控制理论基础-自控a_ch10

1,Ch 10 Design of state variable feed-back system,10.1 Introduction10.2 Controllability10.3 Observability10.4 The Duality Theory10.5 System Structure Decompose10.6 Controllable canonical form and observable canonical form10.7 Cancellation of poles and zeros of scalar system10.8 Controllability of output10.9 The Effect on Controllability and Observability of Feedback Control10.10 Pole placement10.11 Observer design,2,10.1 Introduction,All the state variables are measurable-full state feedback control law.An observer to estimate the states that are not directly sensed.Full-state feedback control law + the observerControllability and Observability,3,Controllability and observability are the inner structure of linear system,4,Example 10.1 the system electric network shown in the following Fig.,RC=1/3,State equation:,10.2 Controllability,5,Thus solution of state equation:,1)x1(0)=x2(0) locates on the line x1=x2,Also locates on the line x1=x2,6,2)x1(0)=-x2(0) locates on the line x1=-x2,Also locates on the line x1=-x2,7,Conclusion: To x1(0)=x2(0), we can find u(t), which can make x(t)=0 in limited time. But to x1(0)=-x2(0), we can find u(t), which cant make x(t)=0 in limited time. System state is not controllable completely.,This examples are just a kind of approximate illustration but not strictly describing, so it is only utilized to interpret or determine controllability of system which is very simple or very obvious.,8,A system is completely controllable if there exists an unconstrained control u(t) that can transfer any limited initial state x(t0) to final-state vector x(t)=0 in the limited time region t0 , t.,Definition,9,A system is completely reachability if there exists an unconstrained control u(t) that can transfer initial state x(t0)=0 to any limited final-state vector x(t) in the limited time region t0 , t.,Definition,10,Discussion:,1). Controllable state x(t0)0, x(t)=0 (Controllability) Reachable state x(t0) =0, x(t)0 (Reachability)2). Controllability: (1) controllable in any state (2) all the nonzero limited point in the state space can reach the origin of the state space.3). Controllability concern with the exist of u(t) other than determining the value of u(t) .4). For the continuous linear time-invariant system, controllability is equivalent to reachability.,11,1) The Algebra Criterion of Controllability,The continuous linear time-invariant system A, B is completely controllable if and only if its controllability matrix has rank Qc =rank B AB An-1B=n,12,Example 10.2,The system is not controllable.,13,Example 10.3,The system is controllable.,14,Example 10.4,15,When R1C1=R2C2, rankQc2, the system is not controllable.,例子可见系统达到不完全可控的条件十分苛刻，因此一般实际系统往往是可控的。,16,2) The PBH Criterion of Controllability,The continuous linear time-invariant system A, B is completely state-controllable if and only if all its eigenvalues li(i=1,2,n) satisfy: rankliI-A B=n, i=1,2,n,17,Example 10.5,The eigenvalues of the system are,18,So as l1,2=0,as,The system is controllable.,19,3) The Eigenvalue Criterion of Controllability Suppose the system A, B has different eigenvalues l1, l2,., ln, and we can get diagonal formafter the nonsingular linear transfer of the system. Then its state is completely controllable if and only if the row vectors of control matrix are not zero.,20,21,If,Then can not be controlled by,directly or indirectly. So the systems,state xi is uncontrollable state and is a uncontrollable mode.,22, If diagonal form of system A, B has more than one eigenvalue be same, the sufficient and necessary condition of state completely controllable is that the row vectors of control matrix B corresponding to the same eigenvalue of A is linear independent, and all other row vectors of the control matrix B is not zero.,23, Suppose the system A, B has repeated eigenvalues, where ，and every repeated eigenvalue is different, and we can only get Jordan standard form after the nonsingular linear transfer of the system, where,24,Then its state is completely controllable if and only if the row vectors of control matrix corresponding to the last row of every Jordan block Ji (i=1,2,k) in the control matrix is not zero.,25, If Jordan form of system A, B has more than one Jordan blocks have the same eigenvalue, the sufficient and necessary condition of state completely controllable is that the row vectors of the control matrix B corresponding to the last row of every Jordan block Ji (i=1,2,k) which has the same eigenvalue is linear independent, and all of the row vectors corresponding to the last row of other Jordan block in the control matrix is not zero.,26,Example 10.6,27,Examples 10.7,28,29,30,31,32,33,10.3 Observability,Example 10.8 the system electric network shown in the following Fig.,34,State equation:,Thus solution of free motion:,35,1) x1(0)=x2(0) locates on the line x1=x2,So we cant get x(0) from y(t), just knowx1(0)=x2(0).,36,2) x1(0)=-x2(0) locates on the line x1=-x2,So if we know y(t), we also can know x(0).,3) Other conditions,If we know y(t), we can only know x1(0)-x2(0).,So the system is not completely state-observable.,37,A linear system is completely observable if and only if there exists a finite time T such that the initial state x(0) can be determined from the observation history y(t) given the control u(t), 0 t T.,Definition,38,Some explanations,If states can be identified by y(t) in free motion, then it must be able to identified by y(t) in forced motion.To linear constant system, if the initial state can be observed during t0, ta, then any initial state can also be observed.,39,Linear nonsingular transform to system can not change the properties of system observabilitly .State observability determined by matrix A and C.If the system can be observed, then the state in any time can also be observed.,40,1) The Algebra Criterion of Observability,The continuous linear time-invariant system A,C is completely observable if and only if its observability matrix has rank,41,Example 10.8,The system is observable.,42,If R1C1=R2C2, rank Qo2, the system is not observable.,In example 10.4, if the output is,43,2) The PBH Criterion of Observability,The continuous linear time-invariant system A, C is completely observable if and only if all its eigenvalues li(i=1,2,n) satisfy:,44,3) The Eigenvalue Criterion of Observability Suppose the system A,C has different eigenvalues l1, l2,., ln, and we can get diagonal formafter the nonsingular linear transfer of the system. Then its state is completely observable if and only if the column vectors of output matrix are not zero.,45,46,If,Then can not be observed by,directly or indirectly. So the systems,state xi is unobservable state and is a unobservable mode.,47, If diagonal form of system A, C has more than one eigenvalue be same, the sufficient and necessary condition of state completely observable is that the column vectors of output matrix C corresponding to the same eigenvalue of A is linear independent, and all other column vectors of the output matrix C is not zero.,48, Suppose the system A, C has repeated eigenvalues, where ，and every repeated eigenvalue is different, and we can only get Jordan standard form after the nonsingular linear transfer of the system, where,49,Then its state is completely controllable if and only if the column vectors of output matrix corresponding to the first column of every Jordan block Ji (i=1,2,k) in the control matrix is not zero.,50, If Jordan form of system A, C has more than one Jordan blocks have the same eigenvalue, the sufficient and necessary condition of state completely observable is that the column vectors of the output matrix C corresponding to the first column of every Jordan block Ji (i=1,2,k) which has the same eigenvalue is linear independent, and all of the column vectors corresponding to the first column of other Jordan block in the control matrix is not zero.,51,Example 10.9,52,10.4 The Duality Theory,The two linear system can be described by state space equations as follows:,We call they are duality system.,53,The relationship of coefficient matrixes of duality system:,54,Conclusion: The controllability of system state is corresponding to the observability of duality system state. The observability of system state is corresponding to the controllability of duality system state.,55,10.5 System Structure Decompose,Example:,56,Therefore Sub-system S1(c+, o+) has state variable x2 Sub-system S2(c+, o-) has state variable x1 Sub-system S3(c-, o+) has state variable x4 Sub-system S4(c-, o-) has state variable x3,57,1. The system A, B,C is not completely controllable, where controllability matrix has rankQc=ln. Then we can use nonsingular transform to get standardization by controllable analyzes.,is l controllable state and is n-l uncontrollable state.,58,59,Construct T-1 We can select , which are l linear independent column vectors of . Then we select n-l column vectors , which are random, but they must be linear independent between each other, to form nonsingular transformation matrix,60,Example10.10,61,几点说明：1、 从传递函数的角度分析时， A, B,C 和 是等价的。2、当 含稳定的特征根时，系统可镇定。3、可控性规范分解不是唯一的，但不同分解的子系统的特征根相同。,62,2. The system A,B,C is not completely observable, where observable matrix has rankQo=rn. Then we can use nonsingular transform to get standardization by observable analyzes is r observable state and is n-r unobservable state.,63,64,We can select , which are r linear independent row vectors of . Then we select n-r row vectors , which are random, and must be linear independent between each other, to form nonsingular transformation matrix T.,Construct T,65,Example 10.11,66,3. The system A, B, C is not completely controllable and observable, where controllability matrix has rankQc = l and observability matrix has rankQo= r. Then we can use controllable and observable nonsingular transformation in turns to get standardization by controllable and observable analyzes.,67,(1) Use nonsingular transform to get standardization by controllable analyzes:,(2) To the controllable sub-system, use nonsingular transform to get standardization by observable analyzes:,68,(3) To the uncontrollable sub-system, use nonsingular transform to get standardization by observable analyzes:,(4) The three nonsingular transforms are as:,69,The system is transformed to the following canonical form,70,71,Example 10.12,72,The controllable sub-system,The uncontrollable sub-system is observable, so the transform should be To2=1.,73,So,74,10.6 Controllable canonical form and observable canonical form,1) Controllable canonical form,75,If a single input linear time-invariable systemis controllable, we can use nonsingular transformation to get controllable canonical form,76,可控规范型的另一种求法是通过传递函数的分母，即特征多项式的系数直接获得。,Construct P,77,Example 10.13,78,79,2) observable canonical form,80,Second observable canonical form,81,If a single output linear time-invariable systemis observable, we can use nonsingular transformation to get observable canonical form,82,Construct T-1,83,Example 10.14,84,85,Note:Suppose the controllable system duality with S1 is . Apply the duality theory, then the transformation matrix needed by observable standard form of S1 is the transposed matrix of the transformation matrix needed by controllable standardization of the duality system S2.,86,10.7 Cancellation of poles and zeros of scalar system,Judge controllability and observability from transfer function.,Example 1,87,Definition: Transfer function from input to state,-1 has been cancelled from input to state, called input decoupling zero, -1 is uncontrollable pole.,88,Example 2,Definition: Transfer function from state to output,89,-1 has been cancellation from state to output, called output decoupling zero, -1 is unobservable pole.,90,If matrix have (occur ) pole-zero canceling ,the SISO system must be uncontrollable or unobservable.,When the same duplicated root correspond to more than on Jordan block, must have pole-zero canceling. So this kind of system must neither controllable nor observable.,Note:,91,Conclusion: 1.The sufficient and necessary condition of linear scalar system A, B state completely controllable is the input-state transfer function Gb(s) has no cancellation of zero and pole. 2. The sufficient and necessary condition of linear scalar system A, C state completely observable is the state-output transfer function Gc(s) has no cancellation of zero and pole.,92,3. The sufficient and necessary condition of linear scalar system A,B,C state completely controllable and observable is the input-output transfer function G(s) has no cancellation of zero and pole.,93,10.8 Controllability of output,Definition: For the linear time-invariant continuous system, we assume that we can find a continuous or subsection-continuous control-function u(t), and in the limited time region t0,t, we can make the system transfer from an random nonzero limited initial-output y(t0) to any final-output y(t), thus the system is completely output-controllable in t0 , ta.,94,Note: output-controllable is not equivalent to state-controllable.,Algebra criterionThe system A, B, C is completely output-controllable if and only if rank of its output controllability matrix P is m. rankP =rankCB CAB CAn-1B D =m,95,10.9 The Effect on Controllability and Observability of Feedback Control,1. Output feedback,u=Rr-LyR: ll nonsingular constant matrix.L: l m output feedback constant matrix.,96,Conclusion:Output feedback can change the poles, but do not change the systems order.,97,2. state feedback,u=Rr-KxR: ll nonsingular constant matrix;K: l n state feedback constant matrix;,98,Conclusion:State feedback can change the poles, but do not change the systems order.,99,3. The relationship between output feedback and state feedback.,Conclusion:Output feedback is special condition of state feedback(K=LC), it isnt complete information feedback. Sometimes it cannt gain satisfied control performance.,100,3. The Effect on Controllability and Observability of Feedback Control,(1) State feedback,State feedback have no effect on state controllability. State feedback possibly have effect on state observability.,101,102,(A-BK)B的列是B AB的列的线性组合，同理可推出QcK的每一列都为Qc的列的线性组合。因此：rank QcK rank Qc由于A B也是(A-BK) B的状态反馈系统，故 rank Qc rank QcK因此： rank Qc =rank QcK,103,Example: State feedback maybe affect system observability.,104,Conclusion:State feedback closed-loop system have same observability as plant (original system) if and only if the closed-loop system have no zero-poles cancellation .,105,(2) Output feedback,Conclusion:Output feedback have no effect on state controllability and observability.,106,对系统稳定性的影响：状态反馈和输出反馈都能影响系统的稳定性。通过反馈使闭环系统稳定，称之为镇定。状态反馈具有很多优越性，且输出反馈总可以找到与之性能相同的状态反馈，因此一般只讨论状态反馈。状态反馈系统 的(A-BK) 的特征根都具有负实部，则称系统实现了状态反馈镇定。,107,定理: 当且仅当线性定常系统的不可控部分渐进稳定时，系统是状态反馈可镇定的。证明： A B不完全可控，则可控性分解：,可见状态反馈对不可控部分的极点毫无影响。,108,10.10 Pole placement,1. State feedback control of SISO system,109,2. Pole placementThe closed-loop system is So the characteristic equation isTheorem: the eigenvalues of (A-BK) can arbitrarily be assigned if and only if the system (A, B) is controllable.,110,If the system is controllable, we can use nonsingular transformation to get controllable canonical form:,If the state variable feedback matrix is,111,So the state feedback matrix is,112,The characteristic equation is,If we want the roots be placed to meet the desired response, the desired characteristic equation should be,113,Therefore, the state variable feedback matrix is,114,Example 10.15 Consider the third-order system:,If we want the roots to be placed at -5, -2+2j, -2-2j, so the desired characteristic equation is,So the transfer function of the closed-loop system is,115,3. Ackermanns formula:The LTI SISO system is (A, B), if the state variable feedback matrix is,Given the desired characteristic equation,The state feedback gain matrix is,where,116,Example 10.16 Consider the system:,Determine the feedback gain to place the closed-loop poles at s= -11j. So the desired characteristic equation,117,So,118,4. 状态反馈对传递函数零点的影响,传递函数实现为可控规范型：,引入状态反馈后,(A-BK)仍然是友矩阵，因此系统仍然符合可控规范型。可见状态反馈不改变传递函数的分子多项式，即不改变系统的零点。但存在这种情况：即引入状态反馈改变系统极点后与零点对消，则造成对消的极点不可观测。,119,10.11 Observer design,In general, only a subset of the states are measurable and available for feedback.If the system is completely observable, then it is possible to determine (or to estimate) the states that are not directly measured.,120,1. Full-dimensional state estimator,Consider the n-dimensional state equationWe can duplicate the original system as,It is an open-loop estimator, and there are two disadvantages: Have to compute the initial state each time we use the estimator. Because of disturbance or imperfect of modeling, the difference between x(t) and will grow up with time.,121,An closed-loop estimator:,122,The difference between y(t) and is used as a correcting term:,So if (A-LC) is stable, will approach to zero.,123,Note:The speed of approaching to zero is determined by the eigenvalues of (A-LC) .Theorem: The eigenvalues of (A-LC) can be assigned arbitrary by selecting a real constant vector L if and only if (A, C) is observa