Chapter 1 Matrices and Systems of EquationsLinear systems arise in applications to such areas as engineering, physics, electronics, business, economics, sociology（社会学）, ecology （生态学）, demography(人口统计学), and genetics（遗传学）, etc. §1. Systems of Linear EquationsNew words and phrases in this section:Linear equation 线性方程Linear system，System of linear equations 线性方程组Unknown 未知量Consistent 相容的Consistence 相容性Inconsistent不相容的Inconsistence 不相容性Solution 解Solution set 解集Equivalent 等价的Equivalence 等价性Equivalent system 等价方程组Strict triangular system 严格上三角方程组Strict triangular form 严格上三角形式Back Substitution 回代法Matrix 矩阵Coefficient matrix 系数矩阵Augmented matrix 增广矩阵Pivot element 主元Pivotal row 主行Echelon form 阶梯形1.1 DefinitionsA linear equation (线性方程) in n unknowns（未知量）is1122...n na x a x a x b+++=A linear system of m equations in n unknowns is11112211211222221122...... .........n n n n m m m n n m a x a x a x b a x a x a x b a x a x a x b+++=⎧⎪+++=⎪⎨⎪⎪+++=⎩ This is called a m x n (read as m by n) system.A solution to an m x n system is an ordered n-tuple of numbers （n 元数组）12(,,...,)n x x x that satisfies all the equations.A system is said to be inconsistent （不相容的） if the system has no solutions.A system is said to be consistent （相容的）if the system has at least one solution.The set of all solutions to a linear system is called the solution set（解集）of the linear system.1.2 Geometric Interpretations of 2x2 Systems11112212112222a x a xb a x a x b +=⎧⎨+=⎩ Each equation can be represented graphically as a line in the plane. The ordered pair 12(,)x x will be a solution if and only if it lies on bothlines.In the plane, the possible relative positions are(1) two lines intersect at exactly a point; (The solution set has exactly one element)(2)two lines are parallel; (The solution set is empty)(3)two lines coincide. (The solution set has infinitely manyelements)The situation is the same for mxn systems. An mxn system may not be consistent. If it is consistent, it must either have exactly one solution or infinitely many solutions. These are only possibilities.Of more immediate concerns is the problem of finding all solutions to a given system.1.3 Equivalent systemsTwo systems of equations involving the same variables are said to be equivalent（等价的，同解的）if they have the same solution set.To find the solution set of a system, we usually use operations to reduce the original system to a simpler equivalent system.It is clear that the following three operations do not change the solution set of a system.(1)Interchange the order in which two equations of a system arewritten;(2)Multiply through one equation of a system by a nonzero realnumber;(3)Add a multiple of one equation to another equation. (subtracta multiple of one equation from another one)Remark: The three operations above are very important in dealing with linear systems. They coincide with the three row operations of matrices. Ask a student about the proof.1.4 n x n systemsIf an nxn system has exactly one solution, then operation 1 and 3 can be used to obtain an equivalent “strictly triangular system ”A system is said to be in strict triangular form (严格三角形) if in the k-th equation the coefficients of the first k-1 variables are all zero and the coefficient ofkx is nonzero. (k=1, 2, …,n)An example of a system in strict triangular form:123233331 2 24x x x x x x ++=⎧⎪-=⎨⎪=⎩Any nxn strictly triangular system can be solved by back substitution (回代法).(Note: A phrase: “substitute 3 for x ” == “replace x by 3”)In general, given a system of linear equations in n unknowns, we will use operation I and III to try to obtain an equivalent system that is strictly triangular.We can associate with a linear system an mxn array of numbers whose entries are coefficient of theix ’s. we will refer to this array as thecoefficient matrix (系数矩阵) of the system.111212122212.....................n nm m m n a a a a a a a a a ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭A matrix （矩阵） is a rectangular array of numbersIf we attach to the coefficient matrix an additional column whose entries are the numbers on the right-hand side of the system, we obtain the new matrix11121121222212n n s m m m na a ab a a a b b a a a ⎛⎫ ⎪ ⎪ ⎪⎝⎭We refer to this new matrix as the augmented matrix （增广矩阵） of a linear system.The system can be solved by performing operations on the augmented matrix. i x ’s are placeholders that can be omitted until the endof computation.Corresponding to the three operations used to obtain equivalent systems, the following row operation may be applied to the augmented matrix.1.5 Elementary row operationsThere are three elementary row operations:(1)Interchange two rows;(2)Multiply a row by a nonzero number;(3)Replace a row by its sum with a multiple of another row.Remark: The importance of these three operations is that they do not change the solution set of a linear system and may reduce a linear system to a simpler form.An example is given here to illustrate how to perform row operations on a matrix.★Example:The procedure for applying the three elementary row operations:Step 1: Choose a pivot element (主元)(nonzero) from among the entries in the first column. The row containing the pivotnumber is called a pivotal row（主行）. We interchange therows (if necessary) so that the pivotal row is the new firstrow.Multiples of the pivotal row are then subtracted form each of the remaining n-1 rows so as to obtain 0’s in the firstentries of rows 2 through n.Step2: Choose a pivot element from the nonzero entries in column 2, rows 2 through n of the matrix. The row containing thepivot element is then interchanged with the second row ( ifnecessary) of the matrix and is used as the new pivotal row.Multiples of the pivotal row are then subtracted form eachof the remaining n-2 rows so as to eliminate all entries belowthe pivot element in the second column.Step 3: The same procedure is repeated for columns 3 through n-1.Note that at the second step, row 1 and column 1 remain unchanged, at the third step, the first two rows and first two columns remain unchanged, and so on.At each step, the overall dimensions of the system are effectively reduced by 1. (The number of equations and the number of unknowns all decrease by 1.)If the elimination process can be carried out as described, we will arrive at an equivalent strictly triangular system after n-1 steps.However, the procedure will break down if all possible choices for a pivot element are all zero. When this happens, the alternative is to reduce the system to certain special echelon form（梯形矩阵）. AssignmentStudents should be able to do all problems.Hand-in problems are: # 7--#11§2. Row Echelon FormNew words and phrases:Row echelon form 行阶梯形Reduced echelon form 简化阶梯形 Lead variable 首变量 Free variable 自由变量Gaussian elimination 高斯消元Gaussian-Jordan reduction. 高斯-若当消元 Overdetermined system 超定方程组 Underdetermined systemHomogeneous system 齐次方程组 Trivial solution 平凡解2.1 Examples and DefinitionIn this section, we discuss how to use elementary row operations to solve mxn systems.Use an example to illustrate the idea.★ Example : Example 1 on page 13. Consider a system represented by the augmented matrix111111110011220031001131112241⎛⎫ ⎪--- ⎪ ⎪-- ⎪- ⎪ ⎪⎝⎭ 111111001120002253001131001130⎛⎫⎪ ⎪ ⎪ ⎪- ⎪ ⎪⎝⎭………..(The details will given in class)We see that at this stage the reduction to strict triangular form breaks down. Since our goal is to simplify the system as much as possible, we move over to the third column. From the example above, we see that the coefficient matrix that we end up with is not in strict triangular form,it is in staircase or echelon form (梯形矩阵).111111001120000013000004003⎛⎫ ⎪ ⎪ ⎪ ⎪- ⎪ ⎪-⎝⎭The equations represented by the last two rows are:12345345512=0 2=3 0=4 03x x x x x x x x x ++++=⎧⎪++⎪⎪⎨⎪-⎪=-⎪⎩Since there are no 5-tuples that could possibly satisfy these equations, the system is inconsistent.Change the system above to a consistent system.111111110011220031001133112244⎛⎫ ⎪--- ⎪ ⎪-- ⎪ ⎪ ⎪⎝⎭ 111111001120000013000000000⎛⎫⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭The last two equations of the reduced system will be satisfied for any 5-tuple. Thus the solution set will be the set of all 5-tuples satisfying the first 3 equations.The variables corresponding to the first nonzero element in each row of the augment matrix will be referred to as lead variable .（首变量） The remaining variables corresponding to the columns skipped in the reduction process will be referred to as free variables （自由变量）.If we transfer the free variables over to the right-hand side in the above system, then we obtain the system:1352435451 2 3x x x x x x x x x ++=--⎧⎪+=-⎨⎪=⎩which is strictly triangular in the unknown 1x 3x 5x . Thus for each pairof values assigned to 2xand4x , there will be a unique solution.★Definition: A matrix is said to be in row echelon form (i) If the first nonzero entry in each nonzero row is 1.(ii)If row k does not consist entirely of zeros, the number of leading zero entries in row k+1 is greater than the number of leading zero entries in row k.(iii) If there are rows whose entries are all zero, they are below therows having nonzero entries.★Definition : The process of using row operations I, II and III to transform a linear system into one whose augmented matrix is in row echelon form is called Gaussian elimination （高斯消元法）.Note that row operation II is necessary in order to scale the rows so that the lead coefficients are all 1.It is clear that if the row echelon form of the augmented matrix contains a row of the form (), the system is inconsistent.000|1Otherwise, the system will be consistent.If the system is consistent and the nonzero rows of the row echelon form of the matrix form a strictly triangular system (the number of nonzero rows<the number of unknowns), the system will have a unique solution. If the number of nonzero rows<the number of unknowns, then the system has infinitely many solutions. (There must be at least one free variable. We can assign the free variables arbitrary values and solve for the lead variables.)2.2 Overdetermined SystemsA linear system is said to be overdetermined if there are more equations than unknowns.2.3 Underdetermined SystemsA system of m linear equations in n unknowns is said to be underdetermined if there are fewer equations than unknowns (m<n). It is impossible for an underdetermined system to have only one solution.In the case where the row echelon form of a consistent system has free variables, it is convenient to continue the elimination process until all the entries above each lead 1 have been eliminated. The resulting reduced matrix is said to be in reduced row echelon form. For instance,111111001120000013000000000⎛⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭ 110004001106000013000000000⎛⎫⎪- ⎪ ⎪ ⎪ ⎪ ⎪⎝⎭Put the free variables on the right-hand side, it follows that12345463x x x x x =-=--=Thus for any real numbersαandβ, the 5-tuple()463ααββ---is a solution.Thus all ordered 5-tuple of the form ()463ααββ--- aresolutions to the system.2.4 Reduced Row Echelon Form★Definition : A matrix is said to be in reduced row echelon form if :(i)the matrix is in row echelon form.(ii) The first nonzero entry in each row is the only nonzero entry in its column.The process of using elementary row operations to transform a matrix into reduced echelon form is called Gaussian-Jordan reduction.The procedure for solving a linear system:(i) Write down the augmented matrix associated to the system; (ii) Perform elementary row operations to reduce the augmented matrix into a row echelon form;(iii) If the system if consistent, reduce the row echelon form into areduced row echelon form. (iv) Write the solution in an n-tuple formRemark: Make sure that the students know the difference between the row echelon form and the reduced echelon form.Example 6 on page 18: Use Gauss-Jordan reduction to solve the system:1234123412343030220x x x x x x x x x x x x -+-+=⎧⎪+--=⎨⎪---=⎩The details of the solution will be given in class.2.5 Homogeneous SystemsA system of linear equations is said to be homogeneous if theconstants on the right-hand side are all zero.Homogeneous systems are always consistent since it has a trivial solution. If a homogeneous system has a unique solution, it must be the trivial solution.In the case that m<n (an underdetermined system), there will always free variables and, consequently, additional nontrivial solution.Theorem 1.2.1 An mxn homogeneous system of linear equations has a nontrivial solution if m<n.Proof A homogeneous system is always consistent. The row echelon form of the augmented matrix can have at most m nonzero rows. Thus there are at most m lead variables. There must be some free variable. The free variables can be assigned arbitrary values. For each assignment of values to the free variables, there is a solution to the system.AssignmentStudents should be able to do all problems except 17, 18, 20.Hand-in problems are 9, 10, 16,Select one problem from 14 and 19.§3. Matrix AlgebraNew words and phrases:Algebra 代数Scalar 数量，标量Scalar multiplication 数乘 Real number 实数 Complex number 复数 V ector 向量Row vector 行向量 Column vector 列向量Euclidean n-space n 维欧氏空间 Linear combination 线性组合 Zero matrix 零矩阵Identity matrix 单位矩阵 Diagonal matrix 对角矩阵 Triangular matrix 三角矩阵Upper triangular matrix 上三角矩阵 Lower triangular matrix 下三角矩阵 Transpose of a matrix 矩阵的转置(Multiplicative ) Inverse of a matrix 矩阵的逆 Singular matrix 奇异矩阵 Singularity 奇异性Nonsingular matrix 非奇异矩阵 Nonsingularity 非奇异性The term scalar (标量，数量) is referred to as a real number (实数) or a complex number （复数）. Matrix notationAn mxn matrix, a rectangular array of mn numbers.111212122212.....................n nm m m n a a a a a a a a a ⎛⎫⎪ ⎪ ⎪ ⎪⎝⎭()ij A a =3.1 VectorsMatrices that have only one row or one column are of special interest since they are used to represent solutions to linear systems.We will refer to an ordered n-tuple of real numbers as a vector （向量）.If an n-tuple is represented in terms of a 1xn matrix, then we will refer to it as a row vector . Alternatively, if the n-tuple is represented by an nx1 matrix, then we will refer to it as a column vector . In this course, we represent a vector as a column vector.The set of all nx1 matrices of real number is called Euclidean n-space (n 维欧氏空间) and is usually denoted by nR.Given a mxn matrix A, it is often necessary to refer to a particular row or column. The matrix A can be represented in terms of either its column vectors or its row vectors.12(a ,a ,,a )n A = ora (1,:)a(2,:)a(,:)A m ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭3.2 EqualityFor two matrices to be equal, they must have the same dimensions and their corresponding entries must agree★Definition : Two mxn matrices A and B are said to be equal ifij ij a b =for each ordered pair (i, j)3.3 Scalar MultiplicationIf A is a matrix,αis a scalar, thenαA is the mxn matrix formed by multiplying each of the entries of A byα.★Definition : If A is an mxn matrix, αis a scalar, thenαA is themxn matrix whose (i, j) is ij a αfor each ordered pair (i, j) .3.4 Matrix AdditionTwo matrices with the same dimensions can be added by adding their corresponding entries.★Definition : If A and B are both mxn matrices, then the sum A+B is the mxn matrix whose (i,j) entry isij ija b + for each ordered pair (i, j).An mxn zero matrix (零矩阵) is a matrix whose entries are all zero. It acts as an additive identity on the set of all mxn matrices.A+O=O+A=AThe additive of A is (-1)A since A+(-1)A=O=(-1)A+A.A-B=A+(-1)B-A=(-1)A3.5 Matrix Multiplication and Linear Systems3.5.1 MotivationsRepresent a linear system as a matrix equationWe have yet to defined the most important operation, the multiplications of two matrices. A 1x1 system can be writtena xb =A scalar can be treated as a 1x1 matrix. Our goal is to generalize the equation above so that we can represent an mxn system by a single equation.A X B=Case 1: 1xn systems 1122... n n a x a x a x b +++=If we set()12n A a a a =and12n x x X x ⎛⎫ ⎪⎪= ⎪ ⎪⎝⎭, and define1122...n n AX a x a x a x =+++Then the equation can be written as A X b =。
Linear System Theory andDesignLecturer Ma Hongjun Long LijunPart 4Chapter 6 Controllability andObservability6.1 IntroductionThis chapter introduces the concepts of controllability and observability.Controllability deals with whether or not the state-space equation can be controlled from the input.Observability deals with whether or not the initial state can be observed from the output.These concepts can be illustrated using the network shown in Fig. 6.1Fig. 6.1 NetworkFrom the network we know that\The input has no effect on x 2or can not control x 2 because of the open circuit across y.\The current passing through the resistoralways equals the current source u; therefore the response excited by the initial state x 1will not appear in y, that is the initial state x 1can not be observed from the output y.Ω−26.2 ControllabilityConsider the n-dimensional p-input state equation(6.1)where A and B are and real constantmatrices, respectively.Bu Ax x +=&n n ×p n ×Definition 6.1: The state equation (6.1) or the pair (A ,B ) is said to be controllable if for any initial state x (0)=x 0and any final state x 1, there exists an input that transfers initial state x 0to x 1in a finite time. Otherwise (6.1) is said to be uncontrollable.Example 6.1Consider the network shown in Fig.6.2 (a)Fig. 6.2 Uncontrollable networkIf x(0)=0, then x(t)=0 for all t≥0 no matter what input u is applied because of the symmetry of the network.So the input u has no effect on the voltage across the capacitorThat is the system is uncontrollable.Next we consider the network shown in Fig.6.2 (b)If x 1(0)=x 2(0)=0, no matter what input isapplied, x1(t) always equals x2(t) for all t ≥0. This means that there exist no control u which transfers 0 to state with x 1≠x 2.Fig. 6.2 Uncontrollable networkTheorem 6.1The following statements are equivalent.1.The n-dimensional pair (A , B ) is controllable.2.The n ×n matrix3.The n ×np controllability matrix4.The n ×(n+p) matrix has full row rank atevery eigenvalue, , of A .∫∫−−==t t T t t T c d e e d e e t T T 0)()(0)(ττττττA A A A BB BB W (6.2)is nonsingular for any t>0 .[]B A B A AB B 12−=n C L (6.3)has rank n (full row rank).[]B λI A −λ5.If, in addition, all eigenvalues of A have negative real parts, then the unique solution ofAW c +W c A T =-BB T (6.4)is positive definite. The solution is called the controllability Gramian given by∫∞=0τττd e e T T C A A BB W (6.5)Proof: 2 1: The response of (6.1) at time t 1is⇒∫−+=1110)(1)()0()(t t t d e x e t τττBu x A A (6.6)We verify that for any x (0)=x 0and any x (t 1)=x 1, the input(6.7)will transfer x 0to x 1at time t 1. In fact, substituting (6.7) into (6.6) yields1 2: We proof this by contradictive method. Suppose W c (t 1) is not nonsingular.])[()(1011)(11x x W B u A A −−=−−t Ct t Te t et T ∫−−=−−−111111011)()(01])[()(t t Ct Tt t d e t eee t T τττx x W BB x x A A A A ])[(10110)()(011111x x W BB x A A A A −⋅−=−−−∫t Ct t Tt t e t d eee T τττ1101110])[()(11x x x W W x A A =−−=−t CC t e t t e ⇒And this means that there exists an n ×1 nonzero vector v such thatwhich implies that(6.8)Because (6.1) is controllable, there must exist an input that transfers the initial stateto 0at time t 2.At this time, (6.6) becomes∫∫===112)(0t T t TTC Td ed ee t TTτττττv B v BB v v W v A A A ],0[all for or 1t e eT TT ∈≡≡τττ0B v 0v B A A vx A 1)0(t e−=. But actually . This contradictionshows that W c (t 1) is nonsingular.2 3: Suppose that the controllability matrix Cdoesn’t have full row rank, then there exists an n-dimensional nonzero column vector v such that v T C =0, i.e.,∫−+==220)(2)()(t t d et τττBu v x 0A 0)(2)8.6(0)(22+=+=∫−v Bu v v v 0A t t TTd eτττ⇒0v =⇒0v ≠⇒)1,,2,1,0(−==n k kTL 0B A vBased on Kelai-Hamilton Theorem, we knowthat can be expressed as alinear combination of ,For this reason, the above equations can beexpanded assoholds for all .),2,1(L ++=n n k kA 12,,,,−n A A A I L ),2,1,0(L ==k kTB A v ),2,1,0(!)(L ==−k k t kk T0B A v ],0[1t t ∈The summation of above equations is Then we haveThis contradicts the preassumption of W c (t)being nonsingular.3 2: Suppose that C has full row rank but W c (t) is singular, then there exists a nonzero vector v such that (6.8) holds.],0[all for !)(10t t e k t tT k kk T ∈=−=−∞=∑B v B A v 0A )(101t d eeC t TTT W BB v0A A ==∫−−τττ⇒Differentiating (6.8) for times, wehaveSetting to above equatins leads toand they are equivalent toThis contradicts the preassumption of C havingfull row rank. So we have obtained 2 3.)1,,2,1,0(−=n k k L 0B A v A =τe k T 0=τ)1,,2,1,0(0−==n k kTL B A v []C v B AB A AB B v ==−Tn T12L ⇔4 3: Suppose that Rank(C )<n, we want toprove that Rank for some eigenvalueof A .By Theorem 6.2 which will be given late, weknow that there is a equivalnet transformationsuch that the equivalent system iswith⇒[]n <−B I A λ1λPxx =},{B A ⎥⎦⎤⎢⎣⎡=⎥⎦⎤⎢⎣⎡==−0B B A 0A A AP P A 1C C,2212where is an m ×m matrix with m<n. Let be an eigenvalue of and q 1be acorresponding 1×m nonzero left eigenvector or , that is, .For 1×n nonzero vecotr q=[0,q 1], it is not difficult to verify thatSo, we have C A 1λC A 111q A q λ=C 0I A q =−)(11λC [][]00I A 0B A I A q 0B I A q =⎥⎦⎤⎢⎣⎡−−=−112111λλλC C C C [][]n-λn -λ<⇒<B I A B IA 11Rank RankExample 6.2Consider the inverted pendulum studied in Example 2.8. Its state equation wasdeveloped in (2.27). Suppose for a given pendulum, the equation becomes(6.11)[]xx x000120100500100001000010=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=y u &We computeClearly Rank(C)=4, so the system is controllable.Example 6.3Consider linear system[]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−==01002100200201201032B A B A AB B CFor this system, we haveu x x⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=021*******00100001000010&[]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−=−02500101000101010001λλλλλB A IThe eigenvalues of A are When ,When 5,5,04321−====λλλλ021===λλλ[]4020500101000010100100010Rank Rank =⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−=−B A I λ53==λλ[]4025510150001015010015Rank Rank =⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡−−−−=−B A I λWhen So by Statement 4 of Theorem 6.1, we knowthat the system is controllable.53−==λλ[]40255101500010150100015Rank Rank =⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎣⎡−−−−−−−−=−B A I λ6.2.1 Controllability IndicesLet A and B be n ×n and n ×p constantmatrices and we assume that Rank(B )=p (that is it is full column rank). We also assume that (A , B ) is controllable, that isLet b i be the ith column of B . Then C can be rewritten as[]nC n ==−B A B A AB B 12Rank Rank L []pn n p p C b A b A Ab Ab b b 11111|||−−=L L L L (6.13)Let us search linearly independent columns ofC from left to right and let be the number ofthe linearly independent columns associated with b m in C. That is, the columnsare linearly independent in C andare linearly dependent. Obviously, we havem μmm m m b AAb b 1,,,−μL )(m m jj μ≥b A np =+++μμμL 21(6.14)Remark:Because of the pattenof C, if A i b mdepends on its left-hand-side (LHS) columns, A j b m (j>i) will also depend on its LHS columns.Definition 6.A1: The set is calledthe controllability indices andis called the controllability index of (A , B ).Definition 6.A2: The controllability index can also be defined as the least integer such that},,,{21p μμμL ),,,max(21p μμμμL =μ[]()nB A AB BC ==−1)(μμρρLExample 6.5: Consider system(6.18)The controllability matrix isx u x x⎥⎦⎤⎢⎣⎡=⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡+⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=010*******0001000020100020030010y &[]⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−−−−−−−=0240021040021000200120010120010032B A B A AB Bsearching linearly independent columns of C from left to right, we obtainSo the controllability indices are andand the controllability index is 2.Theorem 6.A1: Let be the degree of theminimal polynomial of A , then we have,00101⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=b ,10002⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=b ,20011⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡−=Ab ⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=01202Ab 21=μ22=μn )1,min(+−≤≤p n n pnμ(6.16)Proof: Consider that is a matrix withdimensions of and , so we know that , that is .On the other hand, suppose thatthenSuppose that the minimal polynomial of A isSinceμC p n μ×n C =)(μρp n μ≤p n /≥μ121),,,max(μμμμμ==p L 1)1()11()(121+−=−−=++−≤++−==−p n p n n n p p 43421L L μμμμnn n nαλαλαλλφ++++=−−111)(L 0)(=A φ⇒IA AA n n n nααα−−−−=−−111Lwhich implies that can be written as alinear combination ofThus we have .Corollary 6.1: The n-dimensional pair (A , B )is controllable if and only if the matrixC n-p+1:=[B AB …A n-pB ]where Rank(B )=p, has rank n or the n ×nmatrix C n-p+1C T n-p+1is nonsingular.B A n}{12B AB A AB B −n L n ≤μTheorem 6.2: The controllability is invariant under any equivalence transformation. Theorem 6.3: The set of the controllability indices of (A,B) is invariant under any equivalence transformation and any reordering of the columns of B.6.3 ObservabilityThe concept of observability is dual to that of controllability.Controllability studies the possibility of steering the state from the input;Observability studies the possibility of estimating the state from the output. These two concepts are defined under the assumption that the state equation or, equivalently, all A , B , C and D are known.Consider the n-dimensional p-input q-output state equation (6.22) where A, B, C and D are, respectively, n ×n, n ×p, q ×n and q ×p constant matrices.DuCx y Bu Ax x +=+=&The response of (6.22) excited by the initial state x (0) and the input u (t) is(6.23)orDefinition 6.O1The state equation (6.22) is said to be observable if for any unknown initial state x (0), there exists a finite t 1>0 such that the knowledge of the input u and the output y over [0, t 1] suffices to determine uniquely the initial state x (0). Otherwise, the equation is said to be unobservable.)()()0()(0)(t d e e t t t t Du Bu C x C y A A ++=∫−τττ(6.23)where (6.24)is known function.)()0(t e t y x C A =)()()(:)(0)(t d e t t t t Du Bu C y y A −−=∫−τττDefinition 6.A1The state equation (6.22) is said to be observable if and only if the initial state x (0) can be determined uniquely from its zero-input response over a finite time interval.Theorem 6.4The state equation (6.22) is observable if and only if the n×n matrix(6.25) is nonsingular for any t>0.Proof:We premultiply(6.24) by and then integrate it over [0, t1] to yieldif Wo (t1) is nonsingular, then∫=t Todeet T)(τττAA CCWTt Te CA∫∫=⎟⎠⎞⎜⎝⎛11)()0(t Ttt tTt dttedtee TT yCxCC AAA(6.26)this yields a unique x (0) and means that if W o (t ) is nonsingular for any t>0, then (6.22) is observable.By the form of W o (t ), we know that if W o (t ) is singular, it is positive semidefinite. So, there exists an n ×1 nonzero constant vector v such that ∫−=1011)()()0(t T t dt t e t T y C W x A ∫=101)(t t T t T o T dt e e t T v C C v v W v A A ∫=102t tdt e v C Awhich impliesC e A t v≡0 (6.27) for all t in [0, t1]. If u≡0, then two differentinitial states, x1(0)=v and x2(0)=0, both yieldthe same zero-input response of 0:y(t)=C e A t xi(0) ≡0By Theorem 6.A1, we know that (6.22) is not observable and this completes the proof of the Theorem 6.4.Observability is a property of the pair (A, C) and is independent of B and D.If W o(t) is nonsingular for some t, then it is nonsingular for every t and the initial state can be computed form (6.26) by using any nonzero time interval.Theorem 6.5(Theorem of duality)The pair (A, B) is controllable if and only if the pair (A T, B T) is observable.Theorem 6.O1The following statements are equivalent.1.The n-dimensional pair (A , B ) is observable.2.The n ×n matrix3.The nq ×n observability matrix∫=t T o d e e t T 0)(τττA A C C W (6.28)is nonsingular for all t>0⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎣⎡=−1n O CA CA C M (6.29)4.The (n+q ) ×n matrix5.If, in addition, all eigenvalues of A have negative real parts, then the unique solution ofhas rank n (full column rank)⎥⎦⎤⎢⎣⎡−C I A λ(6.30)has full column rank at every eigenvalue, , of A .λC C A W W A T o o T −=+is positive definite. The solution is calledobservability Gramian and can be expressed as∫=t T o d e e t T 0)(τττA A C C W (6.31)6.4 Canonical DecompositionThis section discusses canonical decomposition of system equations. This fundamental result will be used to establish relationship between the state-space description and the transfer-matrix description. Consider& = Ax + Bu x y = Cx + Du(6.38)Theorem 6.6 Consider the n-dimensional state equation in (6.38) withρ (C ) = ([B AB L A n −1B ]) = n1 < nWe definite the n×n matrixP = q1 L q n1 L q n[]−1where the first n1 columns are any n1 linearly independent columns of C, and the remaining columns can arbitrarily be chosen as long as P is nonsingular.Then the equivalent transformation transfer (6.38) into& C ⎤ ⎡AC ⎡x ⎢ ⎥=⎢ ⎣xC ⎦ ⎣ 0y = CCx = PxwillA12 ⎤ ⎡ x C ⎤ ⎡ BC ⎤ ⎥ ⎢ ⎥ + ⎢ ⎥u A C ⎦ ⎣x C ⎦ ⎣ 0 ⎦(6.40)[⎡ xC ⎤ CC ⎢ ⎥ + Du ⎣x C ⎦]where A C is n1×n1 and A C is (n-n1)×(n-n1). The n1-dimensional subequation of (6.40),& c = A c x c + Bc u x y = Cc x c + Du(6.41)is controllable and has the same transfer matrix as (6.38). Proof: LetT ⎡p1 ⎤ ⎢ ⎥ −1 P=Q =⎢ M ⎥ ⎢pT ⎥ ⎣ n⎦Because PQ=I and it is actuallyT ⎡p1 ⎤ ⎢ ⎥ ⎢ M ⎥ q1 L q n1 ⎢pT ⎥ ⎣ n⎦[T T ⎡p1 q1 p1 q2 ⎢ T T p q p 2 1 2 q2 ⎢ L qn = ⎢ L L ⎢ T T p q p ⎢ n q2 ⎣ n 1]T qn ⎤ L p1 ⎥ T L p2 qn ⎥ =I ⎥ L L ⎥ T L pn qn ⎥ ⎦so we havepT i q j = 0, ∀i ≠ j(6.42)By the form of C we know that for any a which is a column of C, Aa can be shown as linearly combination of basis of columns of C, that is {q1,q2,…,qn1}. For this reason, Aqj is a linearly combination of {q1,q2,…,qn1} for j=1,2,…n1. Consider (6.42), we havepT i Aq j = 0,i = n1 + 1, n1 + 2,L, n; j = 1,2,L, n1Now we can compute outT T T T ⎡p1 ⎤ Aq1 L p1 Aq n1 M p1 Aq n1 +1 L p1 Aq n ⎢ ⎥ M M M M ⎢ M ⎥ ⎢pT Aq L pT Aq M pT Aq ⎥ T L p Aq n1 n1 n1 n1 +1 n1 n ⎢ n1 1 ⎥ ⎡A C ⎢ ⎥ A = LLLLLLLLLMLLLLLLLLL =⎢ ⎢ T ⎥ ⎣ 0 T T T ⎢ p n1 +1Aq1 Lp n1 +1Aq n1 M p n1 +1Aq n1 +1 L p n1 +1Aq n ⎥ ⎢ ⎥ M M M M M ⎢ ⎥ T ⎢pT Aq L pT Aq M pT Aq ⎥ L p Aq n n1 n n1 +1 n n ⎣ n 1 ⎦0A12 ⎤ ⎥ AC ⎦Consider also that the any columns of B can be shown as the linearly combination of {q1,q2,…,qn1}, we can compute out thatT ⎡ p1 B⎤ ⎢ ⎥ ⎢ M ⎥ ⎢ pT ⎥ n1 B ⎢ ⎥ ⎡ BC ⎤ B = PB = ⎢ LL ⎥ = ⎢ ⎥ ⎢pT B ⎥ ⎣ 0 ⎦ ⎢ n1 +1 ⎥ ⎢ M ⎥ ⎢ pT B ⎥ ⎣ n ⎦The C has no special form and it can be described byC = CP −1 = [Cq1 L Cq n1 M Cq n1 +1 L Cq n ] = CC[CC]and this end of our proof. The system (6.41) is called the controllable subsystem of (6.38). Example 6.8 Consider the three-dimensional state equation⎡0 1 ⎤ ⎡1 1 0⎤ & = ⎢0 1 0⎥ x + ⎢1 0⎥u x ⎢ ⎥ ⎢ ⎥ ⎢ ⎢ ⎣0 1 ⎥ ⎦ ⎣0 1 1 ⎥ ⎦ y = [1 1 1]x(6.43)use Cn-p+1=C2 instead of C to check the controllability of (6.43). Since⎡0 1 1 1 ⎤ ⎥=2<3 ρ (C 2 ) = ρ ([B AB]) = ρ ⎢ 1 0 1 0 ⎥ ⎢ ⎢ ⎦ ⎣0 1 1 1 ⎥the state equation (6.43) is not controllable.Let us choose⎡0 1 1 ⎤ ⎥ 1 0 0 P −1 := Q = ⎢ ⎢ ⎥ ⎢ ⎣0 1 0 ⎥ ⎦ ⎡0 1 0 ⎤ ⇒ P = ⎢0 0 1 ⎥ ⎢ ⎥ ⎢ ⎣1 0 − 1⎥ ⎦Letx = Px .We compute⎡0 1 0 ⎤ ⎡1 1 0⎤ ⎡0 1 1⎤ ⎥ ⎢ 0 1 0 ⎥ ⎢1 0 0 ⎥ 0 0 1 =⎢ ⎥ ⎥⎢ ⎥⎢ ⎢ ⎢ ⎦ ⎣0 1 0 ⎥ ⎦⎢ ⎣0 1 1 ⎥ ⎦⎢ ⎣1 0 − 1⎥⎡1 0 M 0⎤ ⎢1 1 M 0⎥ ⎥ =⎢ ⎢L L L L⎥ ⎥ ⎢ M 0 0 1 ⎦ ⎣A = PAP −1。
