CONTROLLABILITY. Chapter Reachable Set and Controllability. Suppose we have a linear system described by the state equation


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1 Chapter 2 CONTROLLABILITY 2 Reachable Set and Controllability Suppose we have a linear system described by the state equation ẋ Ax + Bu (2) x() x Consider the following problem For a given vector x in R n, does there exist a time t, < t < and a piecewise continuous control input u, defined on [,t, such that the solution of () x(t) x? We shall refer to this problem as the reachability problem If for x, there exists an input u which solves the reachability problem, we then say that the state x is reachable from at time t We refer to such an input u as the control which transfers the state from the origin to x at time t Denote the set of states reachable from by R If R R n, then every state is reachable from by suitable control One motivation for studying reachability is the interception or rendezvous problem Suppose you know the trajectory of some vehicle and you want to maneuver your own vehicle to intercept it If R R n, the interception problem is solvable To prepare for the discussion later in the chapter, let us give a slightly more abstract but compact description of the reachability problem Using the variation of parameters formula, x is reachable from at time t if there exists an input u such that x e A(t τ) Bu(τ)dτ (22) Fix the time t Let U be the space of all input signals defined on [ t Define the linear operator L which maps U to R n by Lu e A(t τ) Bu(τ)dτ (23) Recall that if A is an m n matrix (a linear transformation) mapping R n into R m, the range of A, denoted R(A), is the set {y R m y Ax, for some x R n } Analogously, R(L), the range of L, is defined to be the set {x R n x Lu, for some u U} Using this formulation, x is reachable from if and only if x R(L) There is nothing particularly special about the initial state x In fact, if R R n, then every state x is reachable from any initial state x To see this, write the variation of parameters formula for the solution of (2) in the form x(t) e At x 27 e A(t τ) Bu(τ)dτ (24)
2 28 CHAPTER 2 CONTROLLABILITY We see that x reachable from x is equivalent to x e At x is reachable from We now define controllability for the system described by (2) Definition: The system (2) is said to be controllable at time t if for every vector x R n and every initial condition x, there exists an input u which transfers the state from x to x at time t In view of the preceding discussion, studying the controllability property is equivalent to studying whether R R n Example: Consider the following network + R i i 2 R 2 v i  R 3 + C v  v C 2 Figure 2: An RC bridge circuit It is straightforward to verify, using Kirchhoff s current law, that v, v 2 satisfy the following differential equations: [ In state space form, denoting v v v ( C R + C v v v 2 R 3 + v i v R C 2 v 2 v v 2 R 3 + v i v 2 R 2 v 2 T, we have C R 3 ) C R 3 C 2 R 3 ( C 2 R 2 + C 2 R 3 ) v + C R C 2 R 2 Put α C R, α 2 C 2 R 2, β C R 3, β 2 C 2 R 3 We can write the differential equation in the form v [ (α + β ) β v + β 2 (α 2 + β 2 ) Notice that if α α 2, ie, C R C 2 R 2, the difference v v 2 satisfies [ α α 2 d dt (v v 2 ) (α + β + β 2 )(v v 2 ) v i v i (25) which is a homogeneous equation with no input In this case, we cannot manipulate v and v 2 arbitrarily, so that the reachable set is not the whole R 2 To study the reachability problem, we introduce 2 important matrices arising from the system (2)
3 2 REACHABLE SET AND CONTROLLABILITY 29 (i) For t >, define the controllability Gramian at time t W t e Aτ BB T e AT τ dτ (26) e A(t s) BB T e AT (t s) ds (27) Finding W t requires computation of e At and integration Example: A (si A) [ 2 [ s + s + 2 [ s+ (s+)(s+2) s+2 [ s + 2 s + (s + )(s + 2) [ s+ s+ s+2 s+2 [ e At L (si A) e t e e 2t e 2t [ B [ e τ e W t e 2τ [ [ e 2τ [ [ e τ e e 2τ [ e 2τ e τ e 2τ e 2τ [ e 2τ 2e 3τ + e 4τ e 3τ e 4τ e τ e τ e 2τ e 2τ dτ dτ dτ e 3τ e 4τ e 4τ [ 2 ( e 2t ) 2 3 ( e 3t ) + 4 ( e 4t ) 3 ( e 3t ) 4 ( e 4t ) 3 ( e 3t ) 4 ( e 4t ) 4 ( e 4t ) Note that W t is a symmetric n n matrix (ii) Let C AB denote the n nm matrix C AB [B AB A 2 B A n B (28) C AB is called the controllability matrix For m, C AB is a square matrix For m >, C AB is a rectangular, wide matrix Recall that the rank of a p q matrix A is the number of independent columns of A, which is the same as the number of independent rows of A Note that rank(c AB ) n If rank(c AB ) n, we say C AB has full rank Let us first illustrate the computation of C AB in some examples
4 3 CHAPTER 2 CONTROLLABILITY Example : For the circuit described by Figure 2, we have C AB α (α 2 + α β ) + α 2 β α 2 (α α 2β 2 ) + α β 2 Note that det C AB (α α 2 + α β 2 + α 2 β )(α α 2 ) so that C AB is singular (equivalently C AB does not have full rank) if and only if α α 2 Example 2: Consider the following system of 2 carts connected by a spring y u 2 K u M M 2 y 2 Figure 22: A coupled 2cart system Newton s law gives M ÿ K(y y 2 ) + u M 2 ÿ 2 K(y 2 y ) + u 2 [ T Let us recast the equations in state space form Define x y ẏ y 2 ẏ 2 Then ẋ K K M M K M 2 K M 2 x x 2 x 3 x 4 + M M 2 u u 2 The controllability matrix for this system is given by K M M 2 K K M C AB M 2 M M 2 K M 2 M M 2 K M 2 M M 2 K M 2 2 K M M 2 K M 2 2 B AB A 2 B A 3 B Note that C AB has 4 linearly independent columns so that it is full rank Before we discuss the main result of Chapter 2, let us review some geometric ideas from linear algebra Suppose V is a subspace of R n The orthogonal complement of V, denoted by V, is the set of all vectors w such that w T v for all v V
5 2 REACHABLE SET AND CONTROLLABILITY 3 Example: Suppose V is the x y plane, a subspace of R 3 Then V is the zaxis Some important properties of orthogonal complements are: V is a subspace 2 The only vector that lies in both V and V is the zero vector 3 Every vector x R n can be uniquely decomposed as x u + w with u V and w V We refer to the decomposition as R n is the direct sum of V and V, written as R n V V 4 (V ) V 5 Let V and W be subspaces of R n V W if and only if V W, and V W if and only if W V The following is the main result of this chapter It shows how C AB and W t feature in the solution of the reachability problem Theorem 2: R(L) R(W t ) R(C AB ) Proof: The theorem will be proved using the following 2 steps () Prove R(L) R(W t ) (2) Prove R(C AB ) R(W t ) Step : Prove R(L) R(W t ) Using property (5) of orthogonal complements, proving R(L) R(W t ) is equivalent to proving R(L) R(W t ) (a) R(L) R(W t ) : Let v R(L) Then for all u U, This is true for all u if and only if v T e A(t τ) Bu(τ)dτ v T e Aτ B, all τ, τ t But then so that v R(W t ) v T W t v T e Aτ BB T e AT τ dτ (b) R(W t ) R(L) : Let v R(W t ) Then, in particular, v T W t v But v T W t v v T e Aτ BB T e AT τ dτ v v T e Aτ B 2 dτ Thus v T W t v holds if and only if v T e Aτ B, τ t This implies v R(L)
6 32 CHAPTER 2 CONTROLLABILITY This completes Step of the proof Note that in the proof, we have also shown that, for t >, v T e Aτ B, τ t if and only if v T W t Step 2: Prove R(C AB ) R(W t ) First note that we can express e At in the form e At ϕ (t)i + ϕ (t)a + + ϕ n (t)a n (29) for certain functions {ϕ i (t)} An outline of the proof of this result is given in the problem sets Intutively, we can see that this is a consequence of the CayleyHamilton Theorem, which states: then If an n n matrix A has characteristic polynomial p(s) given by p(s) s n + p s n + + p n s + p n p(a) A n + p A n + + p n A + p n I CayleyHamilton Theorem implies A k,k n is expressible as a linear combination of A j, j n This means that the infinite series that defines e At should be expressible in the form of (29) We can now carry out the proof of Step 2 We first show v T C AB if and only if for t > Suppose v T C AB Then v T e Aτ B for all τ, τ t v T e Aτ B v T [ϕ (τ)i + + ϕ n (τ)a n B ϕ (τ)i v T [B AB A n ϕ (τ)i B ϕ n (τ)i Conversely, suppose v T e Aτ B for all τ t Setting τ gives v T B For k,,n, take the kth derivative of v T e Aτ B with respect to τ and evaluate the result at τ This gives successively v T AB v T A 2 B v T A n B so that v T C AB
7 2 REACHABLE SET AND CONTROLLABILITY 33 Now in the proof of Step, we have shown that v T e Aτ B, τ t holds if and only if holds This allows us to conclude that if and only if v T W t v T C AB v T W t Therefore v R(C AB ) if and only if v R(W t ) Hence R(C AB ) R(W t ), which is equivalent to R(C AB ) R(W t ) This completes the proof of Step 2 and the proof of the Theorem Observe that since R(C AB ) is independent ot t, Theorem 2 implies that the reachable set and reachability properties are in fact independent of t We can therefore talk about controllability without reference to t Furthermore, note that for an n n matrix M, R(M) R n if and only if M is nonsingular, and for an n nm matrix N, R(N) R n if and only rank(n) n Combining these observations and the Theorem, we obtain the following Corollary: The following statements are equivalent: (a) R R n (b) W t is nonsingular for any t > (c) C AB has rank n (full rank) Since the linear system (2) is controllable if and only if R R n, the equivalence of statements (a) and (c) of the Corollary can be restated as the following important theorem Theorem 22: The linear system (2) is controllable if and only if Rank [B AB A n B n Theorem 22 allows us to check the controllability property using given data A and B It is a very important result Based on the Theorem 22, we are justified in saying: The pair (A,B) is controllable if Rank[B AB A n B n Suppose x is reachable at time t from x It is not difficult to write down explicitly he control that achieves the transfer from x to x at time t In fact, a control input that achieves the transfer can be verified to be given by u(τ) B T e AT (t τ) ξ (2) where ξ is the solution of W t ξ x e At x If (A,B) is controllable, Wt exists, and (2) can be written explicitly as u(τ) B T e AT (t τ) Wt (x e At x ) (2) Note also that if (A,B) is controllable, the state transfer can be accomplished for any t >, no matter how small, since W t is nonsingular for any t > However, if t is small, Wt will be large, and the control input to achieve the transfer will be large too
8 34 CHAPTER 2 CONTROLLABILITY 22 Some Properties of Controllability In this chapter, we discuss 2 properties of controllability: invariance under a change of basis and invariance under state feedback We also introduce the controllable canonical form for single input systems, which will be very useful for the pole assignment problem discussed in the next chapter () Invariance under change of basis: Recall that if x is a state vector, so is V x for any nonsingular matrix V In fact, if we let z V x, ż V x V AV z + V Bu so that with z as the state vector, the system matrices change from (A,B) to (V AV,V B) We refer to this as a change of basis because if we let the columns of V form a new basis for R n, z is then the representation of x in this new basis (See the Appendix to Chapter 2 for more details on change of basis) We call the transformation V featured in the change of basis a similarity transformation Theorem 23: (A,B) is controllable if and only if (V AV,V B) is controllable for every nonsingular V Proof: C V AV,V B [V B V AV V B V [B AB V C AB Since Rank(V C AB ) Rank(C AB ) the result is proved (2) Invariance under state feedback: A control law of the form u(t) Kx(t) + v(t) (22) with v(t) a new input, is referred to as state feedback The closedloop system equation is given by ẋ (A BK)x(t) + Bv(t) It is an important property that controllability is also unaffected by state feedback Theorem 24: (A,B) is controllable if and only if (A BK,B) is controllable for all K A proof of Theorem 24 is discussed in the problem sets Intuitively, the result seems reasonable since the state feedback law (22) can be reversed by setting v Kx + u Therefore, the ability to control the system using the input v should be the same as that of using input u
9 23 DECOMPOSITION INTO CONTROLLABLE AND UNCONTROLLABLE PARTS Decomposition into Controllable and Uncontrollable Parts If the pair (A, B) is not controllable, there is a particular basis in which the controllable and uncontrollable parts are displayed transparently We illustrate the choice of basis and the computation involved using the circuit example in Section 2 Example: We know from Section 2, that the circuit described by Figure is not controllable if R C R 2 C 2 Using the notation from Section 2, and setting α R C R 2 C 2, we can rewrite (25) as [ [ (α + β ) β v α v + v β 2 (α + β 2 ) α i (23) The controllability matrix is given by [ α α 2 C AB α α 2 so that s [ T is a basis for R(CAB ) Pick a second vector s 2 so that s and s 2 form a basis for R 2, for example, s 2 [ T Set V [ s s2 We then have and V AV [ [ [ [ (α + β ) β β 2 (α + β 2 ) [ [ α β α (α + β 2 ) [ α β (α + β + β 2 ) V B [ [ α α [ α Note that the change of basis produces the following system equation ż Ãz + Bu with [Ã Ã V Ã AV 2 Ã 22 [ B B V B (24) (25) The part corresponding to (Ã, B ), with state component z, is controllable, while the part corresponding to the Ã22, with state component z 2, is uncontrollable, since the z 2 equation is decoupled from z, and z 2 is unaffected by u In general, whenever C AB is not full rank, we can find a basis so that in the new basis, A and B take the form given in (24) and (25), respectively, with (Ã, B ) controllable The procedure is: Find a basis for R(C AB ) Denote the vectors in this basis by {v,v 2,,v q }, where q rank(c AB )
10 36 CHAPTER 2 CONTROLLABILITY 2 Select an additional n q linearly independent vectors v q+ v n so that {v,v 2, v n } form a basis for R n Define the matrix V by V [ v v 2 v n 3 Compute Ã V AV and B V B Ã will take the form (24) and B will take the form (25) For more details, the appendix to Chapter 2 contains a formal proof that the above procedure yields Ã and B as described 24 The PBH Test for Controllability While we normally check controllability by determining the rank of C AB, there is an alternative useful test for controllability, referred to as the PBH test We state the result as a theorem Theorem 25 (PBH Test): (A,B) is controllable if and only if Rank[A λi B n for all eigenvalues λ of A Proof: The statement is equivalent to: (A,B) is not controllable if and only if rank[a λi B < n for some eigenvalue λ of A (i) We first prove rank[a λi B < n implies (A, B) is not controllable Suppose rank[a λi B < n for some eigenvalue λ, possibly complex Thus there exists a vector x, possibly complex such that x [A λi B where x is complex conjugate transpose of x This results in x A λx and Then and x B x AB λx B x A k B λ k x B Thus x [B ABA n B so that [Re x [B ABA n B and (A,B) is not controllable (ii) We now show (A,B) not controllable implies rank[a λi B < n for some eigenvalue λ of A Assume (A,B) is not controllable By the results of Section 23, we can find a nonsingular V so that (V AV,V B) (Ã, B), where Ã is of the form (24) and B is of the form (25), with Ã q q and Ã22 n q n q Now note that for any λ an eigenvalue of Ã22, Rank(Ã22 λi) < n q Hence [Ã λi B Rank [ Ã λi Ã 2 B Ã 22 λi < n
11 24 THE PBH TEST FOR CONTROLLABILITY 37 Next observe that Since V and [ V I [ V V [A λi B I are both nonsingular, V [AV λv B [V AV λi V B [Ã λi B Rank[Ã λi B Rank[A λi B < n for λ an eigenvalue of Ã 22 Finally, note that by the structure of Ã, an eigenvalue of Ã 22 is an eigenvalue of Ã Since a change of basis does not change the eigenvalues, λ is also an eigenvalue of A This concludes the proof that (A, B) not controllable implies there exists an eigenvalue of A, λ, such that rank[a λi B < n It is useful to note that Rank[A λi B n for all eigenvalues λ of A if and only if Rank[A λi B n for all complex numbers λ This is because for λ not an eigenvalue of A, Rank(A λi) n We refer to the eigenvalue which causes Rank[A λi B < n as an uncontrollable eigenvalue We will see in Chapter 3 that such an eigenvalue is in some sense fixed and not movable The PBH test is especially useful for checking when a system is not controllable We give a couple of examples to illustrate its use Example : Consider the following (A,B) pair [ 3 2 A 2 B [ This corresponds to the bridge circuit example withe the following parameter values: α α 2, β 2, and β 2 We know that since α α 2, this system is not controllable We can also verify that the controllability matrix is [ C AB which is singular Let us apply the PBH test First det(si A) s 2 + 5s + 4 so that the eigenvalues are 4 and For the eignenvalue, PBH test gives [ [ 2 2 A ( )I B which has rank 2 On the other hand, for the eigenvalue 4, PHB test gives [ [ 2 A ( 4)I B 2 which has rank By the PBH test, the system is not controllable and 4 is an uncontrollable eigenvalue Example 2: Consider the following (A,B) pair A B 2 2 2
12 38 CHAPTER 2 CONTROLLABILITY By direct expansion along the first row of (si A), we find s det(si A) det s s 2 2 s s 2 (s 2 + 2) + det s 2 2 s s 2 (s 2 + 2) + s 2 s 4 + 3s 2 By inspection, we immediately see that with the eigenvalue, Rank[A (A,B) pair is not controllable B < 4 By the PBH test, this 25 Controllable Canonical Form for SingleInput Controllable Systems For single input systems, there is a special form of system matrices for which controllability always holds This special form is referred to as the controllable canonical form Using a lower case b to indicate explicitly that the input matrix is a column vector for a single input system, the controllable canonical form is given by A (26) α n α n α b (27) It is easy to verify that the controllability matrix for this pair (A,b) always has rank n, regardless of the values of the coefficients α j Hence the name controllable canonical form An A matrix taking the form given in (26) is referred to as a companion form matrix It is straightforward to show that the characteristic polynomial of the companion form matrix is given by det(si A) s n + α s n + + α n The description of the controllable realization in Section 7 reflects in effect the properties of the pair (A,b) given by (26) and (27) It will be seen in the next chapter that for applications to pole assignment in singleinput systems, the controllable canonical form is particularly convenient for control design To prepare for that discussion, we now show that if (A,b) is controllable, but not in controllable canonical form, we can always find a similarity transformation V so that (V AV,V b) will be in controllable canonical form
13 25 CONTROLLABLE CANONICAL FORM FOR SINGLEINPUT CONTROLLABLE SYSTEMS 39 Consider the matrix V [A n b A n 2 b b [b A n b [v v n α α 2 α n 2 α n α n 2 α α n α n 2 α α n 2 α α By controllability, V exists, so that its columns v,,v n form a basis of R n Note that v v 2 v n v n A n b + α A n 2 b + + α n b A n 2 b + α A n 3 b + + α n 2 b Ab + α b b and that Av A n b + + α n Ab + α n b α n b α n b by the CayleyHamilton Theorem α n v n Av 2 v α n v n Av n v n α v n Thus the matrix representation of A with respect to the basis {v v n } looks like [A v α n α n α Similarly, the vector b looks like [b v
14 4 CHAPTER 2 CONTROLLABILITY But [A v and [b v are then related to the original matrices through [A v [b v V AV V b so that they are related by a similarity transformation Thus the new system z(t) V x(t) will satisfy an equation of the form ż A c z + b c u with (A c,b c ) in controllable canonical form
15 APPENDIX TO CHAPTER 2 4 Appendix to Chapter 2: Change of Basis and System Decomposition Let us first quickly review the operation of a change of basis in linear algebra and how it affects the representation of vectors and linear transformations Let E {e,,e n } and V {v,,v n } be two bases in R n Any vector x R n can be represented with respect to either the basis E or the basis V Thus we can write x ξ i e i i η i v i The coefficient ξ i is the ith coordinate of x with respect to the E basis To emphasize this dependence on the basis, we write [x E [ξ ξ 2 ξ n T Now let A be a linear transformation mapping R n to R n Its action on on the basis vector e i can be represented by Ae i a ki e k k The coefficients {a ij } are then the ijth element of the matrix reprentation of A with respect to the basis E We write this as [A E {a ij } Similarly, if Av i i β ki v k k then [A V {β ij } We now describe the operation of a change of basis Suppose we wish to change the basis for R n from E to V Define the linear transformation P by Pe i v i so that {α ij } is the matrix of P in the basis E Then α ki e k k x Hence we have which can be written as or ξ i e i i η i Pe i i ξ k k i η i α ki e k α ki η i i [x E [P E [x V [x V [P E [x E [ α ki η i e k k i
16 42 CHAPTER 2 CONTROLLABILITY For linear transformations, we have Av i β ki v k β ki Pe k β ki α jk e j k k k j but also Av i APe i A α ki e k α ki a jk e j k k j On comparing the two representations for Av i, we get α jk β ki a jk α ki In matrix notation, this corresponds to k k so that [P E [A V [A E [P E [A V [P E [A E[P E We are now ready to describe the details of system decomposition for an uncontrollable pair (A, B), described in Section 23 Let Rank[B ABA n B q < n, and let {v v q } be a basis for Range[B ABA n B M We can pick additional basis vectors {v q+,,v n } so that {v,,v q,v q+,,v n } form a basis for R n Now observe that by the CayleyHamilton Theorem, AM M Hence for i q, q Av i α ji v j for some scalars α ji, j,,q With respect to the basis V {v,,v n }, A takes the form [ [A V Ã Ã Ã 2 Ã 22 Similarly the ith column of B, Thus, we obtain and b i j q β ji v j j V AV Ã [ Ã Ã 2 Ã 22 V B B where Ã is a q q matrix and B is a q m matrix We claim that (Ã, B ) is controllable For, [ Rank[ B Ã B B Ã Ã n B Rank B Ã n B q But by the CayleyHamilton theorem, for k, Ã q+k B is linearly dependent on Ãj B, j,,q Hence Rank[ B Ã B Ã q B q so that (Ã, B ) is controllable [ B
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