CN105159090A - PI tracking controller designing method for stochastic system based on Markov models - Google Patents

Abstract

The invention discloses a PI tracking controller designing method for a stochastic system based on Markov models. According to the method, the stochastic system with time lag and unknown nonlinearity, which is changed on structure due to sudden environmental disturbance, change of association between sub-systems and the like, is described by continuous Markov models, the stochastic system is subjected to model transformation based on the traditional PI control strategy, and then a multi-target controller design scheme with stochastic stability and tracking performance is proposed according to the stochastic stability theory of a Markov jump system, the Lyapunov Theory and the linear matrix inequality (LMI) algorithm. A designing method for a controller with a PI structure is provided, to ensure the stochastic stability and good tracking performance of the nonlinear time lag Markov stochastic system.

Description

Based on the PI Tracking Control Design method of the stochastic system of Markov model

Technical field

The present invention relates to a kind of PI Tracking Control Design method of the stochastic system based on Markov model, it is the stochastic system for having time lag and unknown nonlinear, continuous Markov model is utilized to be described, based on PI control strategy, model conversion is carried out to stochastic system,, Li Ya theoretical according to Markov jump system stochastic stability composes promise husband theory and LMI (LMI) algorithm, provide the adequate condition that stochastic system has Stochastic stable performance, good tracking performance, and provide the method for designing of the controller with PI structure.Belong to automatic control technology field.

Background technology

In industrial processes, many real systems all can because of the breakdown maintenance of internal part, to receive between sudden environmental perturbation, subsystem the reasons such as association changes and change on recurring structure.1961, Krasivskii and Lidskii first time introduced linear switching model, and carry out the switching between the different model structure of descriptive system by the Markov chain of continuous time, we call Markov Jump Systems this type systematic.Widely, in biochemical system, manufacturing system, Circuits System, the industries such as even economic projection, wagon control and flying vehicles control are visible at random in the application of Markov Jump Systems.In addition, time lag and uncertainty are the subject matter often faced in Practical Project, the existence of both often causes the instability of system and poor system performance, also make the analysis of system become complex, therefore there is the concern that the stability of time lag and probabilistic Stochastic Markov Jump Systems and Controller gain variations problem cause people day by day.

Part China applies for a patent and in stability analysis, Iamge Segmentation, recognition of face, automobile sound identification etc., achieves certain achievement in research, but existing technology relates to time lag and parameter uncertainty seldom simultaneously;

As everyone knows, PI control has been widely used in the middle of the analysis of engineering and many theoretical methods.Under this technical background, the present invention provides a kind of PI Tracking Control Design method of the stochastic system based on Markov model.Utilize continuous Markov model to describe and there is state and input delay simultaneously, unknown nonlinear, the stochastic system of external interference, based on traditional PI control strategy, stochastic system is carried out to the conversion of model, theoretical according to Markov jump system stochastic stability again, Li Ya composes promise husband theory and LMI (LMI) algorithm, propose one and there is Stochastic stable performance, the multi-objective controller design proposal of tracking performance, provide the method for designing of the controller with PI structure, and then ensure that the Stochastic stable performance of Nonlinear Delay markov stochastic system and good tracking performance.

Summary of the invention

Goal of the invention: for many breakdown maintenances because of internal part in industrial processes, to receive between sudden environmental perturbation, subsystem the reasons such as association changes and change on recurring structure, and often exist and cause the unstable and time lag of poor system performance of system and the actual stochastic system of unknown nonlinear, utilize continuous Markov model to be described, do not consider time lag and non-linear at existing stochastic system tracing control simultaneously, and Controller gain variations is complicated, not easily solve and have model on the basis of a definite limitation, based on traditional PI control strategy and LMI (LMI) algorithm, theoretical according to Markov jump system stochastic stability, Li Ya composes promise husband theory proposition one and has Stochastic stable performance, the multi-objective controller design proposal of tracking performance, construct the tracking control unit with PI structure, ensure that the Stochastic stable performance of Nonlinear Delay markov stochastic system and good tracking performance.

Technical scheme: the present invention is a kind of PI Tracking Control Design method of the stochastic system based on Markov model, and the method concrete steps are as follows:

The first step utilizes continuous Markov model to be described to the labile real system of the structure with Time-varying time-delays and unknown nonlinear

$\begin{array}{c}\stackrel{\·}{Z}\left(t\right)=A_0\left(r\right(t\left)\right)Z\left(t\right)+A_{0d}\left(r\right(t\left)\right)Z(t-\mathrm{\τ}(t\left)\right)+B_{01}\left(r\right(t\left)\right)u\left(t\right)+B_{02}\left(r\right(t\left)\right)u(t-\mathrm{\τ}(t\left)\right)+\\ F_0\left(r\right(t\left)\right)f_0\left(Z\right(t\left)\right)+B_{0v}\left(r\right(t\left)\right)v\left(t\right)\end{array}---\left(1\right)$

Wherein, Z (t) ∈ R ⁿsystem state vector, u (t) ∈ R ^mfor control inputs, v (t) ∈ R ^pbelong to L ₂[0, ∞) on meet bounded Perturbations.τ (t) is satisfied 0 < τ (the t)≤τ of Time-varying time-delays ^*< ∞, wherein border τ ^*, τ ⁺for known constant .{r (t), t>=0} is that value is in finite state collection fight continuity Markov chain, its state transition rates Π=(π _ij) _{n × N}, (i, j ∈ S) is determined by following formula:

$P\{r(t+\mathrm{\&Delta;})=j|r\left(t\right)=i\}=\left\{\begin{array}{cc}{\mathrm{\&pi;}}_{ij}\mathrm{\&Delta;}+o\left(\mathrm{\&Delta;}\right),& i\&NotEqual;j\\ 1+{\mathrm{\&pi;}}_{ii}\mathrm{\&Delta;}+o\left(\mathrm{\&Delta;}\right),& i=j\end{array}\right.$

Wherein: lim _{Δ → 0}o (Δ)/Δ=0 (Δ > 0), π _ijfor the rate of transform from mode i to mode j, and meet

${\mathrm{\&pi;}}_{ii}=-\underset{j\&NotEqual;i}{\&Sigma;}{\mathrm{\&pi;}}_{ij},({\mathrm{\&pi;}}_{ij}\&GreaterEqual;0,j\&NotEqual;i)$

To any given r (t)=i ∈ S, A ₀(i), A _0d(i), B ₀₁(i), B ₀₂(i), B _0v(i), F ₀i () is the constant matrices of suitable dimension.F ₀(V (t)) meets f for unknown nonlinear function ₀(0)=0 and Lipschitz condition, namely there is known matrix U ₀following formula is set up || f ₀(V ₁(t))-f ₀(V ₂(t)) ||≤|| U ₀(V ₁(t)-V ₂(t)) ||.

Second step utilizes PI strategy to carry out model conversion to stochastic system (1)

In order to realize following the tracks of, hypothetical reference Dynamic Signal is x _r(t) ∈ x ⁿ, and have x for all t > 0 _r(t) ∈ L ₂[0, ∞).Target of the present invention is the reference Dynamic Signal x of the state vector tracing preset as far as possible that design PI tracking control unit makes system (1) _rt (), define tracking error is e (t)=Z (t)-x for this reason _r(t).

Based on non-linear stochastic Markovian Jumping model (1), introduce state variable new as follows then stochastic system (1) can be converted into following Stochastic Markov hopping model

$\left\{\begin{array}{c}\stackrel{\&CenterDot;}{x}\left(t\right)=A_{i}x\left(t\right)+A_{di}x(t-\mathrm{\&tau;}(t\left)\right)+B_{1i}u\left(t\right)+B_{2i}u(t-\mathrm{\&tau;}(t\left)\right)+F_{i}f\left(x\right(t\left)\right)+B_{vi}v\left(t\right)+{\mathrm{Hx}}_r\left(t\right)\\ x\left(t\right)=\mathrm{\&phi;}\left(t\right),t\&Element;\&lsqb;-{\mathrm{\&tau;}}^{*},0\&rsqb;,r\left(0\right)=r_0\end{array}\right.---\left(2\right)$

Wherein φ (t) is for being defined in interval [-τ ^*, 0] on initial vector continuous function, r ₀∈ S is initial mode, f (x (t)) meet f (0)=0 and wherein U=diag{U ₀, 0}. and

$A_i=\left[\begin{array}{cc}{A}_{0i}& 0\\ I& 0\end{array}\right],A_{di}=\left[\begin{array}{cc}{A}_{0di}& 0\\ 0& 0\end{array}\right],B_{1i}=\left[\begin{array}{c}{B}_{01i}\\ 0\end{array}\right],$

$B_{2i}=\left[\begin{array}{c}{B}_{02i}\\ 0\end{array}\right],F_i=\left[\begin{array}{c}{F}_{0i}\\ 0\end{array}\right],B_{vi}=\left[\begin{array}{c}{B}_{0vi}\\ 0\end{array}\right],H=\left[\begin{array}{c}0\\ -I\end{array}\right]$

The design of the 3rd step PI tracking control unit

This step composes promise husband theory and LMI algorithm according to robust control theory, Li Ya, design PI tracking control unit makes the closed-loop system of Stochastic Markov Jump Systems (2) be robust convergency, and the reference Dynamic Signal x of state vector Z (t) tracing preset of Stochastic Markov Jump Systems (1) _rt (), tracking error is little as far as possible.Employing theorem is provided Stochastic Markov closed-loop system (2) robust convergency to this step and tracking control problem can sufficient conditions for solution.

In order to solve tracking control problem, we adopt direct PI control strategy, choose controller

$u\left(t\right)=K_{Pi}Z\left(t\right)+K_{Ii}{\&Integral;}_0^{t}e\left(t\right)d\mathrm{\&tau;}$

Wherein K _pi, K _iifor ride gain to be determined, based on Stochastic Markov Jump Systems (2), PI controller can be by further simplified characterization

4th step controller performance inspection

Whether the design of inspection controller meets the demands by this step, carries out by means of conventional numerical evaluation and Control System Imitation instrument Matlab.

5th step design terminates.

Whole design process emphasis considers the Robust tracking control problem of the Stochastic Markov Jump Systems with time lag and unknown nonlinear.First in the above-mentioned first step, continuous Markov model is utilized to be described to the real system that the easy recurring structure with time lag and unknown nonlinear changes; Control to convert stochastic system model to realize PI in second step; PI tracking control unit is devised, to reach the object making system robust Stochastic stable He there is good tracking performance in 3rd step; After above steps, design terminates.

Beneficial effect: the present invention is a kind of PI Tracking Control Design method of the stochastic system based on Markov model, for the multi objective control of the robust stability and tracing property that realize stochastic system.The advantage of the method comprises three aspects: one, utilizes Markov model to describe the labile real system of structure simultaneously with time lag, unknown nonlinear; Its two, designed PI tracking control unit can realize robust stability and good tracing property two control objectives, and the restriction that PI tracking control algorithm form is simple, model is fixed, do not had to structure.Its three, the solution of whole problem finally only needs to solve one group of LMI group, and than solving on going result, coupling inequality is simpler easy.

Accompanying drawing explanation

The state diagram of Fig. 1 mode 1 time stochastic system (1);

The status tracking error curve diagram of Fig. 2 mode 1 time stochastic system (2);

The state diagram of Fig. 3 mode 2 times stochastic systems (1);

The status tracking error curve diagram of Fig. 4 mode 2 times stochastic systems (2).

Embodiment

Below in conjunction with concrete simulation example, set forth the present invention further.Design object of the present invention is design PI tracking control unit: 1. the robust stability realizing closed-loop system; 2. make the constant of system state vector tracing preset, and tracking error is little as far as possible.In concrete enforcement, the stability of system and the emulation of tracing property and inspection all realize by means of Matlab.

First step supposing the system has two mode, and state transition probability is

$\&Pi;=\left[\begin{array}{cc}-933& 933\\ 988& -988\end{array}\right]$

Second step is with the structure of the stochastic system model of time lag and unknown nonlinear, and in system, the model of controlled device is as shown in (1), wherein

$A_{01}=\left[\begin{array}{cc}-140& 35\\ 69.5& -20\end{array}\right],A_{0d1}=\left[\begin{array}{cc}0.3& 8.6\\ -0.5& -0.5\end{array}\right],B_{011}=\left[\begin{array}{c}-252\\ -13.5\end{array}\right],B_{021}=\left[\begin{array}{c}0.7\\ -0.8\end{array}\right],B_{0v1}=\left[\begin{array}{c}0\\ 0.1\end{array}\right]$

$F_{01}=\left[\begin{array}{cc}93& 40\\ -20& 4\end{array}\right],A_{02}=\left[\begin{array}{cc}5& 54\\ -76& -81\end{array}\right],A_{0d2}=\left[\begin{array}{cc}1.6& -62\\ -13& -0.1\end{array}\right],B_{012}=\left[\begin{array}{c}-103\\ 37\end{array}\right]$

Selecting All Parameters

u ₁₁＝-13.8,u ₁₂＝0.3,u ₂₁＝22.5,u ₂₂＝3,u ₃₁＝8,u ₃₂＝7,λ＝0.18.

3rd step design PI tracking control unit

According to theorem 1, the state transition probability Π that utilization obtains above, and the LMI tool box solving state feedback controller utilizing MATLAB, obtaining controller gain is

K _P1＝[5.633811.4967],K _I1＝[22.016629.6201]

K _P1＝[-0.6278-39.3451],K _I2＝[55.0390-123.8101]

4th step robustness and tracking performance inspection

Whether the robustness of checking system and tracing property meet design requirement by this step, carry out by means of conventional numerical evaluation and Control System Imitation instrument Matlab.

If starting condition is Z=[-0.2-0.5] ^t, given reference vector is x _r(t)=[0.30.7] ^teven if can find out to there is Time-varying time-delays and unknown nonlinear by Fig. 1-4, under the effect of designed controller, the state of stochastic system can reach robust convergency, and the state of stochastic system (2) can trace into given constant, and tracking error is less.Thus illustrate that method proposed by the invention is effective.

5th step design terminates

Whole design process emphasis considers the robust control and tracing control with time lag and unknown nonlinear stochastic system.Around these two emphasis, first in the above-mentioned first step, the continuous Markov model of the labile real system of the structure with time lag and unknown nonlinear is described; Control to convert the model of stochastic system to realize PI in second step; For the stochastic system simultaneously with time lag and unknown nonlinear in 3rd step, devise PI tracking control unit, make stochastic system meet good robust performance and tracking performance; After above steps, design terminates.

Claims (5)

1. the PI Tracking Control Design method based on the stochastic system of Markov model, it is characterized in that: utilize continuous Markov model to describe because receiving sudden environmental perturbation, between subsystem, association such as to change at reason and the stochastic system with time lag and unknown nonlinear that recurring structure changes, based on traditional PI control strategy, stochastic system is carried out to the conversion of model, theoretical according to Markov jump system stochastic stability again, Li Ya composes promise husband theory and LMI (LMI) algorithm, propose one and there is Stochastic stable performance, the multi-objective controller design proposal of tracking performance.

2. provide the method for designing of the controller with PI structure, and then ensure that the stochastic stability of Nonlinear Delay markov stochastic system and good tracking performance.

3. the PI Tracking Control Design method of the stochastic system based on Markov model according to claim 1, it is characterized in that: the PI controller designed by PI control strategy not only ensure that stochastic system has good stability, and achieves good tracking performance.

4. the PI Tracking Control Design method of the stochastic system based on Markov model according to claim 1, it is characterized in that: when to Controller gain variations, make use of LMI algorithm give be convenient to solve ensure that stochastic system has the adequate condition of Stochastic stable performance and good tracking performance and the method for designing of control law, and consider the major issue often run in time lag and these two actual industrial systems non-linear simultaneously.

5. the PI Tracking Control Design method of the stochastic system based on Markov model according to claim 1, is characterized in that: the restriction that PI tracking control algorithm form is simple, model is fixed, do not had to structure of design.