Channel Uncertainties Vs. Mean-square Stability Of Networked Control Systems

Networked control system is the deep integration of information technology and control technology.Its fundamental characteristic is that the data of each component in the control sys-tem is transmitted through the network communication channel.This will not only improve the flexibility and expansibility of the control system,but also bring new problems and challenges to the control system.One of the important problem is the uncertainties caused by transmission channel,such as quantization error,time delay and data dropouts.Moreover,the performance of systems will be affected by these uncertainties.A large number of scholars have made a profound study on these issues from the perspective of information theory and control theory.Therefore,many progresses have been made in the past twenty or thirty years.Researches show that the white multiplicative noise is suitable to model memoryless channels such as ran-dom packet dropouts and fading channel.On this basis,we can use stochastic theory and robust control techniques to analyze the stability and stabilizability of the system.The optimal con-trol of the white multiplicative noise system can also be studied.It has important theoretical significance and application value to establish the quantitative relationship between the key pa-rameters of channels and the performance in white multiplicative noise systems.On the other hand,there is an urgent need for a model which is in line with the data transmission mechanism and can effectively describe the input output relationship and dynamic characteristics of the channel.The analytical relationship between the performance of colored multiplicative noise systems and key parameters of the channel should be established through this model.To this end,the main contents of the thesis are as follows.1)The models of channel uncertainties in the networked control systems are established.·The memoryless channels such as random packet dropouts,fading channel and quan-tization error,additive noise with power constraints and a mixture of various uncertainties are decomposed by analyzing the random statistical characteristics of the signals.As a result,the channel is modeled by the white multiplicative noise.Moreover,the key parameter–signal-to-noise ratio（SNR）of the white multiplicative noise channel is discussed.This key parameter can not only characterize the white multiplicative noise channels,but also play a significant role in the analysis of system performances.·An analytical description of input-output relationship for the random transmission de-lay channel by finite impulse response（FIR）is proposed.According to the impulse response of channel and the probability distribution of random transmission delay,the channel is decom-posed into two parts,namely mean channel and deviation channel.Then,the statistical charac-teristics of the channel impulse response and input-output signals are studied based on stochastic analysis.Thus,the key parameter–coefficient of frequency variation which can characterize the random transmission delay is introduced.The coefficient of frequency variation is similar with SNR in the white multiplicative noise channels.On the other hand,we establish state equations for mean channel and deviation channel respectively.The equivalence between input-output description and state equation description of the channel is analyzed simultaneously.The ran-dom delay channel is wholly described from two aspects,one is frequency domain model based on the input-output relationships,the other is time domain model based on state equations.2)Input-output mean-square stability and stabilizability criteria for systems over memo-ryless channels are established.Based on the mean-square small gain theory,the necessary and sufficient conditions for mean-square stability are developed for both minimum phase systems and non-minimum phase systems.These conditions clarify the importance of SNR in white multiplicative noise systems.Then,the mean-square stabilizability problem is transform to a H₂optimal control problem by taking advantages of controller parametrization.Furthermore,the solution of the H₂optimal control problem is provided based on inner-outer factorization and the projection theorem in H₂space.It’s proved that the minimum allowable SNR under the condition of mean-square stabilizability is determined by the linear function of systems’unstable poles and relative degree.It also reveals the analytical relationship between the mean-square stabilizability and the system’s relative degree（or fixed delay）,unstable poles,non-minimum phase zeros and SNR.3)Mean-square stability and stabilizability criterias for the systems over memory channels are established.According to the input-output model of stochastic transmission delay given by FIR,the sufficient and necessary conditions for the input-output mean-square stability are proved through spectral analysis and the Toeplitz matrix which describe the relationships be-tween the input and output signals’variances.It is also proved that the variance of the output signal converges as long as the variance of the input signal converges in the mean-square stable system.On the other hand,the sufficient and necessary conditions for mean-square stability of the system are established according to state equations of the stochastic transmission delay system and Lyapunov stochastic equations.Further,the equivalence between the input-output mean-square stability and the mean-square stability is proved.On this basis,the significance of coefficient of frequency variation in mean-square stabilizability is provided.Moreover,the fundamental constraints of mean-square stabilizabilty is imposed by system’s relative degree,unstable poles and coefficient of frequency variation.4)Mean-square stability and stabilizability of the Multi-Input Multi-Output（MIMO）non-minimum phase system is analysed.The control signals are transmitted to the plant over a set of parallel communication channels with packet dropouts.Each non-minimum zero of the plant is only associated with one of control input channels.Based on the mean-square small gain theorem and the spectral radius theorem,the quantitative relationship between the random packet drop rate,non minimum phase zero,unstable poles,and the mean-square stabilizability of the closed-loop system is obtained.Further,the non-convex property is analyzed by Wonham upper triangular coprime factorization.The admissible region of packet dropout probabilities in the mean-square stabilizability sense is also introduced.5)According to features of the networked control system,a new controller structure is pro-posed for the asymptotic tracking problem of the networked control system.Then,the mean-square optimal asymptotic tracking problem is transformed into the mean-square stabilizability problem by illustrate the equivalence between the state equations and the deviation state equa-tions.Further,we propose a new algorithm for the solution of a Modified Algebraic Riccati Equation（MARE）.Thus,a feasible numerical algorithm is raised for this kind of optimal design problem.