MIMO different-factor compact-form model-free control

Abstract

The invention discloses a MIMO different-factor compact-form model-free control method. In view of the limitations of the existing MIMO compact-form model-free control method with the same-factor structure, namely, at time k, different control inputs in the control input vector can only use the same values of penalty factor and step-size factor, the invention proposes a MIMO compact-form model-free control method with the different-factor structure, namely, at time k, different control inputs in the control input vector can use different values of penalty factors and/or step-size factors, which can solve control problems of strongly nonlinear MIMO systems with different characteristics between control channels widely existing in complex plants. Compared with the existing control method, the inventive method has higher control accuracy, stronger stability and wider applicability.

Claims

1. A method of MIMO different-factor compact-form model-free control, executed on a hardware platform for controlling a controlled plant being a multi-input multi-output (MIMO) system, wherein the MIMO system having a predetermined number of control inputs and a predetermined number of system outputs, said controlled plant comprises at least one of: a reactor, a distillation column, a machine, a device, a set of equipment, a production line, a workshop, and a factory, said hardware platform comprises at least one of: an industrial control computer, a single chip microcomputer controller, a microprocessor controller, a field programmable gate array controller, a digital signal processing controller, an embedded system controller, a programmable logic controller, a distributed control system, a fieldbus control system, an industrial control system based on internet of things, and an industrial internet control system, said MIMO different-factor compact-form model-free control method comprising: calculating the i-th control input u.sub.i(k) at time k as follows: $u_i\left(k\right)=u_i\left(k-1\right)+\frac{{\rho }_i\underset{j=1}{\overset{n}{.Math.}}{\varphi }_{j,i}\left(k\right)e_j\left(k\right)}{{\lambda }_i+{.Math.\Phi \left(k\right).Math.}^2}$ where k is a positive integer; n is the total number of system outputs in said MIMO system, n is a positive integer; i denotes the i-th of the total number of control inputs in said MIMO system, i is a positive integer, 1≤i≤m, where m is the total number of control inputs in said MIMO system and m is a positive integer greater than 1; j denotes the j-th of the total number of system outputs in said MIMO system, j is a positive integer, 1≤j≤n; u.sub.i(k) is the i-th control input at time k; e.sub.j(k) is the j-th error at time k, namely the j-th element in the error vector e(k)=[e.sub.1(k), . . . , e.sub.n (k)].sup.T; Φ(k) is the estimated value of pseudo partitioned Jacobian matrix for said MIMO system at time k, ϕ.sub.ji(k) is the j-th row and the i-th column of matrix Φ(k), ∥Φ(k)∥ is the 2-norm of matrix Φ(k); λ.sub.i is the penalty factor for the i-th control input; ρ.sub.i is the step-size factor for the i-th control input; for said MIMO system, calculating a control input vector u(k)=[u.sub.1(k), . . . ,u.sub.m(k)].sup.T, by traversing all values of i in the positive integer interval [1, m]; said MIMO different-factor compact-form model-free control method has a different-factor characteristic; said different-factor characteristic is that at least one of the following two inequalities holds true for any two unequal positive integers i and x in the positive integer interval [1, m] during controlling said MIMO system by using said MIMO different-factor compact-form model-free control method:
λ.sub.i≠λ.sub.x;ρ.sub.i≠ρ.sub.x; obtaining the system outputs from the MIMO system by adjusting the control inputs of the MIMO system based on the calculated control input vector, such that the system outputs of the MIMO system approach desired system outputs to be received by the hardware platform.

2. The method as claimed in claim 1 wherein said j-th error e.sub.j(k) at time k is calculated by the j-th error function; independent variables of said j-th error function comprise the j-th desired system output and the j-th actual system output.

3. The method as claimed in claim 2 wherein said j-th error function adopts at least one of: e.sub.j(k)=y.sup.*.sub.j(k)−y.sub.j(k), e.sub.j(k)=y.sup.*.sub.j(k+1)−y.sub.j(k), e.sub.j(k)=y.sub.j(k)−y.sup.*.sub.j(k), and e.sub.j(k)=y.sub.j(k)−y.sup.*.sub.j(k+1), where y.sup.*.sub.j(k) is the j-th desired system output at time k, y.sup.*.sub.j(k+1) is the j-th desired system output at time k+1, and y.sub.j (k) is the j-th actual system output at time k.

Description

BRIEF DESCRIPTION OF THE DRAWINGS

(1) FIG. 1 shows a schematic diagram according to the embodiments of the invention;

(2) FIG. 2 shows the tracking performance of the first system output when controlling the two-input two-output MIMO system in the first exemplary embodiment by using the inventive MIMO different-factor compact-form model-free control method;

(3) FIG. 3 shows the tracking performance of the second system output when controlling the two-input two-output MIMO system in the first exemplary embodiment by using the inventive MIMO different-factor compact-form model-free control method;

(4) FIG. 4 shows the control inputs when controlling the two-input two-output MIMO system in the first exemplary embodiment by using the inventive MIMO different-factor compact-form model-free control method;

(5) FIG. 5 shows the tracking performance of the first system output when controlling the two-input two-output MIMO system in the first exemplary embodiment by using the existing MIMO compact-form model-free control method with the same-factor structure;

(6) FIG. 6 shows the tracking performance of the second system output when controlling the two-input two-output MIMO system in the first exemplary embodiment by using the existing MIMO compact-form model-free control method with the same-factor structure;

(7) FIG. 7 shows the control inputs when controlling the two-input two-output MIMO system in the first exemplary embodiment by using the existing MIMO compact-form model-free control method with the same-factor structure;

(8) FIG. 8 shows the tracking performance of the first system output when controlling the two-input two-output MIMO system of coal mill in the second exemplary embodiment;

(9) FIG. 9 shows the tracking performance of the second system output when controlling the two-input two-output MIMO system of coal mill in the second exemplary embodiment;

(10) FIG. 10 shows the first control input when controlling the two-input two-output MIMO system of coal mill in the second exemplary embodiment;

(11) FIG. 11 shows the second control input when controlling the two-input two-output MIMO system of coal mill in the second exemplary embodiment.

DETAILED DESCRIPTION OF THE INVENTION

(12) The invention is hereinafter described in detail with reference to the embodiments and accompanying drawings. It is to be understood that other embodiments may be utilized and structural changes may be made without departing from the scope of the invention.

(13) FIG. 1 shows a schematic diagram according to the embodiments of the invention. For a MIMO system with m inputs (m is a positive integer greater than 1) and n outputs (n is a positive integer), the MIMO different-factor compact-form model-free control method is adopted to control the system; for the i-th control input u.sub.i(k) (i=1, . . . ,m), set the parameters value including penalty factor λ.sub.i and step-size factor ρ.sub.i in the mathematical formula for calculating u.sub.i(k) using the MIMO different-factor compact-form model-free adaptive control method; mark the current moment as time k; define the difference between the j-th desired system output y*.sub.j(k) and the j-th actual system output y.sub.j(k) as the j-th error e.sub.j(k); traverse all values of j in the positive integer interval [1, n] and obtain the error vector e(k)=[e.sub.1(k), . . . ,e.sub.n(k)].sup.T; based on the error vector e(k)=[e.sub.1(k), . . . ,e.sub.n(k)].sup.T at time k, the penalty factor λ.sub.i, and the step-size factor ρ.sub.i, calculate the i-th control input u.sub.i(k) at time k using the MIMO different-factor compact-form model-free adaptive control method; for the MIMO system, traverse all values of i in the positive integer interval [1, m] and calculate the control input vector u(k)=[u.sub.1(k), . . . ,u.sub.m(k)].sup.T at time k using the MIMO different-factor compact-form model-free control method; obtain the n actual system outputs at next time by applying the control input vector u(k) into the controlled plant, and then carry out the MIMO different-factor compact-form model-free control method for the next sampling time.

(14) Two exemplary embodiments of the invention are given for further explanation.

(15) The first exemplary embodiment:

(16) A two-input two-output MIMO system, which has the complex characteristics of non-minimum phase nonlinear system, is adopted as the controlled plant, and it belongs to the MIMO system that is particularly difficult to control:

(17) $y_1\left(k\right)=\frac{2.5y_1\left(k-1\right)y_1\left(k-2\right)}{1+y_1^2\left(k-1\right)+y_2^2\left(k-2\right)+y_1^2\left(k-3\right)}+0.09u_1\left(k-1\right)u_1\left(k-2\right)+1.2u_1\left(k-1\right)+1.6u_1\left(k-3\right)+0.5u_2\left(k-1\right)+0.7\sin \left(0.5\left(y_1\left(k-1\right)+y_1\left(k-2\right)\right)\right)\cos \left(0.5\left(y_1\left(k-1\right)+y_1\left(k-2\right)\right)\right)$$y_2\left(k\right)=\frac{5y_2\left(k-1\right)y_2\left(k-2\right)}{1+y_2^2\left(k-1\right)+y_1^2\left(k-2\right)+y_2^2\left(k-3\right)}+u_2\left(k-1\right)+1.1u_2\left(k-2\right)+1.4u_2\left(k-3\right)+0.5u_1\left(k-1\right)$

(18) The desired system outputs y*(k) are as follows:
y*.sub.1(k)=5 sin(k/50)+2 cos(k/20)
y*.sub.2(k)=2 sin(k/50)+5 cos(k/20)

(19) In this embodiment, m=n=2.

(20) In view of the above specific embodiment, two experiments are carried out for comparison and verification. The first experiment adopts the inventive control method, and the second experiment adopts the existing control method. In order to compare the control performance of the two experiments clearly, root mean square error (RMSE) is used as the control performance index for evaluation:

(21) $\mathrm{RMSE}\left(e_j\right)=\sqrt{\frac{1}{N}\underset{k=1}{\overset{N}{.Math.}}{e}^2\left(k\right)}$

(22) where e.sub.j(k)=y*.sub.j(k)−y.sub.j(k), y*.sub.j(k) is the j-th desired system output at time k, y.sub.j(k) is the j-th actual system output at time k. The smaller the value of RMSE(e.sub.j) is, the smaller the error between the j-th actual system output and the j-th desired system output is, and the better the control performance gets.

(23) The hardware platform for running the inventive control method is the industrial control computer.

(24) The first experiment (RUN1): the inventive MIMO different-factor compact-form model-free control method is adopted to control the above two-input two-output MIMO system; set the parameters value for calculating the first control input: the penalty factor λ.sub.1=0.01, the step-size factor ρ.sub.1=0.37; set the parameters value for calculating the second control input: the penalty factor ρ.sub.2=0.24, the step-size factor ρ.sub.2=0.49; the tracking performance of the first system output and the second system output are shown in FIG. 2 and FIG. 3, respectively, and the control inputs are shown in FIG. 4; evaluate the control method from the control performance indexes: the RMSE(e.sub.1) of the first system output in FIG. 2 is 2.2947, and the RMSE(e.sub.2) of the second system output in FIG. 3 is 1.1935.

(25) The second experiment (RUN2): the existing MIMO compact-form model-free control method with the same-factor structure is adopted to control the above two-input two-output MIMO system; set the penalty factor λ=0.01, the step-size factor ρ=0.50; the tracking performance of the first system output and the second system output are shown in FIG. 5 and FIG. 6, respectively, and the control inputs are shown in FIG. 7; evaluate the control method from the control performance indexes: the RMSE(e.sub.1) of the first system output in FIG. 5 is 7.8933, and the RMSE(e.sub.2) of the second system output in FIG. 6 is 10.5405.

(26) The comparison results of control performance indexes of the two experiments are shown in Table 1; the results of the first experiment (RUN1) using the inventive control method are superior to those of the second experiment (RUN2) using the existing MIMO compact-form model-free control method with the same-factor structure, and the control performance improvement is significant, indicating that the inventive MIMO different-factor compact-form model-free control method has higher control accuracy, stronger stability and wider applicability.

(27) TABLE-US-00001 TABLE 1 Comparison Results of The Control Performance The first system output The second system output RMSE(e.sub.1) Improvement RMSE(e.sub.2) Improvement RUN1 2.2947 70.926% 1.9135 80.846% RUN2 7.8933 Baseline 10.5405 Baseline

(28) The second exemplary embodiment:

(29) A coal mill is a very important set of equipment that pulverizes raw coal into fine powder, providing fine powder for the pulverized coal furnace. Realizing the control of coal mill with high accuracy, strong stability and wide applicability is of great significance to ensure the safe and stable operation of thermal power plant.

(30) The two-input two-output MIMO system of coal mill, which has the complex characteristics of nonlinearity, strong coupling and time-varying, is adopted as the controlled plant, and it belongs to the MIMO system that is particularly difficult to control. Two control inputs u.sub.1(k) and u.sub.2(k) of the coal mill are hot air flow (controlled by the opening of hot air gate) and recycling air flow (controlled by the opening of recycling air gate), respectively. Two system outputs y.sub.1(k) and y.sub.2(k) of the coal mill are outlet temperature (° C.) and inlet negative pressure (Pa), respectively. The initial conditions of the coal mill are: u.sub.1(0)=80%, u.sub.2(0)=40%, y.sub.1(0)=70° C., y.sub.2(0)=−400 Pa. At the 50th second, in order to meet the needs of on-site conditions adjustment in thermal power plant, the desired system output y*.sub.1(50) is adjusted from 70° C. to 80° C., and the desired system output y*.sub.2(k) is required to remain unchanged at −400 Pa. In view of the above typical conditions in thermal power plant, two experiments are carried out for comparison and verification. In this embodiment, m=n=2. The hardware platform for running the inventive control method is the industrial control computer.

(31) The third experiment (RUN3): the inventive MIMO different-factor compact-form model-free control method is adopted to control the above two-input two-output MIMO system; set the parameters value for calculating the first control input: the penalty factor λ.sub.1=0.05, the step-size factor ρ.sub.1=1.95; set the parameters value for calculating the second control input: the penalty factor λ.sub.2=0.05, the step-size factor ρ.sub.2=1.92; the tracking performance of the first system output is shown as RUN3 in FIG. 8, the tracking performance of the second system output is shown as RUN3 in FIG. 9, the first control input is shown as RUN3 in FIG. 10, and the second control input is shown as RUN3 in FIG. 11; evaluate the control method from the control performance indexes: the RMSE(e.sub.1) of the first system output, RUN3 in FIG. 8, is 2.6338, and the RMSE(e.sub.2) of the second system output, RUN3 in FIG. 9, is 0.1882.

(32) The fourth experiment (RUN4): the existing MIMO compact-form model-free control method with the same-factor structure is adopted to control the above two-input two-output MIMO system; set the penalty factor λ=0.07, the step-size factor ρ=2; the tracking performance of the first system output is shown as RUN4 in FIG. 8, the tracking performance of the second system output is shown as RUN4 in FIG. 9, the first control input is shown as RUN4 in FIG. 10, and the second control input is shown as RUN4 in FIG. 11; evaluate the control method from the control performance indexes: the RMSE(e.sub.1) of the first system output, RUN4 in FIG. 8, is 2.8286, and the RMSE(e.sub.2) of the second system output, RUN4 in FIG. 9, is 0.2266.

(33) The comparison results of control performance indexes of the two experiments are shown in Table 2; the results of the third experiment (RUN3) using the inventive control method are superior to those of the fourth experiment (RUN4) using the existing MIMO compact-form model-free control method with the same-factor structure, and the control performance improvement is significant, indicating that the inventive MIMO different-factor compact-form model-free control method has higher control accuracy, stronger stability and wider applicability.

(34) TABLE-US-00002 TABLE 2 Comparison Results of The Control Performance of Coal Mill The first system output The second system output RMSE(e.sub.1) Improvement RMSE(e.sub.2) Improvement RUN3 2.6338 6.887% 0.1882 16.946% RUN4 2.8286 Baseline 0.2266 Baseline

(35) Furthermore, the following three points should be noted in particular:

(36) (1) In the fields of oil refining, petrochemical, chemical, pharmaceutical, food, paper, water treatment, thermal power, metallurgy, cement, rubber, machinery, and electrical industry, most of the controlled plants, such as reactors, distillation columns, machines, equipment, devices, production lines, workshops and factories, are essentially MIMO systems; some of these MIMO systems have the complex characteristics of non-minimum phase nonlinear system, which belong to the MIMO systems that are particularly difficult to control; for example, the continuous stirred tank reactor (CSTR), commonly used in oil refining, petrochemical, chemical, etc., is a two-input two-output MIMO system, where the two inputs are feed flow and cooling water flow, and the two outputs are product concentration and reaction temperature; when the chemical reaction has strong exothermic effect, the continuous stirred tank reactor (CSTR) is a MIMO system with complex characteristics of non-minimum phase nonlinear system, which is particularly difficult to control. In the first exemplary embodiment, the controlled plant with two inputs and two outputs also has the complex characteristic of non-minimum phase nonlinear system and belongs to the MIMO system that is particularly difficult to control; the inventive controller is capable of controlling the plant with high accuracy, strong stability and wide applicability, indicating that it can also achieve high accuracy, strong stability and wide applicability control on complex MIMO systems such as reactors, distillation columns, machines, equipment, devices, production lines, workshops, factories, etc.

(37) (2) In the first and second exemplary embodiments, the hardware platform for running the inventive controller is the industrial control computer; in practical applications, according to the specific circumstance, a single chip microcomputer controller, a microprocessor controller, a field programmable gate array controller, a digital signal processing controller, an embedded system controller, a programmable logic controller, a distributed control system, a fieldbus control system, an industrial control system based on internet of things, or an industrial internet control system, can also be used as the hardware platform for running the inventive control method.

(38) (3) In the first and second exemplary embodiments, the j-th error e.sub.j(k) is defined as the difference between the j-th desired system output y*.sub.j(k) and the j-th actual system output y.sub.j(k), namely e.sub.j(k)=y*.sub.j(k)−y.sub.j(k), which is only one of the ways for calculating the j-th error; the j-th error e.sub.j(k) can also be defined as the difference between the j-th desired system output y.sub.j(k+1) at time k+1 and the j-th actual system output y.sub.j(k), namely e.sub.j(k)=y*.sub.j(k+1)−y.sub.j(k); the j-th error e.sub.j(k) can also be defined by other ways whose independent variables include the j-th desired system output and the j-th actual system output, for example,

(39) $e_j\left(k\right)=\frac{y_j^{*}\left(k+1\right)+y_j^{*}\left(k\right)}{2}-y_j\left(k\right);$
for the controlled plants in the first and second exemplary embodiments, all different definitions of the error function can achieve good control performance.

(40) It should be appreciated that the foregoing is only preferred embodiments of the invention and is not for use in limiting the invention. Any modification, equivalent substitution, and improvement without departing from the spirit and principle of this invention should be covered in the protection scope of the invention.