecting a performance index function:
when the interference is non-repetitive, at infinity, [ t, ] and [ k, ] times, [ t, ] are, a "worst case" performance indicator is defined as the kth batch at time t of the uncertainty system:
wherein V is m (z (t+N|t, k+N|k)) is referred to as a terminal constraint
The constraint conditions are as follows:
wherein Y is 1 ＝[D c1 G C c1 ],Y 2 ＝[D c2 G C c2 ]，Q 1 ,Q 2 R is a corresponding weight matrix, gamma is more than 0, R m ,y m The variables r (t+l|t, k+m|k) and y, respectively e Upper bound of (t+l|t, k+m|k) [ A B C]e.OMEGA.where Ω is an uncertainty set.
Step 3.2: the stability of the system (13) is proved by utilizing the Lyapunov stability theorem, and the Lyapunov function is defined as follows:
ΔV(z(t+l+1|t,k+m|k))＝ΔV h (z(t+l+1|t,k+m|k))+ΔV v (z(t+l+1|t,k+m|k))
＝V h (z(t+l+1|t,k+m|k))-V h (z(t+l|t,k+m|k))+V v (z(t+l+1|t,k+m|k))-V v (z(t+l+1|t,k+m-1|k))
V(z(t+l|t,k+m|k))＝V h (z(t+l|t,k+m|k))+V v (z(t+l|t,k+m|k))
wherein P is 1 ,P 2 Are all undetermined positive definite matrixes, and satisfy the following conditions:
αP 1 +βP 2 ＜P,α＞1,β＞1,P＞0,P＝θ 1 L -1 (15)
the closed loop system (13) is sufficiently robust to be conditioned by the presence of a positive definite symmetric matrix P, P 1 ,P 2 The method (16) is established,
order the
Converting (16) to the following formula:
with respect to the input limit of the expression (13),can be converted into:
with respect to the output limit of the expression (13),can be converted into:
let theta 1 As the upper boundary of V (z (t, k)), the following formula is required to hold:
if X, Y,is the matrix inequality (17 a) - (17 d)If a solution is possible, the parameters of the controller (10) outputting the feedback can be designed as follows:
wherein the full order matrix M, N satisfies the condition XY+MN T =i, obtainable by singular value decomposition of matrix I-XY.
Step 3.3: and optimizing the controller parameters obtained based on the particle swarm algorithm.
v i,k+1 ＝wv i,k +c 1 ξ(p i,k -x i,k )+c 2 η(p g,k -x i,k )
x i,k+1 ＝x i,k +v i,k+1 (18)
Wherein v is i,k Velocity vector v for the ith particle at the kth iteration i,0 For a given initial iteration speed of the ith particle, x i,k For the position of the ith particle at the kth iteration, w is the inertial weight, c 1 And c 2 For learning factors or acceleration factors, ζ and η are two [0,1]]Random numbers uniformly distributed on, p i,k For the optimal position of the ith particle at the kth iteration, p g ,k Is the global optimum at the kth iteration.
The step 3.3 of optimizing the controller parameters based on the particle swarm algorithm comprises the following steps:
3.31: is provided with x Is composed of a controller A c1 ,B c1 ,C c1 ,D c1 ,A c2 ,B c2 ,C c2 ,D c2 Vectors of all elements in the list; it is a decision variable in the optimization problem, and xε R 40 。
3.32: determining an objective function J (x) to be optimized: our goal is: obtainingj=1, 2,..40, where x is j Is the j-th component of x->Is x * The jth component x * Controller parameters determined for conventional methodsVectors of all elements, i.e. x * The optimal position mentioned above is the position at which the performance index J (x) takes the minimum value, which is the initial iterative position in (18).
3.33: initializing a particle swarm: population size N, position x of each particle i And velocity v i And with constraints for each particle positionIs the jth component of the position of the ith particle at the kth iteration, with the number of iterations k=300.
3.34: and solving a local optimal solution of x.
In short, a better solution is found near the controller obtained by the traditional method by using the particle swarm optimization algorithm, so that the function value J of the performance index is as small as possible. In the above constraint particle swarm algorithm, the total number of initial particles is 100, the inertia weight is set to 0.5, and the learning factor c is set 1 And c 2 And k is equal to or less than 1.5 and is equal to or less than 300 (the iteration number is 300).
Example 1:
the injection molding process is a typical chemical industrial production process mainly comprising a multi-stage production mode, and each product produced mainly comprises five steps, namely a mold closing section, an injection section, a pressure maintaining section, a cooling section and a mold opening section. Parameters such as injection speed of the injection section need high-precision control to realize the increase of the yield of the final product. Here we take the injection section as an example, considering its control effect.
The control speed parameter is taken as a study object. First, the response of the injection speed (here, the process output) of the comparative example valve (here, the input) was determined as an autoregressive model, and a frequency domain mathematical model of the injection section of the injection molding process was established as follows:
wherein, IV injection speed, VO is valve opening.
The state space model of the model (19) is expressed as:
wherein the variable delta (t, k) varies randomly over the [0,1] range.
When the disturbance is a repetitive disturbance, it can be seen from the simulation experiment that the state space model of the injection phase is the formula (20), where ω (t, k) is the disturbance of the injection phase and satisfies ω=cos (t) × [ 0.1.0.2] T . In this case, the external disturbance ω (t, k) is determined only by time.
When the disturbance is a non-repetitive disturbance, in this case, robustness against non-repetitive disturbances will be shown. Assuming that the real-time dynamics of the system is shown in equation (20), where the non-repetitive perturbations ω (t, k) satisfy ω=0.3× [ Δ ] 1 Δ 2 ] T 。Δ i (i=1, 2, 3) in [0,1] along the time direction]Directly randomly and non-repeating in the batch direction, i.e. ω (t, k) depends on both t and k.
Fig. 3, fig. 4, fig. 5, fig. 6, and fig. 7 show the control effects obtained in the simulation of the present embodiment, respectively. Fig. 3 shows a comparative graph of the tracking performance of the system with or without the expansion information under the repetitive disturbance, and it can be seen that the tracking performance of the system with the expansion information is obviously better than that of the system without the expansion information, and the system can converge to a stable state more quickly in about 5 batches and approaches zero error tracking. Therefore, under the condition of repetitive disturbance, the system with the expansion information has better tracking performance. Fig. 4 is a graph comparing tracking performance of the system optimized by the particle swarm algorithm when the system has expansion information under the repetitive disturbance, and it can be seen that the tracking performance of the system optimized by the particle swarm algorithm is obviously better than that of the system not optimized by the particle swarm algorithm, and the system can converge to a stable state more quickly in about 4 batches and is close to zero error tracking. Therefore, when the expansion information exists under the repetitive disturbance, the system optimized by adopting the particle swarm optimization has better tracking performance. The output tracking graphs of fig. 5, 6 and 7 further show that the tracking effect of the controller which is designed in the present text, has expansion information and is optimized by adopting the particle swarm algorithm is better. In summary, the controller obtained by using the intermittent process 2D output feedback prediction control method based on particle swarm optimization has better control effect, has more use value in actual production, can generate smaller working errors while realizing quick tracking performance, ensures the product quality to improve the production efficiency, can save energy and reduce consumption, and provides a practical control method for solving the uncertainty problem of the intermittent production process.
While the preferred embodiments of the present invention have been described above, those skilled in the art may make various modifications and additions to the specific embodiments described above, and such modifications and additions should also be considered as being within the scope of the present invention.
Claims (4)
1. The intermittent process 2D output feedback prediction control method based on particle swarm optimization is characterized by comprising the following steps of:
step 1: establishing an uncertainty two-dimensional intermittent process state space model with a bounded norm;
the two-dimensional intermittent process state space model is as follows:
wherein T is the time, k is the batch, T p To terminate the moment, x (t, k) ∈R n ，y(t，k)∈R l ，u(t，k)∈R m ，w(t，k)∈R n Respectively representing the state, output, input and disturbance signals at time k of the t-th batch, { A, B, C } is an n×n, n×m, l×n-dimensional matrix, respectively, ΔA (t, k), ΔB (t, k) represents the system parametersUncertainty of the number;
step 2: designing a two-dimensional iterative learning output feedback prediction controller, wherein the huge head comprises the following steps:
step 2.1: designing a two-dimensional iterative learning prediction controller:
t＝0，1，2，...，T p ；l，m＝0，1，2，...，
where u (t+1|t, k+m|k) represents the predicted value of the input variable at time t of the k batch, R (t+1|t, k+m|k) ∈R m Representing an iterative update law to be designed at the moment of t of k batches, and r (t|t, k|k) =r (t, k), and u (t, 0) represents an initial value of iteration;
step 2.2: introducing expansion information to obtain an expanded two-dimensional closed-loop state space model;
wherein:
step 2.3: designing an output feedback prediction controller:
wherein x is c (t+l|t，k+m|k)∈R n+21 Is the internal state of the controller, { A ci ，B ci ，C ci ，D ci } i＝1，2 Respectively (n+2 l) × (n+2 l), (n+2 l) ×3l, m× (n+2 l), m×3l controller parameters
Step 3: designing a 2D output feedback robust predictive tracking controller optimized based on a particle swarm algorithm; the method comprises the following steps:
step 3.1: selecting a performance index function:
the performance index function in step 3.1 is:
wherein V is m (z (t+n|t, k+n|k)) is referred to as a terminal constraint;
when the interference is non-repetitive, at infinity, [ t, ] and [ k, ] times, [ t, ] are, a "worst case" performance indicator is defined as the kth batch at time t of the uncertainty system:
the constraint conditions of the performance index function are as follows:
wherein Y is 1 ＝[D c1 G C c1 ]，Y 2 ＝[D c2 G C c2 ]，Q 1 ，Q 2 R is a corresponding weight matrix, gamma is more than 0, R m ，y m The variables r (t+l|t, k+m|k) and y, respectively e Upper bound of (t+l|t, k+m|k) [ A B C]E Ω, Ω is an uncertainty set;
step 3.2: solving a controller;
the output feedback predictive controller parameters of the solving controller are designed as follows:
wherein the full order matrix M, N satisfies the condition XY+MNT= I, and can be obtained according to singular value decomposition of the matrix I-XY;
step 3.3: and optimizing the controller parameters obtained based on the particle swarm algorithm.
2. The intermittent process 2D output feedback prediction control method based on particle swarm optimization according to claim 1, wherein the speed update formula and the position update formula for optimizing the obtained controller parameters based on the particle swarm optimization in step 3.3 are respectively:
v i，k+1 ＝wv i，k +c 1 ξ(p i，k -x i，k )+c 2 η(p g，k -x i，k )
x i，k+1 ＝x i，k +v i，k+1 (18)
wherein v is i，k Velocity vector v for the ith particle at the kth iteration i，0 For a given initial iteration speed of the ith particle, x i，k For the position of the ith particle at the kth iteration, w is the inertial weight, c 1 And c 2 For learning factors or acceleration factors, ζ and η are two [0,1]]Random numbers uniformly distributed on, p i，k For the optimal position of the ith particle at the kth iteration, p g，k Is the global optimum at the kth iteration.
3. The intermittent process 2D output feedback prediction control method based on particle swarm optimization according to claim 1, wherein the optimizing the obtained controller parameters based on the particle swarm optimization in step 3.3 comprises the following steps:
3.31: let x be the controller A c1 ,B c1 ,C c1 ,D c1 ,A c2 ,B c2 ,C c2 ,D c2 Vectors of all elements in the list;
3.32: determining an objective function J (x) to be optimized;
3.33: initializing a particle swarm;
3.34: and solving a local optimal solution of x.
4. The intermittent process 2D output feedback prediction control method based on particle swarm optimization according to claim 1, wherein the initializing the particle swarm in step 3.33 comprises: initializing a population size N, a position x of each particle i And velocity v i And with constraints for each particle positionIs the jth component of the position of the ith particle at the kth iteration, with the number of iterations k=300.
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