系统的能控性和能观性英文版.docx

1、系统的能控性和能观性 英文版Unit 13 Controllability and Observability A system is said to be controllable at timeif it is possible by means of an unconstrainedcontrol vector to transfer the system from any initial stateto any other state in a finiteinterval of time. A system is said to be observable at timeif, wi

2、th the system in state,it is possible to determine this state from the observation of the output over a finite timeinterval. The concepts of the controllability and observability were introduced by Kalman. They play animportant role in the design of control systems in state space. In fact, the condi

3、tions ofcontrollability and observability may govern the existence of a complete solution of the controlsystem design problem. The solution to this problem may not exist of the system considered isnot controllable. Although most physical systems are controllable and observable,corresponding mathemat

4、ical models may not possess the property of controllability andobservability. Complete State Controllability of Continuous-Time SystemsConsider the continuous-time system (13. 1)where X=state vector (n-vector) u=control signal (scalar) A= matrix B= matrix The system described by Equation (13. 1) is

5、said to be state controllable atif it ispossible to construct an unconstrained control signal that will transfer an initial state to anyfinal state in a finite time interval. If every state is controllable, then the system issaid to be completely state controllable. We shall now derive the condition

6、 for complete state of controllability. Without loss ofgenerality, we can assume that the final state is the origin of the state space and that the initialtime is zero,or.The solution of Equation (13. 1) isApplying the definition of complete state controllability just given, we haveor (13. 2)Andcan

7、be written (13. 3)Substituting Equation (13. 3) into Equation (13. 2) gives (13. 4)Let us putThen Equation (13. 4) becomes (13. 5)If the system os completely state controllable, then, given any initial state X(0), Equation(13. 5) must be satisfied. This requires that the rank of thematrixbe n. From

8、this analysis, we can state the condition for complete state controllability as follows.The system given by Equation (13. 5) is completely state controllable if and only if the vectorsare linearly independent, or thematrixis the rank n.The result just obtained can be extended to the case where the c

9、ontrol vector U isr-dimensional. If the system is described byWhere U is an r-vector, then it can be proved that the condition of for complete statecontrollability is that thematrixbe of rank n, or contain n linearly independent column vectors. The matrixis commonly called the controllability matrix

10、. Complete Observability of Continuous-Time System In this section we discuss the observability of linear systems. Consider the unforcedsystem described by the following equations (13. 6) (13. 7)where X=state vector (n-vector) Y=output vector (m-vector) A=matrix C=matrix The system is said to be com

11、pletely observable if every statecan be determined fromthe observation of Y(t) over a finite time interval,. The system is, therefore,completely observable if every transition of the state eventually affects every element of theoutput vector. The concept of observability os useful in solving the pro

12、blem or reconstructingunmeasurable state variable from measurable variables in the minimum possible length of time.In this section we treat only linear, time-invariant systems. Therefore, without loss ofgenerality, we can assume that. The concept of observability is very important because, in practi

13、ce, the difficultyencountered with state feedback control is that some of the state variables are not accessible fordirect measurement, with the result that it becomes necessary to estimate the unmeasurablestate variables in order to construct the control signals. Such estimates of state variables a

14、repossible of and only if the system is completely observable. In discussion observability conditions, we consider the unforced system as given byEquation (13. 6) and (13. 7). The reasons for this are as follows, If the system is described bythenAnd Y(t) isSince the matrices A, B, C, and D are known

15、 and u(t) is also known,the last terms onthe right-hand side of this last equation are known quantities. Therefore, they may besubtracted from the observed value of Y(t). Hence, for investigating a necessary and sufficientcondition for complete observability, it suffices to consider the system descr

16、ibed by Equations(13. 6) and (13. 7). Consider the system described by Equations (13. 6) and (13. 7). The output vector Y(t) isAndcan be written asHence, we obtainor (13. 8)If the system is completely observable, then, given the output Y(t) over a time interval, X(0)is uniquely determined from Equat

17、ion (13. 8). It can be shown that this requires therank of thematrixto be n. From this analysis we can state the condition for complete observability as follows. The system described by Equation (13. 6) and (13. 7) is completely observable of and onlyis the matrixis of rank n or has n linearly indep

18、endent column vectors. This matrix is called theobservability matrix.Key Words and Terms1. controllability n. 可控性2. observability n. 可观测性3. controllable adj. 可控的4. observable adj. 可观测的5. mathematical model 数学模型6. property n. 性质，属性7. continuous-time system 连续时间系统8. generality n. 一般性，普遍性9. rank n. 秩10

19、. linearly independent 线性无关11. time-invariant system 时变系统12. suffice v. 满足Notes Although most physical systems are controllable and observable, correspondingmathematical models may not possess the property of controllability and observability.尽管大多数的物理系统都是可控的和可观测的，它们所对应的数学模型并不一定具有可控性和可观测性。 The system

20、 os said to be completely observable if every statecan be determinedfrom the observation of Y(t) over a finite time interval,.如果在有限的时刻t，从系统的输出Y(t)的观测中能确定每一个状态向量的初值，则称系统是完全可观测的。 The concept of observability is very important because, in practice, the difficultyencountered with state feedback control

21、is that some of the state variables are not accessible fordirect measurement, with the result that it becomes necessary to estimate the unmeasurablestate variables in order to construct the control signals.可观测性的概念非常重要，在实际中，状态反馈控制中所遇到的困难在于，一些状态变量是不能够直接测量的，因此有必要估计不可测量的状态变量来构成控制信号。encountered with stat

22、e feedback control为过去分词作定语，修饰the difficultythat some of the state variables are not accessible for direct measurement为表语从句。that it becomes necessary to estimate the unmeasurable state variables in order to constructthe control signal.为同位语从句，解释the result。Exercises1. Consider the system defined by Is

23、the system completely state controllable?2. Consider the system The output is given by Show that the system is not completely observable.3. Please translate the following paragraph into Chinese. A system is said to be controllable at timeif it is possible by means of an unconstrainedcontrol vector t

24、o transfer the system from any initial stateto any other state in a finiteinterval of time. A system is said to be observable at timeif,with the system in state,it is possible to determine this state from observation of the output over a finite timeinterval.Unit 14 Internal Model Control In the last

25、 chapter we presented several methods for tuning PID controllers and developeda model-based procedure (direct synthesis) to synthesize a controller that yields a desiredclosed-loop response trajectory. In this chapter, we first develop an open-loop control designprocedure that then leads to the deve

26、lopment of an internal model control (IMC) structure.There are a number of advantages to the IMC structure (and controller design procedure),compared with the classical feedback control structure. One is that it becomes very clear howprocess characteristics such as time delays and RHP zeros affect t

27、he inherent controllability ofthe process. IMCs are much easier to tune than are controllers in a standard feedback controlstructure.After studying this chapter, the reader should be able to: Design internal model controllers for stable process (either minimum or non-minimumphase); Sketch the closed

28、-loop response of the model is perfect; Derive the closed-loop transfer functions for IMC; Design IMC improved disturbances for IMC. Introduction to Model-Based Control In the previous chapters we focused on techniques to tune PID controllers. The closed-looposcillation technique developed by Ziegle

29、r and Nichols did not require a mode of the process.Direct synthesis, however, was based the use of a process model and a desired closed-loopresponse to synthesize a control law; often this resulted in a controller with a PID structure.In this chapter we develop a model-based procedure, where a proc

30、ess model is embedded inthe controller. By explicitly using process knowledge, by virtue of the process model,improved performances can be obtained. Consider the stirred-tank heater control problem shown in Figure 14. 1. We can use amodel of the process to decide the heat flow (Q) that needs to be a

31、dded to the process to obtaina desired temperature (T) trajectory, specified by the set-point (). A simple steady-stateenergy balance provides the steady-state heat flow needed to obtain a new steady-statetemperature, for example. By using a dynamic model, we can find the time-dependent heatprofile needed to yield a particular time-dependent temperature profile. Assume that the chemical process is represented by a linear transfer function model, andthat it is open-loop stable. The input-output relationship is shown in Figu