A Parameter Tuning Method for a Three-Phase Model Predictive Controller

Abstract

The present invention discloses a method for tuning the parameters of a three-phase model predictive controller. A penalty term for the increment of the predicted output is added to the optimization proposition compared with the traditional MPC algorithm, improving the control quality and enhancing the robustness of the closed-loop system at the same time. The parameter tuning steps are as follows: conduct an identification experiment on the controlled object, collect input and output data, perform model identification calculations to obtain the object model; determine the model time domain, prediction time domain, and control time domain according to the model of the controlled object; normalize the input and output variables of the controlled object; determine the prediction error weighting coefficient and the predicted output increment weighting coefficient from the desired closed-loop first-order step response time; given the upper and lower bounds of the input overshoot, determine the input increment weighting coefficient through multiple closed-loop step test simulations to ensure that the overshoot of the control quantity is within the constraint range. The parameter tuning method of the present invention is simple, intuitive and effective, greatly simplifying the complex MPC tuning work and realizing high-quality and high-robustness MPC control.

Description

Technical Field

[0001] The present invention belongs to the field of industrial control, relates to a model predictive control (MPC) technology, and particularly relates to a method for tuning parameters of a three-term model predictive controller (Three-term MPC). Background Art

[0002] Model predictive control (MPC) is an advanced process control technology (algorithm). It was proposed by Richalet [1] and Culter [2] in the late 1970s. Now it has become a standard advanced process control technology (APC) that can handle multivariable control problems and constrained control problems. The MPC control technology has been popularized in the refining and petrochemical industries and has begun to be applied in other process industries. Dynamic matrix control (DMC) is a model predictive control (MPC) algorithm. The characteristics of DMC [3] are: (1) using a linear step response model; (2) using a quadratic performance objective function in a finite time domain; (3) the calculation of the unconstrained optimal input variable is a linear least squares problem. The traditional DMC algorithm first establishes a model of the object, uses the output of the model to predict the output of the object, and then obtains the optimal control quantity by minimizing an objective function. The objective function includes: (1) a weighted term of the output error (W(t) - Y P (t)); (2) a weighted term of the input increment ΔU(t). In order to reduce the fluctuation of the closed-loop system output, some scholars have proposed an improved DMC algorithm [4][5] , which adds a weighted term of the predicted output increment ΔY P (t) to the objective function compared with the traditional DMC algorithm. This method can reduce the output fluctuation, but it also generates a problem, that is, there is currently no simple and effective method for theoretical analysis and controller parameter tuning of such three-term DMC controllers. When applied to a multivariable system, the tuning of the weighting coefficients will become very complicated.

[0003] [1]J.Richalet,A.Rault,J.L.Testud and J.Papon(1978).Model predictiveheuristic control:Applications to industrial processes.Automatica,14(5),413-428.

[0004] [2]C.R.Cutler and B.L.Ramaker(1980).Dynamic matrix control-a computer control algorithm.In 1980 Joint Automatic Control Conference,17,72.

[0005] [3]S.Qin and T.A.Badgwell(2003).A survey of industrial model predictive control technology.Control Engineering Practice,11(7),733-764.

[0006] [4]W.Zhang,B.Liu and F.Kang(2008).A Fast GPC Algorithm with Output Penalty.The First International Conference on Intelligent Networks and Intelligent Systems,425-428.

[0007] [5]H.W.Gomma and H.Yu.Derivative derived generalised predictive control(DDGPC):a technique for improving performance(2005).IEEE Conference on Control Applications,1152-1157. Summary of the Invention

[0008] Aiming at the deficiencies of the prior art, the present invention provides a method for tuning the parameters of a three-term model predictive controller. By adding a penalty term for the increment of the predicted output to the optimization proposition of the controller and combining the parameter adjustment rules, the closed-loop response characteristics desired by the user are achieved, and the output variance of the closed-loop system is reduced without sacrificing the robustness of the closed-loop system. The present invention is simple and convenient to implement and has important practical value. Since parameter tuning only affects the dynamic characteristics of the MPC control system, for the sake of simplicity of description, only the unconstrained MPC control algorithm is discussed.

[0009] The optimization proposition of the traditional two-term model predictive controller (MPC) algorithm is as follows:

[0010]

[0011] Among them, W(t) represents the set value (curve), Y P (t) represents the predicted output, ΔU(t) represents the input increment, Q is the prediction error weighting coefficient matrix, and R is the input increment weighting coefficient matrix; the predicted output Y P (t) can be expressed as:

[0012] Y P (t) = Y P0 (t) + AΔU(t) (2)

[0013] Among them, Y P0 (t) is the prediction initial value (assuming that the future input increment ΔU(t) is 0), A is the dynamic matrix, and it is composed of the step response sequence a ij (t) of the controlled object model,

[0014]

[0015]

[0016] Among them, P is the prediction time domain, and M is the control time domain. Under unconstrained conditions, the optimization proposition has an analytical solution:

[0017] ΔU(t) = (A T QA + R) -1 A T Q(W(t) - Y P0 (t)) (5)

[0018] The parameter tuning of the binomial model predictive controller involves the determination of the prediction time domain P, the control time domain M, and the model time domain N, which can be obtained from the step response of the controlled object model. Usually, N and P should be close to the time when the step response reaches the steady state; the prediction error weighting coefficient matrix Q and the input increment weighting coefficient matrix R are given by a certain parameter tuning method.

[0019] The optimization proposition of the three - term model predictive controller (MPC) used in the present invention is as follows:

[0020]

[0021] Among them, W(t) represents the set value, Y P (t) represents the predicted output, ΔU(t) represents the input increment, ΔY P (t) represents the predicted output increment, Q is the prediction error weighting coefficient matrix, R is the input increment weighting coefficient matrix, and S is the predicted output increment weighting coefficient matrix; ΔY P (t) is obtained by taking the difference of the predicted output Y P (t),

[0022] ΔY P (t) = T2Y P (t) - T3Y(t) (7)

[0023] Where Y(t) is the output at time t;

[0024]

[0025] Taking the derivative of the objective function J(t) with respect to the input increment ΔU(t) gives:

[0026]

[0027] Then, in the case of no constraints, the optimization proposition has an analytical solution:

[0028]

[0029] Transforming equation (9) gives the following expression:

[0030]

[0031] Where I is the identity matrix, Continuing to simplify gives:

[0032]

[0033] Let Q' = Q(I + T4T2), W'(t) = (I + T4T2) -1 (W(t) + T4T3Y(t)),

[0034] Where,

[0035]

[0036] w′ i (t) represents the equivalent reference trajectory corresponding to the i-th output, w′ i (t) is a vector, i = 1,..., p.

[0037] Assume that the setpoint w i (t) of the i-th output is a constant, such as w i (t) = r i (t) = [1…1] T , where r i (t) is the unit step vector, then the equivalent reference trajectory w′ i (t) of the i-th output can be expressed as:

[0038] w′ i (t) = (I + T4T2) -1 ·(r i(t)+T4T3y i (t)) (14)

[0039] where y i (t) represents the current value of the i-th output and is a scalar.

[0040] By simple shifting, we can obtain:

[0041] (I + T4T2)w′ i (t) = r i (t)+T4T3y i (t) (15)

[0042] where T2 is a difference matrix, so we have:

[0043] T2w′ i (t) = Δw′ i (t)+T3w′ i (t) (16)

[0044] where Δw′ i (t) is the vector after differentiation:

[0045] Δw′ i (t) = [w′ i (t + 1)-w′ i (t),..., w′ i (t + P)-w′ i (t + P - 1)] T (17)

[0046] Therefore, Equation (15) can be expressed as:

[0047] w′ i (t)+T4Δw′ i (t)+T4T3w′ i (t) = r i (t)+T4T3y i (t) (18)

[0048] In Equation (18), w′ i (t) and y i (t) are two scalars, representing the starting point of the reference trajectory and the current output value respectively. Considering that the starting point of the reference trajectory and the current output value should be equal, therefore, the relevant terms can be eliminated to obtain a simplified form:

[0049] w′ i (t)+T4Δw′ i (t) = r i (t) (19)

[0050] Substitute Substituting into Equation (18), we can obtain:

[0051]

[0052]

[0053] where is an accumulation matrix,

[0054]

[0055] Therefore, the left side of Equation (21) is a difference vector, and the right side is an accumulation vector. The one-to-one correspondence can be expressed by a function, and the functional relationship is shown in italics:

[0056]

[0057] where k = 1,..., P. Assuming that the prediction horizon P → ∞ and the sampling time T s → 0, then Equation (23) is converted into a continuous-time expression:

[0058]

[0059] Differentiating both sides of Equation (24) simultaneously, we can obtain:

[0060]

[0061] The general solution of Equation (25) is:

[0062]

[0063] and w′ i (t + k) = r i (t) is a particular solution. Therefore, the complete solution of Equation (25) is:

[0064]

[0065] The solution conditions are:

[0066]

[0067] Solving, we get C1 = 0, C2 = r i (t) - y i (t). In summary, the expression of the equivalent reference trajectory is:

[0068]

[0069] The equivalent reference trajectory shown in Equation (29) is a first-order response curve, starting from the current output y i (t) and ending at the target set value ri (t), and the weight coefficient and the time constant λ of the first-order response curve i have the following relationship:

[0070]

[0071] Based on the above proof, the present invention proposes a parameter tuning method for a three-term model predictive controller, including the following steps:

[0072] S1. Conduct an identification experiment on the controlled object, collect the input data U(t), t = 1,..., L and the output data Y(t), t = 1,..., L, where L is the number of data sampling points, and perform model identification calculations to obtain the object model G(q);

[0073] S2. Determine the model time domain N, prediction time domain P, and control time domain M according to the model of the controlled object;

[0074] S3. According to the input and output data obtained from the identification experiment, perform the following normalization processing on the input and output variables of the controlled object, and give the corresponding weighted coefficient values:

[0075]

[0076]

[0077] where u j (t) and y i (t) are taken from the data obtained from the identification experiment, representing the values of the jth input variable and the ith output variable at time t, respectively, and var(·) represents the variance; k u and k y are scalars to be tuned;

[0078] S4. For p outputs, set the desired closed-loop first-order step response time T i cl ; the ratio of the predicted output increment weighting coefficient s i to the predicted error weighting coefficient q i determines the equivalent closed-loop response time, and the relationship is:

[0079]

[0080] T i cl = nλ i (34)

[0081] where λ i is the time constant of the first-order system, and n is usually taken as 3 or 4. For setting T i cl to the output y iThe time required for the closed-loop step response of (t) to reach 98% of its steady-state value, then n is taken as 4, for T i cl Set to output y i The time required for the closed-loop step response of (t) to reach 95% of its steady-state value, then n is taken as 3;

[0082] As long as the user gives T i cl 、k u and k y The values of the weighting coefficients of matrices Q and S are determined;

[0083] S5, take a set of k u and k y such that the ratio is relatively small, and perform closed-loop step test simulation; gradually increase the ratio k yu , perform closed-loop step test simulation until some input overshoots in the step test reach or exceed the preset upper and lower bounds; finely adjust the weighting coefficient q i such that most of the input overshoots in the step test are close to or touch the preset upper and lower bounds.

[0084] The beneficial effects of the present invention are as follows: Compared with the traditional MPC algorithm, the present invention adds a penalty term for the prediction output increment in the optimization proposition, improves the control quality while enhancing the robustness of the closed-loop system; combined with the parameter adjustment rule, it realizes the closed-loop response characteristics expected by the user, and reduces the output variance of the closed-loop system without sacrificing the robustness of the closed-loop system. The present invention is simple and convenient to implement. Using the parameter tuning method of the present invention, users can simply, intuitively and effectively determine the parameters of a complex, multi-variable model predictive controller, and obtain a high-quality and robust MPC control system, which has important practical value. BRIEF DESCRIPTION OF THE DRAWINGS

[0085] Figure 1 is a flowchart of the parameter tuning method for the three-term model predictive controller of the present invention;

[0086] Figure 2 is the open-loop step response curve of the controlled object provided by an exemplary embodiment;

[0087] Figure 3 is the output curve provided by an exemplary embodiment;

[0088] Figure 4 is the input curve provided by an exemplary embodiment. DETAILED DESCRIPTION OF THE INVENTION

[0089] To make the above objects, features, and advantages of the present invention more obvious and understandable, the following detailed description of the specific embodiments of the present invention will be given in conjunction with the accompanying drawings.

[0090] In the following description, many specific details are set forth in order to provide a thorough understanding of the present invention. However, the present invention may be implemented in other ways different from those described herein. Those skilled in the art can make similar generalizations without departing from the connotation of the present invention. Therefore, the present invention is not limited by the specific embodiments disclosed below.

[0091] The present invention provides a method for tuning the parameters of a three - term model predictive controller. The optimization proposition of the controller includes three weighting terms, namely: the weighting term for the prediction error (W(t) - Y P (t)), the weighting term for the input increment ΔU(t), and the weighting term for the predicted output increment ΔY P (t). Where W(t) is the setpoint and Y P (t) is the predicted output; considering a multivariable controlled object with p outputs and m inputs, the optimization proposition is:

[0092]

[0093] Where Q is the prediction error weighting coefficient matrix, R is the input increment weighting coefficient matrix, and S is the predicted output increment weighting coefficient matrix.

[0094]

[0095]

[0096] Where q i is the prediction error weighting coefficient, r j is the input increment weighting coefficient, s i is the predicted output increment weighting coefficient, i = 1,..., p; j = 1,..., m; The three weighting coefficient matrices Q, R, and S are determined by the three - term model predictive controller parameter tuning method, as Figure 1 shown, and the steps are as follows:

[0097] S1. Conduct an identification experiment on the controlled object, collect the input data U(t), t = 1,..., L and the output data Y(t), t = 1,..., L, where L is the number of data sampling points, and perform model identification calculations to obtain the object model G(q);

[0098] S2. Determine the model time domain N, prediction time domain P, and control time domain M according to the model of the controlled object;

[0099] S3. According to the input-output data obtained from the identification experiment, perform the following normalization processing on the input-output variables of the controlled object, and give the corresponding weighting coefficient values:

[0100]

[0101]

[0102] where u j (t) and y i (t) are taken from the data obtained from the identification experiment, representing the values of the jth input variable and the ith output variable at time t respectively, and var(·) represents the variance; k u and k y are scalars to be tuned;

[0103] S4. For p outputs, set the desired closed-loop first-order step response time T i cl ; the ratio of the predicted output increment weighting coefficient s i to the predicted error weighting coefficient q i determines the equivalent closed-loop response time, and the relationship is:

[0104]

[0105] T i cl =nλ i (41)

[0106] where λ i is the time constant of the first-order system, and n is usually taken as 3 or 4. For setting T i cl to the time required for the closed-loop step response of the output y i (t) to reach 98% of its steady-state value, then n is taken as 4. For setting T i cl to the time required for the closed-loop step response of the output y i (t) to reach 95% of its steady-state value, then n is taken as 3;

[0107] As long as the user gives the values of T i cl , k u and k y , the weighting coefficient values of matrices Q and S are determined;

[0108] S5. Take a set of k u and k y such that the ratio is relatively small (for example, k yu can be set to 1), and perform closed-loop step test simulation; gradually increase the ratio kyu Perform a closed-loop step test simulation until some input overshoots in the step test (e.g., set to about 50%) reach or exceed the preset upper and lower bounds; finely adjust the weighting coefficient q i such that most of the input overshoots in the step test (e.g., set to more than 90%) are close to or touch the preset upper and lower bounds.

[0109] In some embodiments, the model time domain N should include all the dynamic processes of the controlled object; the prediction time domain P should include more than 70% of the dynamic processes of the controlled object; the control time domain M can be selected as 30% - 50% of the prediction time domain P, and take a larger value as much as possible within the allowable range of computing power.

[0110] In some embodiments, when selecting the desired closed-loop first-order step response time T of the closed-loop system i cl , the steady-state time T of the open-loop step response of the controlled object needs to be considered i op :

[0111] (1) Slow controller: Set T i cl =(0.6 - 0.8)T i op ;

[0112] (2) Medium-speed controller: Set T i cl =(0.3 - 0.6)T i op ;

[0113] (3) Fast controller: Set T i cl =(0.1 - 0.3)T i op .

[0114] In some embodiments, the upper and lower bounds of the input variable overshoot in the closed-loop step test are designed by the user according to actual needs and can be divided into three types:

[0115] (1) Smooth control: No input overshoot;

[0116] (2) Medium overshoot control: Input overshoot of 0 - 50%;

[0117] (3) Large overshoot control: Input overshoot of 50% - 200%.

[0118] Taking the following simulation object as an example, the present invention will be described in detail.

[0119] Consider a 2-input 2-output controlled object, whose transfer function is as follows:

[0120]

[0121]

[0122] where q -1 represents the unit delay operator, e.g., q -1 U(t) = U(t - 1); The step response of the controlled object is as follows Figure 2 shown.

[0123] The user gives the closed-loop step set values r1(t) = 1, r2(t) = 1; The expected closed-loop first-order step response times are T1 cl = 16; T2 cl = 8.

[0124] Step 1: Conduct an identification experiment on the controlled object, collect input and output data, perform model identification calculations, and obtain the controlled object model;

[0125] Step 2: According to Figure 2 the step response curve of the controlled object shown, determine the model time domain N = 60, the prediction time domain P = 40, and the control time domain M = 10;

[0126] Step 3: Normalize the input and output variables of the controlled object;

[0127] Step 4: For p outputs, set the expected closed-loop first-order step response times T i cl , T1 cl = 16, T2 cl = 8, and calculate s1 / q1 = 16, s2 / q2 = 4 according to the formula ;

[0128] Step 5: Predetermine the overshoot upper and lower bounds of input 1 and input 2, which are 50% and 50% respectively. Through multiple closed-loop step test simulations, determine r1 = 1, r2 = 1, q1 = 25, q2 = 50 to ensure that the control quantity overshoot is within the constraint range.

[0129] Figure 3 shows the transition curve graphs of outputs y1 and y2, Figure 4 shows the transition curve graphs of inputs u1 and u2. It can be seen that y1 is very close to the equivalent reference trajectory, and y2 also tracks the equivalent reference trajectory well, basically achieving the expected closed-loop response time. However, due to the constraint on the control quantity overshoot, the tracking effect is worse than that of y1.

[0130] As described above, it is only the specific implementation manner of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention can easily think of changes or substitutions, which should all be covered within the protection scope of the present invention.

Claims

1. A method for tuning the parameters of a three-phase model predictive controller, characterized in that, The optimization proposition of the controller includes three weighted terms, namely: the weighted term for the prediction error (W(t) - Y P (t)), the weighted term for the input increment ΔU(t), and the weighted term for the predicted output increment ΔY P (t). Where W(t) is the set value and Y P (t) is the predicted output; considering a multivariable controlled object with p outputs and m inputs, the optimization proposition is: Among them, Q is the prediction error weighting coefficient matrix, R is the input increment weighting coefficient matrix, and S is the predicted output increment weighting coefficient matrix. where q i is the prediction error weighting coefficient, r j is the input increment weighting coefficient, s i is the predicted output increment weighting coefficient, i = 1, ..., p; j = 1, ..., m; the three weighting coefficient matrices Q, R, and S are determined by the three-term model predictive controller parameter tuning method, including the following steps: S1. Conduct an identification experiment on the controlled object, collect the input data U(t), t = 1, ..., L and the output data Y(t), t = 1, ..., L, where L is the number of data sampling points, and perform model identification calculations to obtain the object model G(q). S2. Determine the model time domain N, prediction time domain P, and control time domain M according to the model of the controlled object. S3. According to the input and output data obtained from the identification experiment, perform the following normalization processing on the input and output variables of the controlled object, and give the corresponding weighting coefficient values: where, u j (t) and y i (t) are taken from the data obtained in the identification experiment, representing the values of the j-th input variable and the i-th output variable at time t respectively, and var(·) represents variance; k u and k y are scalars to be tuned; S4. Set the desired closed-loop first-order step response time T for p outputs i cl ; Predict the ratio s of the output increment weighting factor i and the prediction error weighting factor q i which determines the equivalent closed-loop response time. The relationship is as follows: T i cl = nλ i (7) where λ i is the time constant of the first-order system; for setting T i cl as the time required for the closed-loop step response of the output y i (t) to reach 98% of its steady-state value, n is taken as 4, and for setting T i cl as the time required for the closed-loop step response of the output y i (t) to reach 95% of its steady-state value, n is taken as 3; As long as the user gives T i cl , k u and k y values, the weighted coefficient values of matrices Q and S are determined; S5, take a set of k u and k y such that the ratio is relatively small, and perform a closed-loop step test simulation; gradually increase the ratio k yu and perform a closed-loop step test simulation until the overshoot of some inputs in the step test reaches or exceeds the preset upper and lower bounds; finely adjust the weighting coefficient q i so that the overshoot of most inputs in the step test is close to or touches the preset upper and lower bounds.

2. The method for tuning the parameters of the three-item model prediction controller according to claim 1, wherein The optimized proposition includes a weighted term for the predicted output increment ΔY P (t), where ΔY P (t) represents the difference vector of the predicted output Y P (t).

3. The method for tuning the parameters of the three-item model prediction controller according to claim 1, characterized in that In S1, the identified model will be used in the simulation-based parameter tuning method. In the simulation, the identified model is the controlled object.

4. The method for tuning the parameters of the three-term model predictive controller according to claim 1, characterized in that In S1, the identification experiment conducted on the controlled object is a closed-loop identification experiment.

5. The method for tuning the parameters of the three-term model predictive controller according to claim 1, wherein In S2, the model time domain N should include all the dynamic processes of the controlled object; the prediction time domain P should include more than 70% of the dynamic processes of the controlled object; the control time domain M is set to 30% - 50% of the prediction time domain P, and a larger value should be taken as much as possible within the allowable range of computing power.

6. The three - item model predictive controller parameter tuning method according to claim 1, characterized in that In S3, normalize all input and output variables according to the variances of the input and output data of the identification experiment, and give the corresponding weighting coefficients.

7. The method for tuning the parameters of the three-term model predictive controller according to claim 1, wherein In S4, the control quality of the control system is determined by the expected closed-loop first-order step response time T of the closed-loop system, and then the prediction error weighting coefficient q i cl and the prediction output increment weighting coefficient s i cl are determined by the expected closed-loop first-order step response time T i . The relationship among T i , q i cl , and s i is as follows: i ​ 8. The method for tuning the parameters of the three-term model predictive controller according to claim 1, characterized in that In S4, select the desired closed-loop first-order step response time T of the closed-loop system i cl , and it is necessary to consider the steady-state time T of the open-loop step response of the controlled object i op : (1) Slow controller: Set T i cl = (0.6 to 0.8)T i op ; (2) Medium-speed controller: Set T i cl = (0.3 - 0.6)T i op ; (3) Fast controller: Set T i cl = (0.1 - 0.3)T i op .

9. The method for tuning the parameters of the three-term model predictive controller according to claim 1, wherein, In S5, in order to ensure the robustness of the control system, set an upper bound on the overshoot of the input variable in the closed-loop step test and impose restrictions on it; the overshoot of the input variable refers to the excess value of the highest peak value of the input signal relative to the input steady-state value in the closed-loop step response simulation test.

10. The method for tuning the parameters of the three-term model predictive controller according to claim 1, wherein In S5, the upper and lower bounds of the overshoot of the input variable in the closed-loop step test are designed by the user according to actual needs and are divided into three types: (1) Smooth control: no overshoot of the input. (2) Medium overshoot control: 0 - 50% overshoot of the input. (3) Large overshoot control: 50% - 200% overshoot of the input.

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