IMPROVED MODEL-FREE ADAPTIVE CONTROL METHOD

Abstract

The present invention discloses an improved model-free adaptive control method, in particular to an improved method for a compact dynamic linearization model-free adaptive control based on MIMO systems, and belongs to the field of control algorithm design. Firstly, proportional control is added in CFDL-MFAC to improve the problems of low response speed and large overshoot in the original control system. Secondly, an anti-windup control algorithm of an actuator is added in the above control structure so that the actuator does not conduct transfinite operation when reaching the upper or lower saturation limit and the actuator can quickly make a control response when a control instruction enters an unsaturated region again to improve the control accuracy of the system. Then, it is proved through strict analysis that the improved control algorithm can ensure tracking error and BIBO stability under certain conditions. Finally, the above control algorithm is applied to an aero-engine control system, and the effectiveness and superiority of the above control algorithm can be obtained by numerical experiments.

Claims

1. An improved model-free adaptive control method, comprising steps of: step A: analyzing the existing method for the compact dynamic linearization model-free adaptive control, and from experimental results, finding that the application process has deficiencies in response time and stability; expressing MIMO discrete-time nonlinear systems as follows:
y(k+1)=f(y(k), . . . ,y(k−n.sub.y),u(k), . . . ,u(k−n.sub.u))  (1) wherein u(k) and y(k) are system inputs and system outputs at time k, respectively; n.sub.y and n.sub.u are two unknown integers; f( . . . )=(f.sub.1( . . . ), . . . , f.sub.m( . . . )) is an unknown nonlinear function; when f has a continuous partial derivative condition and formula (1) satisfies a generalized Lipschitz condition, expressing formula (1) as the following CFDL data model form: $\begin{array}{cc}\Delta y\left(k+1\right)={\Phi }_c\left(k\right)\Delta u\left(k\right)\mathrm{wherein}& \left(2\right)\end{array}$ ${\Phi }_c\left(k\right)=\left[\begin{array}{cccc}{\varphi }_{11}\left(k\right)& {\varphi }_{12}\left(k\right)& .Math.& {\varphi }_{1m}\left(k\right)\\ {\varphi }_{21}\left(k\right)& {\varphi }_{22}\left(k\right)& .Math.& {\varphi }_{2m}\left(k\right)\\ .Math.& .Math.& .Math.& .Math.\\ {\varphi }_{m1}\left(k\right)& {\varphi }_{m2}\left(k\right)& .Math.& {\varphi }_{mm}\left(k\right)\end{array}\right]\in R^{m\times m};$ firstly, proposing the following assumptions: assumption 1: Φ.sub.c(k) as a pseudo Jacobian matrix of the system shall be a diagonal dominant matrix which satisfies the following conditions: |ϕ.sub.ij|≤b.sub.1,b.sub.2≤|.sub.ii(k)|≤αb.sub.2,α≥1,b.sub.2>b.sub.1(2α+1)(m−1), i=1, . . . , m, j=1, . . . , m, i≠j; b.sub.1 and b.sub.2 are set as bounded constants, i and j are set as row and column indexes of the matrix; and the signs of all elements in Φ.sub.c(k) remain the same at any time k; expressing a control input criterion function as formula (3):
J(u(k))=∥y*(k+1)−y(k+1)∥.sup.2+λ∥u(k)−u(k−1)∥.sup.2  (3) wherein λ>0 represents a weight factor, which is used to punish the change of excessive control input quantity; y*(k+1) is a desired output signal; substituting formula (2) into formula (3), deriving u(k) and making the equation equal to zero to obtain a control input algorithm as follows: $\begin{array}{cc}u\left(k\right)=u\left(k-1\right)+\frac{{\mathrm{\rho \Phi }}_c^T\left(k\right)\left(y^{*}\left(k+1\right)-y\left(k\right)\right)}{\lambda +{.Math.{\Phi }_c\left(k\right).Math.}^2}& \left(4\right)\end{array}$ considering the following parameter estimation criteria function:
J(Φ.sub.c(k))=∥Δy(k)−Φ.sub.c(k)Δu(k−1)∥.sup.2+μ∥Φ.sub.c(k)−{circumflex over (Φ)}.sub.c(k−1)∥.sup.2  (5) wherein μ is a weight factor used to punish excessive changes in PJM estimates; {circumflex over (Φ)}.sub.c(k) is an estimate of Φ.sub.c(k); deriving Φ.sub.c(k) in formula (5) and making the equation equal to zero to obtain a parameter estimation algorithm as follows: $\begin{array}{cc}{\hat{\Phi }}_c\left(k\right)={\hat{\Phi }}_c\left(k-1\right)+\frac{\eta \left(\Delta y\left(k\right)-{\hat{\Phi }}_c\left(k-1\right)\Delta u\left(k-1\right)\right)\Delta u^T\left(k-1\right)}{\mu +{.Math.\Delta u\left(k-1\right).Math.}^2}& \left(6\right)\end{array}$ conducting parameter estimation in each k by the above control parameter estimation algorithm to provide control inputs at the time; however, the calculation of the parameter estimation algorithm needs to occupy a certain time, causing slow system response and causing the control algorithm to be limited in use for a system with a small requirement for a control period; and the system vibrates greatly under non-ideal conditions from the experimental results; step B: based on the above problems of slow response and vibration, considering the following improved solution;
Δu(k)=Δu.sub.m(k)′+Δu.sub.p(k)  (7) wherein u.sub.m(k)′ is MFAC controller output, and Δu.sub.p(k) is proportional controller output expressed by the following formulas: $\begin{array}{cc}\Delta u_m\left(k\right)=\frac{{\mathrm{\rho \Phi }}_c^T\left(k\right)\left(y^{*}\left(k+1\right)-y\left(k\right)\right)}{\lambda +{.Math.{\Phi }_c\left(k\right).Math.}^2}& \left(8\right)\end{array}$ $\begin{array}{cc}\Delta u_p\left(k\right)=\beta K\left(y^{*}\left(k+1\right)-y\left(k\right)\right)-\beta K\left(y^{*}\left(k\right)-y\left(k-1\right)\right)& \left(9\right)\end{array}$ proposing the following anti-windup algorithm as part of the proposed control algorithm: stopping updating an integrator when an actuator is at an upper saturation limit and there is still a growing trend, or when the actuator is at a lower saturation limit and is still decreasing; otherwise, the integrator works normally; that is, in the case of saturation, only the integral operations that help to reduce the degree of saturation are performed, and expressed by the following formulas: $\begin{array}{cc}\Delta {u_m\left(k\right)}^{\prime }=\Delta u_m\left(k\right)f\left(k\right)& \left(10\right)\end{array}$ $\begin{array}{cc}f\left(k\right)=\{\begin{array}{cc}0,& \begin{array}{c}u\left(k\right)>\mathrm{u\_max}.Math.\Delta u\left(k\right)>0,\\ u\left(k\right)<\mathrm{u\_min}.Math.\Delta u\left(k\right)<0\end{array}\\ 1,& \mathrm{otherwise}\end{array}& \left(11\right)\end{array}$ wherein u_max and u_min are the upper and lower limitations of the actuator; proposing the following control solution based on formulas (6), (7), (8) and (9): $\begin{array}{cc}{\hat{\varphi }}_{\mathrm{ii}}\left(k\right)={\hat{\varphi }}_{\mathrm{ii}}\left(1\right),\mathrm{if}.Math.{\hat{\varphi }}_{\mathrm{ii}}\left(k\right).Math.<b_2\mathrm{or}.Math.{\hat{\varphi }}_{\mathrm{ii}}\left(k\right).Math.>\alpha b_2\mathrm{or}& \left(12\right)\end{array}$ $\mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ii}}\left(k\right)\right)\ne \mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ii}}\left(1\right)\right)i=1,.Math.,m$ $\begin{array}{cc}{\hat{\varphi }}_{\mathrm{ij}}\left(k\right)={\hat{\varphi }}_{\mathrm{ij}}\left(1\right),\mathrm{if}.Math.{\hat{\varphi }}_{\mathrm{ij}}\left(k\right).Math.>b_1\mathrm{or}\mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ij}}\left(k\right)\right)\ne \mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ij}}\left(1\right)\right),i\ne j& \left(13\right)\end{array}$ $u\left(k\right)=u\left(k-1\right)+\frac{{\mathrm{\rho \Phi }}_c^T\left(k\right)\left(y^{*}\left(k+1\right)-y\left(k\right)\right)}{\lambda +{.Math.{\Phi }_c\left(k\right).Math.}^2}f\left(k\right)+\beta K\left(y^{*}\left(k+1\right)-y\left(k\right)\right)-\beta K\left(y^{*}\left(k\right)-y\left(k-1\right)\right)$ wherein {circumflex over (ϕ)}.sub.ij(1) is an initial value of {circumflex over (ϕ)}.sub.ij(k), i=1, . . . , m; j=1, . . . , m; step C: for the above improved control algorithm, analyzing the convergence of tracking error and the stability of bounded input and bounded output through theoretical derivation; firstly, defining the following output errors of the system:
e(k)=y*(k)−y(k)  (15) substituting formula (2) and formula (14) into formula (15), and when f(k)=1, obtaining: $\begin{array}{cc}\begin{array}{c}e\left(k+1\right)=e\left(k\right)-{\Phi }_e\left(k\right)\Delta u\left(k\right)\\ =[I-\left(\frac{{\mathrm{\rho \Phi }}_e\left(k\right){\hat{\Phi }}_c^T\left(k\right)}{\lambda +{.Math.{\hat{\Phi }}_c\left(k\right).Math.}^2}+\beta {\Phi }_c\left(k\right)K\right)\\ e\left(k\right)+{\mathrm{\beta \Phi }}_c\left(k\right)\mathrm{Ke}\left(k-1\right)\end{array}& \left(16\right)\end{array}$ $\begin{array}{cc}{D}_j=\left\{z.Math.z-|1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\Phi \left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right).Math.\le \stackrel{m}{\underset{l=1,i\ne j}{.Math.}}.Math.\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{i=1}{\overset{m}{.Math.}}\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\right\}& \left(17\right)\end{array}$ wherein z is a characteristic value of matrix I−(ρΦ.sub.c(k){circumflex over (Φ)}.sub.c.sup.T(k)/(λ+∥{circumflex over (Φ)}.sub.c(k)∥.sup.2)+βΦ.sub.c(k)K) and D.sub.j, j=1, 2, . . . , m is a Gershgorin disk; formula (17) is equivalent to formula (18); $\begin{array}{cc}{D}_i=\{z.Math.z.Math.\le .Math.1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right).Math.+\stackrel{m}{\underset{l=1,i\ne j}{.Math.}}.Math.\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{i=1}{\overset{m}{.Math.}}\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\}& \left(18\right)\end{array}$ by resetting algorithms (12) and (13), obtaining |{circumflex over (ϕ)}.sub.ij|≤b.sub.1 and b.sub.2≤|{circumflex over (ϕ)}.sub.ii(k)|≤αb.sub.2, i=1, . . . ,m; j=1, . . . ,m; i≠j; from assumption 1, obtaining |ϕ.sub.ij|≤b.sub.1,b.sub.2≤|ϕ.sub.ii(k)|≤αb.sub.2, i=1, . . . ,m; j=1, . . . ,m; i≠j; from the above conditions, obtaining the following inequalities $\begin{array}{cc}1-\left(\frac{\rho {.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{ji}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)\le 1-\left(\frac{\rho .Math.{\varphi }_{\mathrm{jj}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{jj}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)\le 1-\left(\frac{\rho b_2^2}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta K_{\min }{b}_2\right)& \left(19\right)\end{array}$ $\begin{array}{cc}\begin{array}{cc}\stackrel{m}{\underset{l=1,i\ne j}{.Math.}}.Math.\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{i=1}{\overset{m}{.Math.}}\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\le \rho \stackrel{m}{\underset{l=1,i\ne j}{.Math.}}\frac{{.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{li}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\stackrel{m}{\underset{l=1,i\ne j}{.Math.}}\beta .Math.{\varphi }_{\mathrm{jl}}\left(k\right).Math.K_{\mathrm{il}}=\rho \frac{{.Math.}_{i=1,l\ne j}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{lj}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\rho \frac{{.Math.}_{i=1,l\ne j}^m.Math.{\varphi }_{\mathrm{jl}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{li}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\rho \frac{\begin{array}{c}{.Math.}_{l=1,l\ne j}^m{.Math.}_{i=1,i\ne j,l}^m\\ .Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{li}}\left(k\right).Math.\end{array}}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\stackrel{m}{\underset{l=1,l\ne j}{.Math.}}\beta .Math.{\varphi }_{\mathrm{jl}}\left(k\right).Math.K_{\mathrm{il}}\le \rho \frac{\begin{array}{c}2\alpha b_1{b}_2\left(m-1\right)+\\ b_1^2\left(m-1\right)\left(m-2\right)\end{array}}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta K_{\max }{b}_1\left(m-1\right).& \end{array}& \left(20\right)\end{array}$ $\begin{array}{cc}1-\left(\frac{\rho {.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{ji}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)+{.Math.}_{l=1,l\ne j}^m.Math.\frac{{.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+{\mathrm{\beta \varphi }}_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\le 1-\left[\begin{array}{c}\rho \frac{b_2^2-2\alpha b_1{b}_2\left(m-1\right)-b_1^2\left(m-1\right)\left(m-2\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]=1-\left[\begin{array}{c}\rho \frac{b_2\left(b_2-2\alpha b_1\left(m-1\right)\right)-b_1^2\left(m-1\right)\left(m-2\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]<1-\left[\begin{array}{c}\rho \frac{b_2{b}_1\left(m-1\right)-b_1^2\left(m-1\right)\left(m-2\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]<1-\left[\begin{array}{c}\rho \frac{b_2{b}_1\left(m-1\right)-b_1^2\left(m-1\right)\left(m-2\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]=1-\left[\rho \frac{b_1\left(m-1\right)\left(b_2-b_1\left(m-1\right)\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\right]<1-\left[\rho \frac{2\alpha {b_1^2\left(m-1\right)}^2}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+2\beta K_{\min }\alpha b_1\left(m-1\right)\right]& \left(21\right)\end{array}$ from resetting algorithm formula (11) and assumption 1, obtaining {circumflex over (ϕ)}.sub.ji(k)ϕ.sub.ji(k)>0, i=1, . . . , m; j=1, . . . , m; therefore, there is a λ.sub.min, so that when λ>λ.sub.min, the following equality holds: $\begin{array}{cc}\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{lj}}=\frac{\rho {.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{ji}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}=\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{ji}}\le \rho \frac{{\alpha }^2{b}_2^2+b_1^2\left(m-1\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta \alpha b_2{K}_{\max }<\rho \frac{{\alpha }^2{b}_2^2+b_1^2\left(m-1\right)}{{\lambda }_{\min }+{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta \alpha b_2{K}_{\max }<1& \left(22\right)\end{array}$ thus, selecting 0<ρ≤1 and λ>λ.sub.min such that $\begin{array}{cc}.Math.1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{lj}}\right).Math.=1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)& \left(23\right)\end{array}$ for any λ>λ.sub.min, the following inequalities hold obviously $\begin{array}{cc}0<M_1\le \rho \frac{2\alpha {b_1^2\left(m-1\right)}^2}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+2\beta K_{\min }\alpha b_1\left(m-1\right)<\frac{b_2^2}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+2\beta K_{\min }\alpha b_1\left(m-1\right)\le \frac{{\alpha }^2{b}_2^2+b_1^2\left(m-1\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+2\beta K_{\min }\alpha b_1\left(m-1\right)<\frac{{\alpha }^2{b}_2^2+b_1^2\left(m-1\right)}{{\lambda }_{\min }+{.Math.\hat{\Phi }\left(k\right).Math.}^2}+2\beta K_{\min }\alpha b_1\left(m-1\right)<1& \left(24\right)\end{array}$ from formulas (21), (23) and (24), knowing $\begin{array}{cc}.Math.1-\left(\begin{array}{c}\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\end{array}\right).Math.+{.Math.}_{l=1,l\ne j}^m.Math.\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.<1-M_1<1& \left(25\right)\end{array}$ from formulas (18) and (24), obtaining $\begin{array}{cc}s\left[I-\left(\frac{\rho {\Phi }_c\left(k\right){\hat{\Phi }}_c^T\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\Phi }_c\left(k\right)K\right)\right]\le 1-M_1& \left(26\right)\end{array}$ wherein s(M) is the spectral radius of matrix M; letting $A={.Math.I-\left(\frac{\rho {\Phi }_c\left(k\right){\hat{\Phi }}_c^T\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\Phi }_c\left(k\right)K\right).Math.}_v$ and B=∥βΦ.sub.c(k)K)∥.sub.v; from the conclusion of the spectral radius of the matrix, an any small positive number ε.sub.1 exists, such that $\begin{array}{cc}A\le s\left[I-\left(\frac{\rho {\Phi }_c\left(k\right){\hat{\Phi }}_c^T\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\Phi }_c\left(k\right)K\right)\right]+{\epsilon }_1\le 1-M_1+{\epsilon }_1<1& \left(27\right)\end{array}$ wherein ∥M∥.sub.v is the compatible norm of matrix M; β exists such that B satisfies the following inequality:
1>1−A≤M.sub.1−ε.sub.1>B>0  (28) from formulas (16) and (28), obtaining:
∥e(k+1)∥.sub.v≤A∥e(k)∥.sub.v+B∥e(k−1)∥.sub.v<(1−B)∥e(k)∥.sub.v+B∥e(k−1)∥.sub.v  (29) after transposition, obtaining:
∥e(k+1)∥.sub.v−∥e(k)∥.sub.v<−B(∥e(k)∥.sub.v−∥e(k−1)∥.sub.v)  (30) based on formula (30), discussing the form of e(k) from the following four aspects: in a first case, when ∥e(k+1)∥.sub.v>∥e(k)∥.sub.v and ∥e(k)∥.sub.v>∥e(k−1)∥.sub.v, obtaining
∥e(k+1)∥.sub.v−∥e(k)∥.sub.v>−B(∥e(k)∥.sub.v−∥e(k−1)∥.sub.v)  (31) which is the opposite of formula (30); therefore, this assumption does not exist; in a second case, when ∥e(k+1)∥.sub.v>∥e(k)∥.sub.v and ∥e(k)∥.sub.v<∥e(k−1)∥.sub.v from formula (30), obtaining: $\begin{array}{cc}\frac{{.Math.e\left(k+1\right).Math.}_v-{.Math.e\left(k\right).Math.}_v}{{.Math.e\left(k-1\right).Math.}_v-{.Math.e\left(k\right).Math.}_v}<B<1& \left(32\right)\end{array}$ i.e., the decrease of e(k) is larger than the increase in three adjacent sampling points; and as a result, the overall trend is decreasing under this situation; in a third case, when ∥e(k+1)∥.sub.v<∥e(k)∥.sub.v and ∥e(k)∥.sub.v<∥e(k−1)∥.sub.v, obtaining $\begin{array}{cc}\frac{{.Math.e\left(k+1\right).Math.}_v}{{.Math.e\left(k\right).Math.}_v}<1\mathrm{and}\frac{{.Math.e\left(k\right).Math.}_v}{{.Math.e\left(k-1\right).Math.}_v}<1& \left(33\right)\end{array}$ which satisfies formula (30), and e(k) has a decreasing trend in this case; in a fourth case, when ∥e(k+1)∥.sub.v<∥e(k)∥.sub.v and ∥e(k)∥.sub.v>∥e(k−1)∥.sub.v, according to formula (30), this situation may exist; two possibilities exist in the time of k+2 in detail: if ∥e(k+2)∥.sub.v>∥e(k+1)∥.sub.v exists, obtaining the same conclusion as the second case; if ∥e(k+2)∥.sub.v<∥e(k+1)∥.sub.v, obtaining the same conclusion as the third case; in short, e(k) still has a decreasing trend in this case; the above methods of proof are also applicable when f(k)=0; to sum up, the overall trend of error e(k) is decreasing; therefore, the convergence of the error is proved; step D: applying the above control algorithm to control of an aero-engine model, and selecting three different cases for result comparison to verify the effectiveness and superiority of the control algorithm; firstly, comparing the control effects of MFAC+Kp, CFDL-MFAC and PID under the standard conditions to illustrate the effectiveness of an improved controller; and then, comparing the control effects at different heights and different delays to illustrate the superiority of the controller.

Description

DESCRIPTION OF DRAWINGS

[0050] FIG. 1 is a structural diagram of a controller.

[0051] FIG. 2 is an effect comparison diagram of three control algorithms of MFAC+Kp, MFAC and PID.

[0052] FIG. 3 shows comparison of control effects at different flight heights.

[0053] FIG. 4 shows comparison of control effects at different delays.

DETAILED DESCRIPTION

[0054] To make the proposed technical solutions and the technical problems solved by the present invention more clear, the technical solutions of the present invention are illustrated in detail below in combination with the drawings.

[0055] A structural block diagram of the improved control algorithm of the present invention is shown in FIG. 1. The controller mainly comprises three parts: MFAC, proportional control and anti-windup control. The control algorithm combines the advantages of three algorithms, can realize stable and quick control even for a very complex nonlinear model and has good robustness.

[0056] The specific composition of all parts of the control algorithm is as follows:

[0057] (1) MFAC algorithm: at each sampling time point, the parameters of the control algorithm are updated by the estimation algorithm, so that the control algorithm can be changed adaptively to achieve a good control effect on a control object, with certain robustness. However, due to the addition of the estimation algorithm, the response time of the controller becomes slow and easily affected by disturbance. In order to satisfy the requirements for the rapidity and the robustness of the controller, a proportional control link is considered to be added on the basis of the control algorithm.

[0058] (2) Proportional control algorithm: this algorithm is simple in operation and short in time consumption, reduces steady-state errors, accelerates control response, makes up for the deficiency of MFAC algorithm and improves the control performance.

[0059] (3) Anti-windup algorithm: due to the upper and lower limits of the actuator in the control system, the output of the control algorithm may exceed the executive capacity of the actuator, making the actuator fall into saturation, which will affect the response speed and control accuracy of the controller. The anti-windup algorithm can stop operation when the actuator reaches saturation, so that when the control algorithm provides a normal instruction, the actuator can respond as quickly as when the actuator is not saturated.

[0060] The basic standard for measuring the control algorithm is the accuracy, stability and rapidity of control. The present invention also has anti-windup performance while satisfying the above standard. The improved model-free adaptive control method of the present invention mainly has the following advantages:

[0061] (1) Good accuracy. It can be seen from FIG. 3 and FIG. 4 that the control algorithm in the present invention can achieve good control effects at different heights and different delays, which indicates that the algorithm has good accuracy.

[0062] (2) Excellent stability. It can be seen from FIG. 2, FIG. 3 and FIG. 4 that by comparing with MFAC and PID algorithms under the same conditions, the control algorithm in the present invention has excellent stability, and can realize stable control at different flight heights and different delays, and the stability is obviously better than the original MFAC algorithm.

[0063] (3) Excellent rapidity. It can be seen from FIG. 3 and FIG. 4 that by comparing with MFAC and PID algorithms under the same conditions, the control algorithm in the present invention has excellent rapidity, and can realize stable control at different flight heights and different delays, and the stability is obviously better than the original MFAC algorithm.

[0064] (4) Good anti-windup performance. It can be seen from FIG. 4 that the anti-windup algorithm in the present invention may stop accumulating after the actuator is saturated to prevent further saturation. After the controller outputs a normal value, the actuator can respond quickly, and the response speed is higher than that of the original MFAC algorithm.

[0065] The improved model-free adaptive control method proposed in the present invention is provided below, which comprises the following specific steps:

[0066] step A: analyzing the existing method for the compact dynamic linearization model-free adaptive control, and from experimental results, finding that the application process has deficiencies in response time and stability;

[0067] expressing MIMO discrete-time nonlinear systems as follows:

y(k+1)=f(y(k), . . . ,y(k−n.sub.y),u(k), . . . ,u(k−n.sub.u))  (1)

wherein u(k) and y(k) are system inputs and system outputs at time k, respectively; n.sub.y and n.sub.u are two unknown integers; f( . . . )=(f.sub.1( . . . ), . . . , f.sub.m( . . . )) is an unknown nonlinear function;

[0068] when f has a continuous partial derivative condition and formula (1) satisfies a generalized Lipschitz condition, expressing formula (1) as the following CFDL data model form:

[00019]$\begin{array}{cc}\Delta y\left(k+1\right)={\Phi }_c\left(k\right)\Delta u\left(k\right)\mathrm{wherein}& \left(2\right)\end{array}$${\Phi }_c\left(k\right)=\left[\begin{array}{cccc}{\varphi }_{11}\left(k\right)& {\varphi }_{12}\left(k\right)& .Math.& {\varphi }_{1m}\left(k\right)\\ {\varphi }_{21}\left(k\right)& {\varphi }_{22}\left(k\right)& .Math.& {\varphi }_{2m}\left(k\right)\\ .Math.& .Math.& .Math.& .Math.\\ {\varphi }_{m1}\left(k\right)& {\varphi }_{m2}\left(k\right)& .Math.& {\varphi }_{mm}\left(k\right)\end{array}\right]\in R^{m\times m};$

[0069] firstly, proposing the following assumptions:

[0070] assumption 1: Φ.sub.c(k) as a pseudo Jacobian matrix of the system shall be a diagonal dominant matrix which satisfies the following conditions: |ϕ.sub.ij|≤b.sub.1,b.sub.2≤|.sub.ii(k)|≤αb.sub.2,α≥1,b.sub.2>b.sub.1(2α+1)(m−1), i=1, . . . ,m, j=1, . . . ,m, i≠j; b.sub.1 and b.sub.2 are set as bounded constants, i and j are set as row and column indexes of the matrix; and the signs of all elements in Φ.sub.c(k) remain the same at any time k;

[0071] expressing a control input criterion function as formula (3):

J(u(k))=∥y*(k+1)−y(k+1)∥.sup.2+λ∥u(k)−u(k−1)∥.sup.2  (3)

[0072] wherein λ>0 represents a weight factor, which is used to punish the change of excessive control input quantity; y*(k+1) is a desired output signal;

[0073] substituting formula (2) into formula (3), deriving u(k) and making the equation equal to zero to obtain a control input algorithm as follows:

[00020]$\begin{array}{cc}u\left(k\right)=u\left(k-1\right)+\frac{{\mathrm{\rho \Phi }}_c^T\left(k\right)\left(y^{*}\left(k+1\right)-y\left(k\right)\right)}{\lambda +{.Math.{\Phi }_c\left(k\right).Math.}^2}& \left(4\right)\end{array}$

[0074] considering the following parameter estimation criteria function:

J(Φ.sub.c(k))=∥Δy(k)−Φ.sub.c(k)Δu(k−1)∥.sup.2+μ∥Φ.sub.c(k)−{circumflex over (Φ)}.sub.c(k−1)∥.sup.2  (5)

[0075] wherein μ is a weight factor used to punish excessive changes in PJM estimates; {circumflex over (Φ)}.sub.c(k) is an estimate of Φ.sub.c(k);

[0076] deriving Φ.sub.c(k) in formula (5) and making the equation equal to zero to obtain a parameter estimation algorithm as follows:

[00021]$\begin{array}{cc}{\hat{\Phi }}_c\left(k\right)={\hat{\Phi }}_c\left(k-1\right)+\frac{\begin{array}{c}\eta \left(\Delta y\left(k\right)-{\hat{\Phi }}_c\left(k-1\right)\Delta u\left(k-1\right)\right)\\ \Delta u^T\left(k-1\right)\end{array}}{\mu {.Math.\Delta u\left(k-1\right).Math.}^2}& \left(6\right)\end{array}$

[0077] conducting parameter estimation in each k by the above control parameter estimation algorithm to provide control inputs at the time; however, the calculation of the parameter estimation algorithm needs to occupy a certain time, causing slow system response and causing the control algorithm to be limited in use for a system with a small requirement for a control period; and the system vibrates greatly under non-ideal conditions from the experimental results;

[0078] step B: based on the above problems of slow response and vibration, considering the following improved solution;

Δu(k)=Δu.sub.m(k)′+Δu.sub.p(k)  (7)

[0079] wherein u.sub.m(k)′ is MFAC controller output, and Δu.sub.p(k) is proportional controller output expressed by the following formulas:

[00022]$\begin{array}{cc}\Delta u_m\left(k\right)=\frac{{\mathrm{\rho \Phi }}_c^T\left(k\right)\left(y^{*}\left(k+1\right)-y\left(k\right)\right)}{\lambda +{.Math.{\Phi }_c\left(k\right).Math.}^2}& \left(8\right)\end{array}$$\begin{array}{cc}\Delta u_p\left(k\right)=\beta K\left(y^{*}\left(k+1\right)-y\left(k\right)\right)-\beta K\left(y^{*}\left(k\right)-y\left(k-1\right)\right)& \left(9\right)\end{array}$

[0080] proposing the following control solution based on formulas (6) and (7):

[00023]$\begin{array}{cc}{\hat{\varphi }}_{\mathrm{ii}}\left(k\right)={\hat{\varphi }}_{\mathrm{ii}}\left(1\right),\mathrm{if}.Math.{\hat{\varphi }}_{\mathrm{ii}}\left(k\right).Math.<b_2\mathrm{or}.Math.{\hat{\varphi }}_{\mathrm{ii}}\left(k\right).Math.>\alpha b_2\mathrm{or}& \left(10\right)\end{array}$$\mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ii}}\left(k\right)\right)\ne \mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ii}}\left(1\right)\right)i=1,.Math.,m$$\begin{array}{cc}{\hat{\varphi }}_{\mathrm{ij}}\left(k\right)={\hat{\varphi }}_{\mathrm{ij}}\left(1\right),\mathrm{if}.Math.{\hat{\varphi }}_{\mathrm{ij}}\left(k\right).Math.>b_1\mathrm{or}\mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ij}}\left(k\right)\right)\ne \mathrm{sign}\left({\hat{\varphi }}_{\mathrm{ij}}\left(1\right)\right),i\ne j& \left(11\right)\end{array}$$\begin{array}{cc}u\left(k\right)=u\left(k-1\right)+\frac{{\mathrm{\rho \Phi }}_c^T\left(k\right)\left(y^{*}\left(k+1\right)-y\left(k\right)\right)}{\lambda +{.Math.{\Phi }_c\left(k\right).Math.}^2}f\left(k\right)+\beta K\left(y^{*}\left(k+1\right)-y\left(k\right)\right)-\beta K\left(y^{*}\left(k\right)-y\left(k-1\right)\right)& \left(12\right)\end{array}$

wherein {circumflex over (ϕ)}.sub.ij(1) is an initial value of {circumflex over (ϕ)}.sub.ij(k), i=1, . . . , m; j=1, . . . , m;

[0081] proposing the following anti-windup algorithm as part of the proposed control algorithm: stopping updating an integrator when an actuator is at an upper saturation limit and there is still a growing trend, or when the actuator is at a lower saturation limit and is still decreasing; otherwise, the integrator works normally; that is, in the case of saturation, only the integral operations that help to reduce the degree of saturation are performed, and expressed by the following formulas:

[00024]$\begin{array}{cc}\Delta {u_m\left(k\right)}^{\prime }=\Delta u_m\left(k\right)f\left(k\right)& \left(13\right)\end{array}$$\begin{array}{cc}f\left(k\right)=\{\begin{array}{cc}0,& \begin{array}{c}u\left(k\right)>\mathrm{u\_max}.Math.\Delta u\left(k\right)>0,\\ u\left(k\right)<\mathrm{u\_min}.Math.\Delta u\left(k\right)<0\end{array}\\ 1,& \mathrm{otherwise}\end{array}& \left(14\right)\end{array}$

[0082] wherein u_max and u_min are the upper and lower limits of the actuator;

[0083] step C: for the above improved control algorithm, analyzing the convergence of tracking error and the stability of bounded input and bounded output through theoretical derivation;

[0084] firstly, defining the following output errors of the system:

e(k)=y*(k)−y(k)  (15)

[0085] substituting formula (2) and formula (14) into formula (15), and when f(k)=1, obtaining:

[00025]$\begin{array}{cc}\begin{array}{c}e\left(k+1\right)=e\left(k\right)-{\Phi }_c\left(k\right)\Delta u\left(k\right)\\ =\left[I-\left(\frac{{\mathrm{\rho \Phi }}_c\left(k\right){\hat{\Phi }}_c^T\left(k\right)}{\lambda +{.Math.{\hat{\Phi }}_c\left(k\right).Math.}^2}+\beta {\Phi }_c\left(k\right)K\right)\right]\\ e\left(k\right)+\beta {\Phi }_c\left(k\right)Ke\left(k-1\right)\end{array}& \left(16\right)\end{array}$$\begin{array}{cc}{D}_j=\left\{z.Math.z-.Math.1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right).Math.\le \underset{l=1,l\ne j}{\overset{m}{.Math.}}.Math.\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{i=1}{\overset{m}{.Math.}}\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\right\},& \left(17\right)\end{array}$

[0086] wherein z is a characteristic value of matrix I−(ρΦ.sub.c(k){circumflex over (Φ)}.sub.c.sup.T(k)/(λ+∥{circumflex over (Φ)}.sub.c(k)∥.sup.2)+βΦ.sub.c(k)K) and D.sub.j, j=1, 2, . . . , m is a Gershgorin disk;

[0087] formula (17) is equivalent to formula (18);

[00026]$\begin{array}{cc}{D}_j=\{z.Math..Math.z.Math.\le .Math.1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right).Math.+\underset{l=1,l\ne j}{\overset{m}{.Math.}}.Math.\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{i=1}{\overset{m}{.Math.}}\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\}& \left(18\right)\end{array}$

[0088] by resetting algorithms (12) and (13), obtaining |{circumflex over (ϕ)}.sub.ij|≤b.sub.1 and b.sub.2≤|{circumflex over (ϕ)}.sub.ii(k)|≤αb.sub.2, i=1, . . . ,m; j=1, . . . ,m; i≠j; from assumption 1, obtaining |ϕ.sub.ij|≤b.sub.1,b.sub.2≤|ϕ.sub.ii(k)|≤αb.sub.2, i=1, . . . ,m; j=1, . . . ,m; i≠j;

[0089] from the above conditions, obtaining the following inequalities

[00027]$\begin{array}{cc}1-\left(\frac{\rho {.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right){\widehat{.Math..Math.\varphi }}_{\mathrm{ji}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)\le 1-\left(\frac{\rho .Math.{\varphi }_{\mathrm{jj}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{jj}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+{\mathrm{\beta \varphi }}_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)\le 1-\left(\frac{\rho b_2^2}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta K_{\min }{b}_2\right)& \left(19\right)\end{array}$$\begin{array}{cc}\underset{l=1,l\ne j}{\overset{m}{.Math.}}.Math.\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{i=1}{\overset{m}{.Math.}}\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\le \rho \underset{l=1,i\ne j}{\overset{m}{.Math.}}\frac{{.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{li}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{l=1,i\ne j}{\overset{m}{.Math.}}\beta .Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math.K_{\mathrm{li}}=\rho \frac{{.Math.}_{i=1,l\ne j}^m.Math.{\varphi }_{\mathrm{jj}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{lj}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\rho \frac{{.Math.}_{i=1,l\ne j}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{il}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\rho \frac{{.Math.}_{l=1,l\ne j}^m{.Math.}_{i=1,i\ne j,l}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{li}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\underset{i=1,i\ne j}{\overset{m}{.Math.}}\beta .Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math.K_{\mathrm{ii}}\le \rho \frac{\begin{array}{c}2\alpha b_1{b}_2\left(m-1\right)+\\ b_1^2\left(m-1\right)\left(m-2\right)\end{array}}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta K_{\max }{b}_1\left(m-1\right).& \left(20\right)\end{array}$$\begin{array}{cc}1-\left(\frac{\rho {.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right).Math..Math.{\hat{\varphi }}_{\mathrm{ji}}\left(k\right).Math.}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)+{.Math.}_{l=1,l\ne j}^m.Math.\frac{{.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{li}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{ji}}\left(k\right)K_{\mathrm{il}}.Math.\le 1-\left[\begin{array}{c}\rho \frac{b_2^2-2\alpha b_1{b}_2\left(m-1\right)-b_1^2\left(m-1\right)\left(m-2\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]=1-\left[\begin{array}{c}\rho \frac{b_2\left(b_2-2\alpha b_1\left(m-1\right)\right)-b_1^2\left(m-1\right)\left(m-2\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]<1-\left[\begin{array}{c}\rho \frac{b_2{b}_1\left(m-1\right)-b_1^2\left(m-1\right)\left(m-2\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]<1-\left[\begin{array}{c}\rho \frac{b_2{b}_1\left(m-1\right)-b_1^2\left(m-1\right)\left(m-1\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]=1-\left[\begin{array}{c}\rho \frac{b_1\left(m-1\right)(b_2-b_1\left(m-1\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\\ \beta K_{\min }\left(b_2-b_1\left(m-1\right)\right)\end{array}\right]<1-\left[\rho \frac{2\alpha {b_1^2\left(m-1\right)}^2}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+2\beta K_{\min }\alpha b_1\left(m-1\right)\right]& \left(21\right)\end{array}$

[0090] by resetting algorithm formula (11) and assumption 1, obtaining {circumflex over (ϕ)}.sub.ji(k)ϕ.sub.ji(k)>0, i=1, . . . ,m; j=1, . . . ,m; therefore, there is a λ.sub.min, so that when λ>λ.sub.min, the following equality holds:

[00028]$\begin{array}{cc}\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}=\frac{\rho {.Math.}_{i=1}^m.Math.{\varphi }_{\mathrm{ji}}\left(k\right){\widehat{.Math..Math.\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\varphi }_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\le \rho \frac{{\alpha }^2{b}_2^2+b_1^2\left(m-1\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\mathrm{\beta \alpha }{b}_2{K}_{\max }<\rho \frac{{\alpha }^2{b}_2^2+b_1^2\left(m-1\right)}{{\lambda }_{\min }+{.Math.\hat{\Phi }\left(k\right).Math.}^2}+\beta {\mathrm{\alpha b}}_2{K}_{\max }<1& \left(22\right)\end{array}$

[0091] thus, selecting 0<ρ≤1 and λ>λ.sub.min such that

[00029]$\begin{array}{cc}.Math.1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+{\mathrm{\beta \varphi }}_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right).Math.=1-\left(\frac{\rho {.Math.}_{i=1}^m{\varphi }_{\mathrm{ji}}\left(k\right){\hat{\varphi }}_{\mathrm{ji}}\left(k\right)}{\lambda +{.Math.\hat{\Phi }\left(k\right).Math.}^2}+{\mathrm{\beta \varphi }}_{\mathrm{jj}}\left(k\right)K_{\mathrm{jj}}\right)& \left(23\right)\end{array}$