(电气工程与自动化专业英语)Chapter 11Modern Control System Design

1、Add Your Company Slogan( (电气工程与自动电气工程与自动化专业英化专业英语语)Chapter )Chapter 11Modern 11Modern Control System Control System DesignDesign611.1 Controllability of the plant2Concepts of the pole placement The pole placement compensator consists of three parts designed in separate steps: a controller an observe

2、r the matrix N The pole placement design method works only with state-space plant models. The first steps in the design process are to verify that the plant is controllable and observable. Thereafter you must determine a set of closed-loop pole locations in the complex plane that satisfy the system

3、performance specifications11.1 Controllability of the plant3A method for selecting suitable closed-loop pole locations For a given the state-space plant model and the desired pole locations 1、It takes just one Control System Toolbox command to compute the controller gain vector K 2、The second step i

4、s designing the observer. 3、The final design step is determining the feedforward gain N Any linear plant model that passes the tests for controllability and observability can, in principle, produce any desired speed of response. In a real-world implementation, however, effects such as actuator satur

5、ation and plant nonlinearities limit the achievable performance of the system.7811.1 Controllability of the plant3A method for selecting suitable closed-loop pole locations The pole placement design method begins with a plant model in state-space form. Eq. (11.1)xAxBuyCxDu(11.1) This equation is the

6、 state-space equations for a MIMO plant. This discussion is equally applicable to SISO systems.11.1 Controllability of the plant4The test for observability The system controllability is determined from an analysis of the matrices A and B. Example: a second-order state-space model 21001.8100Gss tfpla

7、nt = tf(100, 1 1.8 100); ssplant = ss(tfplant); c0 = ctrb(ssplant) controllable = (rank(c0) = length(c0) The variable controllable will have the value of 1.0 if the plant is controllable and 0 otherwise.911.1 Controllability of the plant the controllability test computes the plant controllability ma

8、trix and tests whether it is of full rank. The controllability matrix rank must equal the number of rows (and columns) of the matrices. The MATLAB function rank() returns an estimate of the rank of a matrix. Because MATLAB uses numerical methods to determine this result, it is subject to the limitat

9、ions of finite-precision point in mathematics. In other words, it is possible for the rank function to return an incorrect result 4The test for observability10 11.1 Controllability of the plantText A: Pole Placement Design2 11.2 Observability of the plant11Outline13411.4 Summary1211.2 Observability

10、of the plant1Observability of the plant A plant is observable if it is possible to form an estimate of all of its internal states only the plant inputs and measured plant outputs. Matrices A and C of the linear plant model is used. The following MATLAB statements determine the observability of the s

11、tate-space plant model contained in ss plant variable. ob = obsv(ssplant) observable = (rank(ob) = length(ob) The variable observable will have value 1 if the plant is observable and 0 if not observable.The test for observability 11.1 Controllability of the plantText A: Pole Placement Design3 11.2 O

12、bservability of the plant13Outline12411.4 Summary After confirming that the plant model is both controllable and observable, the next design step is to select a suitable set of closed-loop pole locations. The number of pole locations that must be selected equals the number of elements in the system

13、state vector.1Observability of the plantSelecting the number of pole locations Function place()in the Control System Toolbox can be used to test the restriction of not allowing more poles than the number of inputs at any location. Function acker()does the same thing except that it is only suitable f

14、or use with SISO plants with orders less than 10 approximately. 142selecting closed-loop system pole locations The rules for selecting closed-loop system pole locations with the pole placement method are the as those used for root locus design. 1、The closed-loop poles must be located in the region o

15、f the complex plane to the left of the origin to ensure stability. 2、They must also be located left of a vertical line in the left half of the complex plane to satisfy settling time constraints. 3、The poles must be located between upper and lower diagonal lines in the left half of the complex emanat

16、ing from the origin to ensure that damping ratio constraints are satisfied. 4、The poles can be placed at points on the negative real axis or in complex conjugate pairs.152selecting closed-loop system pole locations To prevent driving the actuators too hard and to avoid the need for an overly fast di

17、gital controller sampling rate, the poles should not be located any further to the left in the complex plane than the necessity to satisfy the performance specifications. Loading the ss plant model into the SISO Design Tool can be done by the following command. sisotool(rlocus,ssplant) Add design co

18、nstraints for a 1-second settling time and a 0.7 damping ratio. From an examination of the resulting diagram, accep-table real pole locations can be chosen at -6 and -7 on the real axis.162selecting closed-loop system pole locations To design the vector K, form a vector containing these pole locatio

19、ns and compute the gain with the following two commands. p = -6 -7 K = place(ssplant.a, ssplant.b, p) Function place() computes the controller gain given the plants matrices A and B and a vector closed-loop pole locations. The resu-lting controller gain： K = 5.6000 -0.9062.172selecting closed-loop s

20、ystem pole locations For higher order plants, MATLAB provides a function select_poles(). This function computes a set of unique closed-loop pole locations given the plant models order and the closed-loop systems settling time and damping ratio constraints. The following functions can be used for the

21、 pole selection.SELECT_POLES P = SELECT_POLES(ORDER, T_SETTLE, DAMP_RATIO) P = SELECT_POLES(ORDER, T_SETTLE, DAMP_RATIO,SETTLING_PCT)182selecting closed-loop system pole locations For example, the function below computes the poles for a 4th-order system with a settling time of 0.5 seconds, a damping

22、 ratio of 0.8, and a settling error of 2%. p = select_poles(4, 0.5, 0.8, 0.02) p = select_poles (2, 1, 0.7) K = place(ssplant.a, ssplant.b, p) The results of these commands are closed-loop poles located at -4.6052 4.6982i and the controller gain K=3.7052 -0.8862. Till now we complete the design of t

23、he controller gain K. A settlingtime of 1 secondand a damping ratio of 0.7 involves the following steps.193Designing a state estimator If you dont have accurate measurements of all state variables, the next step is to design a state estimator. The observer (or state estimator) is implemented as show

24、n ：xAxBuL yCxDu The observer develops an estimate of the state vector by using the plant input vector u and the measurement y as its input. The term in the parentheses in Eq. (11. 2) is the observer error.Eq. (11. 2) The procedure for determining matrix L in Eq. (11. 2) is similar to the procedure f

25、or determining controller gain matrix K. 203Designing a state estimator As a rule of thumb, the real components of the observer pole locations should be approximately up to as five times large as those of the closed-loop system poles. A simple way to develop a suitable set of observer poles is to mu

26、ltiply all of the closed-loop poles by a factor of 3 to 5. Function place()can also be used for the observer design: q = 3 * p % Observer pole locations L = place(ssplant.a, ssplant.c, q) The results of these commands are observer poles located at -13.8155 10.3616i and the observer matrix3.024716.53

27、19TL 214Computing of the feedforward gain N The final design step is the computation of the feedforward gain NuxNNKN(11. 3) K is the controller gain matrix designed previously. The matrices Nu and Nx in Eq. (11. 3) are determined from the plant model as in Eq. (11. 4).10 xuNABNCDI (11. 4)Matrix 0 co

28、ntains only zeros, and I is an identity matrix225Combined observer-controller The next design step is to combine the observer and controller The complete computation of the state estimate and the plant act-uator u is conducted in Eq. (11. 5), using the reference input r and sensor measurements y as

29、inputs to the algorithm. xAxBuL yCxDuALC xLyBLD uuNrKx(11. 5) Equation (11. 5) requires the plant input vector u and the measu-red plant outputs y as inputs. In state-space models, only the outputs (elements of vector y) of a state-space component are available as inputs to a subsequent component. 2

30、35Combined observer-controller The solution is to form an augmented plant output vector by appe-nding the plant inputs to the vector. To do this, matrices C and D in the plant model are modified as in Eq. (11. 6), showing the modified matrices format.0yCDxuuIy (11. 6)With the result of Eq. (11. 6),

31、the closed-loop system equations consisting of the plant model and the observer-controller are written as follows.0 xAxBuCDxuIxALC xLBLD yuNrK xy(11. 7)246Integral Control An integral term can be added to eliminate steady-state errors.Figure 11.2 shows a control system configuration containing an ob

32、-server and control algorithm similar to those developed previously, along with a term involving the error integral. Subtracting the reference input from the output and integrating the result forms the error integral as in Eq.(11. 8)Ixeyr(11. 8)25 Multiplying the error integral by the gain -KI provi

33、des the integral control term, which is summed the output of a control algorithm sim-ilar to that developed previously. With the use of Eq. (11. 8) and bec-ause of y = Cx + Du, the system state equation with the added integr-al state is shown in Eq. (11. 9).6Integral Control 0100IIxCxDurxAxB (11. 9)

34、Now the control algorithm for the system is given in Eq. (11. 10).00IIxxuKKK N rx (11. 10)26 In designing the controller gain, one additional closed-loop pole location must be specified in the parameter p to accommodate the added state. The design of a SISO system is shown in the following statement

35、s.6Integral Control n = size(ssplant.a, 1) % Number of states a = 0ssplant.c; zeros(n, 1) ssplant.a) b = ssplant.d; ssplant.b p = select_poles(n+1, t_settle, damp_ratio) K = place(a, b, p) Ki = K(1) K0 = K(2:end) Nxu = inv(ssplant.a ssplant.b; ssplant.c splant.d)*zeros(n.1); 1 Nx = Nxu(1:n, :)27 The

36、 next statements break the K vector into the state feedback gain K0 and the integral gain KI for implementation in the controller structure shown in Figure 11.2. The final two statements determine Nx gain.6Integral Control The complete equations for the combined observer-controller this system is in

37、 Eq. (11.11).0IIIxxAxBuL yCxDuxyruK xKN rx (11.11)28 11.1 Controllability of the plantText A: Pole Placement Design3 11.2 Observability of the plant29Outline12411.4 Summary301Summary 11.4 Summary The feedback control algorithm used in the pole placement method is a gain matrix K multiplied by the sy

38、stem state vector x. Some restrictions on the structure of the state-space model must be satisfied for the pole placement design approach to succeed Given a controllable and observable state-space plant model along with a set of desired system and observer pole locations, the design of the controlle

39、r and observer requires only a few Control System Toolbox commands. Pole placement control with an error provides a generalization of the PID control. The pole placement approach described here is appli-cable to both MIMO and SISO systems. The same MATLAB comm-ands are used to design the estimator a

40、nd gain in both cases. 11.5 Concepts of Optimal ControlText B Optimal control design 111.6 Linear Quadratic Regulator (LQR) Design 312Outline11.7 Kalman state estimation11.8 Combined Observer-Controller11.9 Summary345321Concepts of Optimal Control 11.5 Concepts of Optimal Control In these approaches

41、, the controller design effort proceeds until it is good enough in terms of the performance requirements. There has been no attempt to design the best controller for a given application. In fact, the differences between pole placement and optimal control are the techniques used for determining the c

42、ontroller gain matrix K and the observer matrix L. The optimality criterion for the controller is based on a weighted trade-off between the response speed and the actuator effort. Two matrices Q and R specify the weighting function applied to the states and actuator commands. 33111.5 Concepts of Opt

43、imal ControlConcepts of Optimal Control The observer optimality criterion is based on the measurement noise of the sensors and the process noise of the plant itself. The optimal observer gain can be determines on the basis of the plant model, the process noise model, and covariance matrices. The opt

44、imal controller and observer gains are placed in a controller structure to prov-ide generalized forms of PD control and PID control. Note that these techniques are applicable to both SISO and MIMO plants. As with the pole placement method described before, the controller and observer gains for optim

45、al control are also designed in separate steps. 111.5 Concepts of Optimal ControlConcepts of Optimal Control The optimality criterion for the controller design is to minimize the scalar cost function J shown in equation. (11. 12) 0TTJ ux Qxu Ru dt(11. 12)x is the plant state vector with length n, u

46、is the input vector with length r. Q is n n matrix R is an r r matrix. A simple way to ensure this condition is to let Q and R be the diagonal matrices with the diagonal elements greater or equal to zero. When Q and R are diagonal matrices, Eq. (11.12) can be written as in Eq. (11.13). 111.5 Concept

47、s of Optimal ControlConcepts of Optimal Control 22110nriiiiiiiiJ uq xr udt(11. 13) qii and rii represent the diagonal elements of Q and Rxi and ui are the ith element of vectors x and u, Given the plant model and matrices Q and R, a single Control System Toolbox command can compute the optimal contr

48、oller gain K The observer in the presence of process noise and measurement noise is called Kalman filter.11.5 Concepts of Optimal ControlA Kalman filter can be time-varying or steady state. The varying filter is fairly complex and computationally expensive and thus will not be described here. The di

49、scussion below limits to the steady-state Kalman filter. This filter uses matrix L to which the time-varying filter converges after a number of iterations, assuming that the system noise properties remain fixed. In developing the Kalman filter, all of the error sources must be described in the form

50、of truly random white noise passed through a filter.1Concepts of Optimal Control 11.5 Concepts of Optimal Control The designer must develop two covariance matrices describing the properties of the white noise to the process noise and measurement noise models The matrices K and L in optimal control d

51、escribed above must be placed controller structures identical to those described before. These steps include the determination of the feedforward gain N the optional use of integral control. optimal design moves the iteration from the selection of pole locations to the development of cost function w

52、eighting matrices, the selection of a process noise model, and the determination of process and measurement noise covariance matrices.1Concepts of Optimal Control 11.5 Concepts of Optimal ControlText B Optimal control design 111.6 Linear Quadratic Regulator (LQR) Design 382Outline11.7 Kalman state e

53、stimation11.8 Combined Observer-Controller11.9 Summary34511.6 Linear Quadratic Regulator (LQR) Design1 The controller gain K that minimizes the cost function J shown in Eq. (11. 12) is called a linear quadratic regulator. We will use the controller structure in Figure 11.1 to demonstrate the LQR des

54、ign The MATLAB Control System Toolbox provides the function lqr() for the LQR design in time-domain systems. With this function, the work is mainly in the selection of appropriate weighting matrices Q and R. It is often best to set these matrices to be diagonal with the positive elements. 3911.6 Lin

55、ear Quadratic Regulator (LQR) Design1The design procedure of the LQR control is described as follows. 1. Initialize the Q and R as identity matrices.2. Compute a controller gain K with function lqr().3. Evaluate the performance of the resulting controller assuming perfect state estimation. 4. If the

56、 performance is acceptable, work is done. Otherwise, go to step 5.5. (a) Increase the diagonal element weight in Q for states that must respond more rapidly , (b) Increase corresponding elements in R for input signals that are being driven too hard, (c) Decrease the elements in Q and R for input and

57、 output that are of less concern, Note that in initial design iterations, increasing and/or decreasing the weights is usually by a multiplying factor of 10.0 in value.6. Return to step 2.4011.6 Linear Quadratic Regulator (LQR) Design2The plant model 21001.8100GssThe settling time of 1second and the

58、damping ratio of at least 0.7 We begin the LQR design by initializing R = 1 and Q = 1, 1T and find the controller gain by using function lqr() in Control System Toolbox. tfplant = tf(100, 1 1.8 100) ssplant = ss(tfplant); Q = diag(1 1); R = 1; K = lqr(ssplant, Q, R)4111.6 Linear Quadratic Regulator

59、(LQR) Design2This sequence of commands produces the controller gain K = 2.44 0.29. For the moment, assume that all states are available as outputs, which can be done by setting matrix C to be a 2 x 2 identity matrix. That is, ssplant.c = eye(2); The command above forms a closed-loop system with the

60、controller specifications required (i.e., the settling time of 1.0 and damping ratio of 0.7). The resulting plot is shown in Figure 6.1 by the following commands.4211.6 Linear Quadratic Regulator (LQR) Design2 cl_sys = feedback(ssplant, -K, +1): t_settle = 1; damp_ratio = 0.7; plot_poles(cl_sys, t_s