State Space Models Basic Concepts

Transcription

1 Chapter 2 State Space Models Basic Concepts Related reading in Bay: Chapter Section Sbsection 1 (Models of Linear Systems) In this Chapter we provide some fndamental reslts on state space systems which will be sefl later on: canonical forms and changes of state coordinates. Also, we describe relations between transfer matrices and state space models and how to form interconnections of state space models. Finally we end p with a method to compte a linear approximation of a nonlinear system. We recall the LTI state space model (1.4): ẋ = Ax + B y = Cx + D (2.1a) (2.1b) where R m, y R p, x R n. 2.1 State space to transfer fnction (ss2tf) For a LTI state space system (2.1) we can take the Laplace transform of both sides of the state space eqations (2.1) to obtain L{ẋ} = sl{x} x(0 ) = L{Ax} + L{B} = AL{x} + BL{} L{y} = CL{x} + DL{} (2.2a) (2.2b) 21

2 22 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS where L{x 1 } L{x} =. L{x n } Also in (2.2) we have sed the linearity property of the Laplace transform, i.e., L{Ax} = AL{x}, L{B} = BL{}, L{Cx} = CL{x}, L{D} = DL{}. We denote L{x} = X(s), L{} = U(s), L{y} = Y (s) as the Laplace transform of vectors x,, and y respectively, and assming x(0 ) = 0, as is cstomary for transfer fnction models, we have from (2.2) (si A)X(s) = BU(s) Y (s) = CX(s) + DU(s) (2.3a) (2.3b) Solving for X(s) in (2.3a) gives X(s) = (si A) 1 BU(s) which can be sbstitted into (2.3b) to obtain Y (s) = (C(sI A) 1 B + D)U(s) = G(s)U(s) (2.4) where G(s) = (C(sI A) 1 B + D) is the p m transfer matrix from to y. We remark that if we expand B = [B 1,...,B m ], B i R n 1 into its colmns and C T = [C T 1,...,CT p ], C i R 1 n into its rows we have C 1 G(s) =... (si A) [ 1 B 1 C p Hence, the (i, j)th entry of G(s) is given by... B m ] + D G ij (s) = C i (si A) 1 B j + D ij where D ij is the (i, j)th entry of D. We recall Cramer s rle for matrix inverse [Bay99, pg. 518]: A 1 = adj(a) det (A) where adj(a) is adjoint of A and det(a) is the determinant of A. The (i, j)th element of the adjoint matrix of A is the cofactor of (j, i)th element of A. The cofactor, or signed minor, is the ( 1) i+j times the determinant of the sbmatrix obtained by deleting the row and colmn containing the (i, j)th element of A. For example for n = 2 we have [ ] a11 a A = 12, adj(a) = a 21 a 22 [ a22 a 12 a 21 a 11 ], det(a) = a 11 a 22 a 21 a 12

3 2.1. STATE SPACE TO TRANSFER FUNCTION (SS2TF) 23 For n = 3 we have a 22 a 23 a 32 a 33 a 12 a 13 a 32 a 33 a 12 a 13 a 22 a 23 a 11 a 12 a 13 A = a 21 a 22 a 23, adj(a) = a 21 a 23 a a 31 a 32 a 31 a 33 a 11 a 13 a 31 a 33 a 11 a 13 a 21 a a 21 a 22 a 31 a 32 a 11 a 12 a 31 a 32 a 11 a 12 a 21 a 22 ([ ]) ([ ]) ([ ]) a22 a det(a) = a 11 det 23 a21 a a a 32 a 12 det 23 a21 a + a 33 a 31 a 13 det a 31 a 32 (2.5) where we have sed A to denote det(a). Cramer s rle is not an efficient method for compting a matrix inverse and shold only be sed for theoretical argments. From Cramer s rle we can make the following observation abot C(sI A) 1 B = 1 Cadj(sI A)B det(si A) We observe that since s a 11 a 12 a 1n a 21 s a 22 a 2n si A =.... a n1 a n2 s a nn so det(si A), which is called the characteristic polynomial of A, has degree 1 n since its expansion has only one term which is the prodct of the diagonal elements of si A: (s a 11 )(s a 22 ) (s a nn ) A similar argment shows that matrix adj(si A), whose entries are obtained by deleting a row and colmn from si A and compting a determinant, has entries which are polynomials 1 of degree at most n 1. Hence, Cadj(sI A)B is a strictly proper transfer matrix, det(si A) i.e., a transfer matrix whose transfer fnction entries have denominator polynomials of degree strictly greater than their nmerator degree. Therefore, when D = 0, then G(s) is strictly proper. For state space systems with nonzero D matrix, the transfer matrix is proper bt not strictly proper, i.e., there exists at least one entry of G(s) whose nmerator degree eqals its denominator degree. Note that a system with a state space model can not have improper entries in its transfer matrix. We say λ is pole of a transfer matrix if it is a pole of any of its transfer fnction entries. Clearly, the poles of a transfer matrix are the roots of the characteristic eqation of A: 1 The degree (or sometimes order ) of a nivariate polynomial p(λ) of a scalar variable λ is given by its highest power. For example, has a degree of r. We denote this deg(p) = r. p(λ) = α 0 + α 1 λ + + α r 1 λ r 1 + λ r.

4 24 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS det(si A) = 0 which are the eigenvales of A. We remark that an eigenvale of A may not be a pole of a transfer matrix if pole-zero cancelations occr. Chapter 3 covers the necessary backgrond on linear algebra inclding eigenvales. From classical control we recall that for SISO systems z i is a zero if G(z i ) = 0. For the MIMO case this notion is generalized into the so-called transmission zero z which satisfies G(z)w = 0 for some w 0. For p = m or sqare systems this condition can be expressed more conveniently as [ ] zi + A B det( ) = 0. (2.6) C D There are two ways to perform ss2tf in MATLAB: Using [nm,den]=ss2tf (A,B,C,D,i) which comptes the transfer matrix from inpt i to the otpt. nm has p rows for a p-otpt system. The ith row in nm corresponds to the nmerator polynomial for the ith component of the otpt. This reqires nmerical A, B, C, D. Using the Symbolic Math Toolbox. This method does not reqired nmerical A, B, C, D and can work with general MIMO systems. Hence, it is more general than ss2tf. We remark that for a given state space model (2.1) we always obtain a corresponding niqe transfer matrix model. We will see shortly that going from transfer matrix to state space does not share this niqeness. Example 6 From the armatre controlled DC motor in Example 5 when L a = 0 we have [ ] [ ] ẋ = x + 0 K 2 K 1 A B y = [ 1 0 ] (2.7) x }{{} C We have Taking the inverse we obtain (si A) 1 = [ ] s 1 si A = 0 s + K 2 1 s(s + K 2 ) [ ] s + K2 1 0 s (2.8) (2.9) Hence we have the strictly proper transfer fnction model Y (s) U(s) = G(s) = C(sI A) 1 B = K 1 s(s + K 2 ) (2.10)

5 2.1. STATE SPACE TO TRANSFER FUNCTION (SS2TF) 25 We remark that the eigenvales of A are 0 and K 2 which are the poles of the transfer fnction. From (2.6) [ ] zi + A B det( ) = K C D 1 > 0 the system has no zeros. Example 7 We consider the mass spring system in Example 3. Althogh we cold se (2.4) to compte the transfer matrix, this wold involve a 4 4 matrix inverse which is npleasant to perform by hand. Instead we take Laplace transforms of (1.5) and solve 2 linear eqations for Y (s) in terms of U(s). This approach only reqires a 2 2 matrix inverse. From (1.5) we have [ s 2 m 1 + k 1 + k 2 k 2 k 2 s 2 m 2 + k 1 + k 2 ][ Y1 (s) Y 2 (s) ] = [ ] U1 (s) U 2 (s) Solving for Y (s) = (Y 1 (s), Y 2 (s)) T gives 1 Y (s) = (s 2 m 1 + k 1 + k 2 )(s 2 m 2 + k 1 + k 2 ) k2 2 [ ] s 2 m 2 + k 1 + k 2 k 2 k 2 s 2 U(s) m 1 + k 1 + k 2 Using (2.6) we find the system has no transmission zeros as [ ] zi + A B det( ) = 1 > 0 C D m 1 m 2 The MATLAB code for Examples 6 and 7 is % Example of ss2tf sing symbolic math toolbox and SS2TF % Armatre-voltage controlled DC motor with La=0 syms K1 K2 s A=[0 1;0 -K2]; B=[0;K1]; C=[1 0]; D=0; disp( Transfer fnction for Ex #1 (Symbolic Math Comptation) ) G=simple(C*inv(s*eye(size(A))-A)*B+D); pretty(g) syms z disp( Transmission zero comptation ) det([-z*eye(size(a))+a B;C D]) % To se SS2TF we need to assign nmerical vales to K1 and K2. SS2TF does % not accept symbolic A,B,C,D A_v=sbs(A,K2,40); B_v=sbs(B,K1,74); [nm,den]=ss2tf(a_v,b_v,c,d,1);

6 26 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS disp( Transfer fnction for Ex #1 sing SS2TF (K1= 74 rad/(vs^2), K2=40 (1/s)) ) tf(nm,den) % Example of ss2tf sing symbolic math toolbox % 2-DOF mechanical system syms k1 k2 m1 m2 A=[ ;-(k1+k2)/m1 0 k2/m1 0; ;k2/m2 0 -(k1+k2)/m2 0]; B=[0 0;1/m1 0;0 0;0 1/m2]; C=[ ; ]; D=zeros(2); disp( Transfer matrix for mass-spring system (Symbolic Math Comptation) ) pretty(simple(c*inv(s*eye(size(a))-a)*b+d)) disp( Transmission zero comptation ) det([-z*eye(size(a))+a B;C D]) 2.2 Transfer fnction to state space (tf2ss) For a given transfer matrix G(s), if a LTI state space form (2.1) can be fond (i.e.,, some A, B, C, D matrices) so that G(s) = C(sI A) 1 B + D, we say this state space model is a realization of G(s). There are an infinite nmber of realizations for a given transfer matrix and these realization do not even have to have the same state dimension. Here we consider three particlarly sefl state space realizations for SISO LTI systems of the form (2.1) called the Controllable Canonical Form (CCF), Observable Canonical Form (OCF), and Jordan Canonical From (JCF). The CCF and OCF will be especially sefl when discssing state feedback and state observers later. A realization is said to be minimal if the dimension of the state variable is the smallest possible. It trns ot that a realization is minimal iff its dimension is eqal to the degree of the denominator polynomial of the system s transfer fnction. This implies that transfer fnctions where a pole zero cancelation occrs do not have minimal realizations. The Controllable Canonical Form (CCF) The development starts with the simpler SISO system with no zeros which can be written in time domain as G(s) = Y (s) U(s) = 1 s n + α n 1 s n α 1 s + α 0 (2.11) d n y dt n = α n 1 d n 1 y dt n 1 α 1 dy dt α 0y + (2.12)

7 2.2. TRANSFER FUNCTION TO STATE SPACE (TF2SS) 27 The simlation diagram of this system is given in Figre 2.1. We define the state variables as the otpt of the integrators: x 1 = y x 2 = ẏ. x n = y (n 1) (2.13) When states are defined as sccessive time derivatives of a single system variable, the otpt y in this case, they are sometimes referred to as phase variables. In general, as we see below, states cannot defined in this simple manner (see PS1 Q1). Time differentiating x k defined in (2.13) and sing (2.12) we obtain ẋ 1 = x 2 ẋ 2 = x 3. ẋ n 1 = x n ẋ n = α 0 x 1 α 1 x 2 α n 1 x n + y = x 1 or in matrix form (2.1): ẋ =..... x α 0 α 1 α 2 α n 1 1 A B y = [ ] x }{{} C (2.14) Now we can consider the general proper SISO system G(s) = Y (s) U(s) = β ns n + β n 1 s n β 1 s + β 0 s n + α n 1 s n α 1 s + α 0 (2.15) we split the transfer fnction into two factors where G(s) = Y (s) U(s) = Y (s) Z(s) Z(s) U(s) Z(s) U(s) = 1, s n + α n 1 s n α 1 s + α 0 Y (s) Z(s) = β ns n + β n 1 s n β 1 s + β 0

8 28 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS - x n x n 1 x 2 x 1 1/s 1/s 1/s 1/s y α n 1 α n 2 α 1 α 0 Figre 2.1: Simlation diagram of (2.11). Smming jnctions which are not labelled are positive. We have already compted a state space realization for Z(s) U(s) in (2.14) and since d n z y = β n dt + β d n 1 z n n 1 dt + + β dz n 1 1 dt + β 0z we obtain Figre 2.2. Performing some block diagram maniplation which cancel the integrators and differentiators we obtain the simlation diagram in Figre 2.3. From Figre 2.3 the otpt is given by y = [ β 0 α 0 β n β 1 α 1 β n β n 1 α n 1 β n ] x + βn Hence a state space realization of (2.15) is ẋ =..... x α 0 α 1 α 2 α n 1 1 A c B c y = [ ] β 0 α 0 β n β 1 α 1 β n β n 1 α n 1 β n }{{} C c x + β n }{{} D c (2.16) Realization (2.16) is called the Controllable Canonical Form (CCF). We remark that if we had defined x k = y (n k), 1 k n which corresponds to taking the integrator otpts as states, bt defines x 1, x 2,... from left to right in Figre 2.3. This

9 2.2. TRANSFER FUNCTION TO STATE SPACE (TF2SS) 29 definition of state yields the realization α n 1 α n 2 α 2 α 1 α x = x y = [ β n 1 α n 1 β n β 1 α 1 β n β 0 α 0 β n ] x + βn s s s s β n s s s β n 1 y s β 1 - x 1/s n x 1/s n 1 x 1/s 2 x 1/s 1 z α n 1 α n 2 α 1 α 0 β 0 Figre 2.2: Block diagram of (2.15). Smming jnctions which are not labelled are positive. Althogh the MIMO case is more involved, generating a SIMO CCF is a straightforward generalization of the SISO case. For example if we have a two otpt transfer matrix [ ] 1 β1n s G(s) = n + β 1n 1 s n β 11 s + β 10 s n + α n 1 s n α 1 s 1 + α 0 β 2n s n + β 2n 1 s n β 21 s + β 20 we get ẋ =..... x α 0 α 1 α 2 α n 1 1 (2.17) A c B [ c ] [ ] β10 α y = 0 β 1n β 11 α 1 β 1n β 1n 1 α n 1 β 1n β1n x + β 20 α 0 β 2n β 21 α 1 β 2n β 2n 1 α 2n 1 β 2n β 2n C c D c This idea easily generalizes to the p > 2 otpt case.

10 30 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS y β n β n 1 β n 2 β 1 β 0 - x n x n 1 x 2 x 1 1/s 1/s 1/s 1/s z α n 1 α n 2 α 1 α 0 Figre 2.3: Simlation diagram of CCF (2.15). Smming jnctions which are not labelled are positive. The Observable Canonical Form (OCF) Consider again a SISO system with transfer fnction (2.15). For this system we have or (s n + α n 1 s n α 1 s + α 0 )Y (s) = (β n s n + β n 1 s n β 1 s + β 0 )U(s) s n (Y (s) β n U(s)) + s n 1 (α n 1 Y (s) β n 1 U(s)) + + (α 0 Y (s) β 0 U(s)) = 0 Dividing by s n and solving for Y (s) gives Y (s) = β n U(s) + 1 s (β n 1U(s) α n 1 Y (s)) s n(β 0U(s) α 0 Y (s)) which has the simlation diagram shown in Figre 2.4. Defining states as the otpts of the integrators as in Figre 2.4 gives the state space eqations ẋ 1 = x 2 + β n 1 α n 1 y = x 2 α n 1 x 1 + (β n 1 α n 1 β n ) ẋ 2 = x 3 + β n 2 α n 2 y = x 3 α n 2 x 1 + (β n 2 α n 2 β n ). ẋ n 1 = x n α 1 x 1 + (β 1 α 1 β n ) ẋ n = α 0 x 1 + (β 0 α 0 β n ) y = x 1 + β n

11 2.2. TRANSFER FUNCTION TO STATE SPACE (TF2SS) 31 In matrix form we have α n β n 1 α n 1 β n α n β n 2 α n 2 β n ẋ =..... x +. α β 1 α 1 β n α β 0 α 0 β n A o B o y = [ ] x + β n C o D o (2.18) Realization (2.18) is called the Observable Canonical Form (OCF). We remark that if we had defined x k = y (n k), 1 k n which corresponds to taking the integrator otpts as states, bt defines x 1, x 2,... from left to right in Figre 2.4. This definition of state yields the realization α 0 β 0 α 0 β n α 1 β 1 α 1 β n x = x α n 2 β n 2 α n 2 β n α n 1 β n 1 α n 1 β n y = [ ] x + β n Althogh the MIMO OCF case is more involved, generating a MISO OCF is a straightforward generalization of the SISO case. For example if we have a two inpt transfer matrix [ ] 1 β1n s G(s) = n + β 1n 1 s n 1 T + + β 10 s n + α n 1 s n α 1 s 1 + α 0 β 2n s n + β 2n 1 s n β 20 we get α n β 1n 1 α n 1 β 1n β 2n 1 α n 1 β 2n α n β 1n 2 α n 2 β 1n β 2n 2 α n 2 β 2n ẋ =..... x +.. α β 11 α 1 β 1n β 21 α 1 β 2n α β 10 α 0 β 1n β 20 α 0 β 2n A o B o y = [ ] [ ] x + β1n β 2n C o D o This idea easily generalizes to the m > 2 inpt case. Jordan Canonical Form (JCF) Another important canonical form is the Jordan Canonical Form (JCF) which reslts in a diagonal system matrix when the roots of the denominator polynomial are distinct. We

12 32 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS β 0 β 1 β 2 β n 1 β n x n x n 1 x 2 x 1 1/s 1/s 1/s 1/s y α 0 α 1 α 2 α n 1 Figre 2.4: Simlation diagram of OCF (2.15). Smming jnctions which are not labelled are positive. consider the SISO system (2.15) and first assme the poles of the system are distinct. Hence, (2.15) has a partial fraction expansion G(s) = β n + r 1 s s 1 + r 2 s s r n s s n This expansion sggests a block diagram which consists of n first order systems in parallel with a direct path of gain β n. This block diagram is shown in Figre 2.5 where the resides are placed at the otpt of the integrators. (They cold have alternatively been placed on the inpt side of the integrator). Letting the otpts of the integrators be the states we have the following JCF realization eqations ẋ 1 = s 1 x 1 + ẋ 2 = s 2 x 2 +. ẋ n = s n x n + y = r 1 x 1 + r 2 x r n x n + β n

13 2.2. TRANSFER FUNCTION TO STATE SPACE (TF2SS) 33 β n 1/s x 1 r 1 y s 1 x 2 1/s r 2 s 2 1/s x n r n s n Figre 2.5: Jordan Canonical Form block diagram of (2.15) when poles are distinct. Smming jnctions which are not labelled are positive.

14 34 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS In matrix form this looks like s s ẋ = x s n 1 A d B d y = [ ] r 1 r 2 r n x + β n C d D d (2.19) If G(s) has q < n distinct poles s i, 1 i q with the mltiplicity of s i being m i we break G(s) into q + 1 sbsystems Y (s) U(s) = G(s) = β n + G 1 (s) + + G q (s) where the q sbsystems G i (s), 1 i q have transfer fnctions Y i (s) U(s) = G i(s) = r 1i s s i + + r mi i (s s i ) m i (2.20) The realization of G i (s) is shown in Figre 2.6 and sing the integrator otpts as states its state space model is s i s i s i ẋ i = x i s i s i 1 0 (2.21) s i 1 A i B i y i = [ ] i r mi i r mi 1i... r 1i x }{{} C i where x i = (x 1i, x 2i,...,x mi i) T R m i is the state of sbsystem G i (s) and y i is its otpt. The realization of the entire system G(s) is the parallel combination of the q sbsystems. Hence, the JCF is ẋ 1 ẋ q A d x 1 x q B 1 B 2 ẋ 2 ẋ =. = blockdiag{a x 2 1, A 2,...,A q } }{{}. +. y = [ ] C 1 C 2... C q x + β n C d D d B q }{{} B d

15 2.2. TRANSFER FUNCTION TO STATE SPACE (TF2SS) 35 y i r 1i r 2i r mi i 1/s x mi 1 1/s x mi 1i 1/s x 1i s i s i s i Figre 2.6: Jordan Canonical Form block diagram of sbsystem (2.20) with mltiple poles. Smming jnctions which are not labelled are positive. Other system interconnection formla (series, feedback) will be seen below in Section 2.4. Example 8 Consider the doble integrator system Y (s)/u(s) = 1/s 2. Here, the CCF, OCF, and JCF are identical with A c = A o = A d = [ ] 0 1, B 0 0 c = B o = B d = [ ] 0, C 1 c = C o = C d = [ 1 0 ], D c = D o = D d = 0 Example 9 In this example we compte the CCF, OCF, and JCF for the DC motor Y (s) U(s) = K 1 s(s + K 2 ) The entries of the CCF matrices can be simply read off the transfer fnction coefficients. We have [ ] [ ] ẋ = x + 0 K 2 1 A c B c y = [ K 1 0 ] x }{{} C c

16 36 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS Similarly the entries of OCF matrices can also be read off the transfer fnction coefficients. We have [ ] [ ] K2 1 0 ẋ = x K 1 A o B o y = [ 1 0 ] x }{{} C o To obtain the JCF we perform a partial fraction expansion Y (s) U(s) = K 1/K 2 + K 1/K 2 s s + K 2 Hence the JCF is [ ] [ ] ẋ = x + 0 K 2 1 A d B d y = [ ] K 1 /K 2 K 1 /K 2 x }{{} C d 2.3 Linear change of state coordinates (ss2ss) Given (2.1) we can define new state coordinates ξ = Mx (2.22) where M R n n is an invertible matrix. Taking the time derivative of both sides of (2.22) we obtain dξ dt = M dx dt = M(Ax + B) = MAM 1 ξ + MB (2.23) and sbstitting x = M 1 ξ into (2.1b) gives Hence the system in the ξ-coordinates is where y = CM 1 ξ + D (2.24) ξ = Āξ + B y = Cx + D Ā = MAM 1, B = MB, C = CM 1, D = D (2.25) When Ā = MAM 1 we say A is similar to Ā. When transforming state coordinates, confsion sometimes arises de to choice of notation. It shold be noted that the se of symbols ξ, M in (2.22) is arbitrary. Any symbols cold be

17 2.3. LINEAR CHANGE OF STATE COORDINATES (SS2SS) 37 sed instead (e.g. z, T in place of ξ, M, respectively). Also, in some cases new z-coordinates are defined sing zm = x, and this alters the formlas for matrices in the new coordinates (2.25). As long as yo work consistently, no confsion shold arise with any choice of notation or definition of new coordinates. We remark that as there are an infinite nmber of invertible matrices which can be sed to define an infinite nmber of state coordinate transformations. Hence, any given system has an infinite nmber of realizations. We remark that a change of state coordinates does not change the transfer matrix of a system. This can be seen by directly compting the transfer matrices in old and new state coordinates sing (2.4). The transfer matrix of (2.1) is The transfer matrix in the new state coordinates ξ = Mx is G 1 (s) = C(sI A) 1 B + D (2.26) G 2 (s) = C(sI Ā) 1 B + D = CM 1 (si MAM 1 ) 1 MB + D = CM 1 M(sI A) 1 M 1 MB + D = G 1 (s) Above we have sed the fact that for sqare nonsinglar matrices P, Q R n n we have (PQ) 1 = Q 1 P 1. The reslt that G 1 (s) = G 2 (s) is to be expected, as the system does not know which choice of state variables we have made and therefore this choice shold not affect how the inpt determines the otpt. Example 10 Let s reconsider the mass spring system in Example 3 and write its state space model in new state coordinates corresponding to the centre of mass of the system and the difference of the positions of the masses. These are commonly sed coordinates when stdying mass spring systems. That is, the centre of mass is and the difference in position is x = m 1 m 2 y 1 + y 2 m 1 + m 2 m 1 + m 2 δ = y 1 y 2 We define the new state as ξ = ( x, x, δ, δ) T which means m 1 /(m 1 + m 2 ) 0 m 2 /(m 1 + m 2 ) 0 M = 0 m 1 /(m 1 + m 2 ) 0 m 2 /(m 1 + m 2 )

18 38 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS Using (2.25), in the ξ-coordinates the system has the form k 2 1 k m ξ = 1 +m (m 1 m 2 ) (m 1 +m 2 0 ) ξ + 1/(m 1 + m 2 ) 1/(m 1 + m 2 ) 0 0 m k 1 m 2 1 m 1 m 2 0 (k 1+k 2 )m m 1m 2 k 2 +m 2 2 k 1+m 2 2 k 2 m 1 m 2 (m 1 +m 2 0 1/m 1 1/m 2 ) Ā B [ ] 1 0 m2 /(m y = 1 + m 2 ) 0 ξ 1 0 m 1 /(m 1 + m 2 ) 0 }{{} C Notice that for the case where the masses are only attached to each other (i.e., k 1 = 0), the dynamics become decopled. That is, the soltions of δ do not depend on x and vice versa. The MATLAB code is % Define symbolic variables syms m1 m2 k1 k2 % Matrices of realization in original state coordinates A=[ ;... -(k1+k2)/m1 0 k2/m1 0; ;... k2/m2 0 -(k1+k2)/m2 0]; B=[0 0;1/m1 0;0 0;0 1/m2]; C=[ ; ]; M=[m1/(m1+m2) 0 m2/(m1+m2) 0;0 m1/(m1+m2) 0 m2/(m1+m2); ; ] %% disp( Abar ) pretty(simple(m*a*inv(m))) disp( Bbar ) pretty(simple(m*b)) disp( Cbar ) pretty(simple(c*inv(m))) 2.4 System interconnections It is often sefl to interconnect two or more state space systems. The common interconnections are: serial, parallel, and feedback. These interconnections are shown in Figre 2.7.

19 2.4. SYSTEM INTERCONNECTIONS 39 Sys 1 Sys 2 y Sys 1 y Sys 2 v - Sys 1 y Sys 2 Figre 2.7: Three typical system interconnections We consider two systems ẋ i = A i x i + B i i y i = C i x i + D i i i = 1, 2 (2.27) Series interconnections For the series configration we have y = y 2, y 1 = 2, = 1 so ] [ ][ ] [ ] [ẋ1 A1 0 x1 B1 = + ẋ 2 B 2 C 1 A 2 x 2 B 2 D 1 y = [ ] [ ] x D 2 C 1 C D x 2 D 1 2 This same reslt (with states reordered) can be obtained in MATLAB sing s1=ss(a1,b1,c1,d1); % Creates a state space model s2=ss(a2,b2,c2,d2);

20 40 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS s3=s2*s1; However, MATLAB only accepts nmerical-valed (as opposed to symbolic-valed) A, B, C, D. We reconsider the example in Section 1.4 which considered the series interconnection of Y (s) U(s) Y (s) = P(s) = U(s) = 1 U(s) U(s) and = C(s) = s 1 V (s) V (s) = s 1 s + 1 We take CCF realizations for C(s) (System 1) and P(s) (System 2) and apply the series formla to get a realization of the system with inpt v and otpt y. We have A 1 = 1, B 1 = 1, C 1 = 2, D 1 = 1, A 2 = 1, B 2 = 1, C 2 = 1, D 2 = 0 Hence, [ A1 0 ẋ = B 2 C 1 A 2 y = [ D 2 C 1 ] [ ] B1 x + v = B 2 D 1 C 2 ] x + D2 D 1 = [ 0 1 ] x [ ] 1 0 x [ ] 1 v 1 Parallel interconnections For the parallel configration we have y = y 1 + y 2 and 1 = 2 = so that ] [ ][ ] [ ] [ẋ1 A1 0 x1 B1 = + ẋ 2 0 A 2 x 2 B 2 y = [ ] [ ] x C 1 C (D x 2 + D 1 ) 2 This same reslt (with states reordered) can be obtained in MATLAB sing s1=ss(a1,b1,c1,d1); s2=ss(a2,b2,c2,d2); s3=s2+s1; Feedback interconnections For the feedback configration we have 1 = v y 2, y = y 1 = 2. We have which implies 1 = v y 2 = v (C 2 x 2 + D 2 2 ) = v (C 2 x 2 + D 2 (C 1 x 1 + D 1 1 )) = v (C 2 x 2 + D 2 C 1 x 1 + D 2 D 1 1 ) (I + D 2 D 1 ) 1 = v C 2 x 2 D 2 C 1 x 1 1 = (I + D 2 D 1 ) 1 (v D 2 C 1 x 1 C 2 x 2 ) = Q(v D 2 C 1 x 1 C 2 x 2 )

21 2.5. LINEARIZATION OF NONLINEAR STATE SPACE SYSTEMS 41 where Q = (I + D 2 D 1 ) 1. Note that in the expression for Q the dimension of I mst be the same as D 2 D 1 which is a p 2 m 1 matrix, where p 2 = dim(y 2 ), m 1 = dim( 1 ). Hence and For the otpt we have ẋ 1 = A 1 x 1 + B 1 1 = A 1 x 1 + B 1 Q(v D 2 C 1 x 1 C 2 x 2 ) = (A 1 B 1 QD 2 C 1 )x 1 B 1 QC 2 x 2 + B 1 Qv ẋ 2 = A 2 x 2 + B 2 2 = A 2 x 2 + B 2 y 1 = A 2 x 2 + B 2 (C 1 x 1 + D 1 1 ) = A 2 x 2 + B 2 C 1 x 1 + B 2 D 1 Q(v D 2 C 1 x 1 C 2 x 2 ) = (A 2 B 2 D 1 QC 2 )x 2 + (B 2 C 1 B 2 D 1 QD 2 C 1 )x 1 + B 2 D 1 Qv y = y 1 = C 1 x 1 + D 1 1 = C 1 x 1 + D 1 Q(v D 2 C 1 x 1 C 2 x 2 ) = (C 1 D 1 QD 2 C 1 )x 1 D 1 QC 2 x 2 + D 1 Qv In matrix form ] [ [ẋ1 A = 1 B 1 QD 2 C 1 B 1 QC 2 ẋ 2 B 2 C 1 B 2 D 1 QD 2 C 1 A 2 B 2 D 1 QC 2 y = [ ] [ ] x C 1 D 1 QD 2 C 1 D 1 QC D x 1 Qv 2 ][ x1 x 2 ] + [ ] B1 Q v B 2 D 1 Q This same reslt (with states reordered) can be obtained in MATLAB sing s1=ss(a1,b1,c1,d1); s2=ss(a2,b2,c2,d2); s3=feedback(s1,s2); Note that the MATLAB command sysic can be sed to create very general system interconnections. 2.5 Linearization of nonlinear state space systems Many real systems (e.g. the ROTPEN in the lab) are accrately modelled sing a nonlinear time-invariant state space model given by ẋ = f(x, ) y = h(x, ). (2.28a) (2.28b) Here we treat the MIMO case: y R p, R m, x R n and we assme that vector-valed fnctions f and h are smooth. Note that we have ignored any distrbance inpts only to simplify presentation. These cold also be inclded by treating them analogosly to inpt. Althogh many systems are accrately modelled sing (2.28), the nonlinearity of these systems can make analysis or controller design difficlt. Conseqently, we often compte an approximate linear system or so-called linearization which can:

22 42 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS provide a good local description of the original nonlinear system s behavior, be more amenable to analysis and control design methods de to its linearity. The method to obtain this linearization relies on a fnction s Taylor series expansion. We consider two types of linearizations: Case 1: Linearization abot a general (possibly non-constant) nominal state and inpt trajectory denoted (x r (t), r (t)), t 0. Case 2: Linearization abot a constant eqilibrim trajectory (x e (t), e (t)) = const., t 0 sch that f(x e, e ) = 0. We remark that Case 2 is a special case of Case 1. Case 1: Linearization abot a nominal trajectory Given some nominal inpt fnction r (t), t 0, we can compte the system s corresponding state response, denoted by x r (t), t 0, and otpt response, denoted by y r (t), t 0, sing (2.28) for a given initial condition x r (0) = x 0r. That is, the trajectories x r (t) and y r (t) necessarily satisfy the ODE (2.28a) and otpt eqation (2.28b): ẋ r = f(x r, r ) y r = h(x r, r ) (2.29a) (2.29b) Next we expand the fnctions f and h abot the nominal trajectory (x r (t), r (t)) sing a Taylor series f(x, ) = f(x r + x, r + ) = f(x r, r ) + f x (x r, r ) x + f (x r, r ) + H.O.T. h(x, ) = h(x r + x, r + ) = h(x r, r ) + h x (x r, r ) x + h (x r, r ) + H.O.T. (2.30a) (2.30b) where H.O.T. denotes higher order terms in x and, we have defined the deviation variables x = x x r, = r, and we have sed the standard notation f 1 f x f f 1 f x n x = f m..., f n f x 1... n =... f n f x n 1... n m h 1 h x h h 1 h x n x = h m..., =... The n n matrix f x Jacobian matrices. h p x 1..., n m matrix f h p x n h p 1... h p m h h, the p n matrix, and p m matrix x are called

23 2.5. LINEARIZATION OF NONLINEAR STATE SPACE SYSTEMS 43 Hence sbbing (2.30) into (2.28) we have ẋ r + ẋ = f(x r, r ) + f x (x r, r ) x + f (x r, r ) y r + y = h(x r, r ) + h x (x r, r ) x + h (x r, r ) (2.31a) (2.31b) where we have assmed H.O.T. are negligible since x x r, r and have defined y = y y r. The cancelation in (2.31) follows from (2.29). Rewriting (2.31) gives ẋ = f x (x r, r ) x + f }{{} (x r, r ) }{{} A(t) B(t) y = h x (x r, r ) x + h }{{} (x r, r ) }{{} C(t) D(t) (2.32a) (2.32b) Defining the time-varying matrices A(t), B(t), C(t), D(t) in (2.32), we have a linear timevarying state space system in the deviation variables ẋ = A(t) x + B(t) y = C(t) x + D(t) (2.33a) (2.33b) It is important to note that the approximate linear system (2.33) is in terms of deviation variables x,, and y and not the original variables x,, and y. Case 2: Linearization abot an eqilibrim soltion Often we are interested in approximating a nonlinear system (2.28) abot a so-called eqilibrim point 2, i.e., a constant state x e and a corresponding constant inpt e which satisfy f(x e, e ) = 0 (2.34) It is important to note that in general for systems (2.28), there may be an infinite nmber of eqilibrim points, or there may be none. Also, simply taking e constant does not ensre a constant state trajectory. However, for many real-world systems there is at least one eqilibrim point of practical importance where we can hold y and x constant for a given constant. Using Taylor series to expand fnctions f and h abot the constant vale (x e, e ) gives f(x e + x, e + ) = f(x e, e ) + f x (x e, e ) x + f (x e, e ) + HOT h(x e + x, e + ) = h(x e, e ) + h x (x e, e ) x + h (x e, e ) + HOT (2.35a) (2.35b) 2 sometimes referred to as a constant operating point

24 44 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS Similar to (2.31), we obtain ẋ 0 e + ẋ = 0 f(x e, e ) + f x (x e, e ) x + f (x e, e ) y 0 e + y = 0 h(x e, e ) + h x (x e, e ) x + h (x e, e ) (2.36a) (2.36b) Rewriting (2.31) gives ẋ = f x (x e, e ) x + f }{{} (x e, e ) }{{} A B y = h x (x e, e ) x + h }{{} (x e, e ) }{{} C D (2.37a) (2.37b) Defining the constant matrices A, B, C, D in (2.37) we have a LTI system in the deviation variables ẋ = A x + B y = C x + D (2.38a) (2.38b) The linearized system (2.38) is the same as that derived sing different notation in [Bay99, Eqn. (1.62)]. Example 11 Case 1: Simple System Consider a nonlinear state space system model ẋ = f(x, ) = ax + bx y = x Sppose we take a constant nominal inpt r (t) = a/(2b), t 0. This nominal inpt reslts in the nominal state trajectory x r (t) = x 0r e at/2 (2.39) for some initial condition x 0r. As x r (t) varies with time, it is not an eqilibrim soltion even thogh we took a constant inpt. It is worth noting that in general for nonlinear systems, we cannot always obtain closed-form expressions sch as (2.39) for the state for a given inpt and initial state. We have f x = a + b, f = bx, h x = 1 Hence from (2.32) we have A(t) = f x (x r, r ) = a/2, B(t) = f (x r, r ) = bx 0r e at/2, C(t) = h x (x r, r ) = 1 (2.40)

25 2.5. LINEARIZATION OF NONLINEAR STATE SPACE SYSTEMS 45 The linearized system abot the nominal trajectory is therefore ẋ = A(t) x + B(t) y = C(t) x (2.41a) (2.41b) where A(t), B(t), C(t) are defined in (2.40) and x = x x r, = a/(2b), y = y x r Example 12 Case 1: Rocket The height x 1, velocity x 2, and mass x 3 of a rocket are modelled by ẋ 1 = x 2 ẋ 2 = g + a/x 3 ẋ 3 = y = x 1 (2.42a) (2.42b) (2.42c) (2.42d) The SISO system inpt is the time derivative of the mass (i.e., = ẋ 3 ) and the otpt is the height of the rocket y = x 1. The acceleration de to gravity is denoted g, and a is a constant parameter. Clearly the state space model (2.42) is nonlinear and of the form (2.28) with x 2 f(x, ) = g + a/x 3, h(x, ) = x 1 (2.43) We consider the linearization of (2.42) abot a nominal trajectory (i.e., Case 1) corresponding to a constant inpt r (t) = 0 < 0, t 0 and initial state x r (0) = (0, 0, x 30r ) T. We can integrate (2.42) in closed-form for this inpt and initial state to obtain the nominal state trajectory x r = (x 1r, x 2r, x 3r ) T. We obtain x 1r = gt2 2 + x 30ra x 2r = gt + a ln x 3r = x 30r + 0 t 0 (( ( x 30r t ) t x ) 30r ( ln 1 + ) 0 t ) 0 t x 30r x 30r The Jacobian matrices are f x = 0 0 a/(x 2 3 ), f = a/x 3, 1 h x = ( )

26 46 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS Hence we have A(t) = f x (x r, r ) = B(t) = f (x r, r ) = a/(x 30r + 0 t) a/(x 30r + 0 t) 1 C(t) = h x (x r, r ) = ( ) Hence the linearized system abot the nominal trajectory is therefore ẋ = A(t) x + B(t) y = C(t) x (2.45) (2.46a) (2.46b) where A(t), B(t), C(t) are defined in (2.45) and x = x x r, = 0, y = y x 1r Example 13 Case 2: Cart and Pendlm Consider a cart and pendlm system shown m l M y 1 Figre 2.8: Pendlm on cart in Figre 2.8. Here an inpt force is applied to a cart of mass M to balance a pendlm of length l in its pright position. The pendlm is massless and has a point mass m fixed to its end. The otpt of the system is the cart s position and pendlm angle. Note the strong similarity of this system with the ROTPEN which is essentially a rotational version of this

27 2.5. LINEARIZATION OF NONLINEAR STATE SPACE SYSTEMS 47 system. We wish to compte the linearization of this system abot one of its eqilibrim points (i.e., Case 2). The nonlinear ODEs for the pendlm on a cart are given by Mÿ 1 = mÿ 1 ml θ cos θ + ml( θ) 2 sin θ g sin θ = l θ + ÿ 1 cosθ (2.47a) (2.47b) Using the states defined as x 1 = y 1, x 2 = ẏ 1, x 3 = θ, x 4 = θ we pt the system in state space form ẋ = f(x, ), y = h(x, ). We rewrite (2.47) in matrix form ( ) ) ( ) M + m ml cosx3 (ẋ2 x 2 = 4 ml sin x 3 + cos x 3 l ẋ 4 g sin x 3 ) ( ) 1 ( ) (ẋ2 M + m ml cos x3 x 2 = 4 ml sin x 3 + ẋ 4 cosx 3 l g sin x 3 ) ( (ẋ2 1 = ẋ 4 M + m sin 2 x 3 Hence, the state space form of the system is + mlx 2 4 sin x 3 mg sin x 3 cosx 3 ((M + m)g sin x 3 mlx 2 4 sin x 3 cosx 3 cosx 3 )/l ) ẋ 1 = x 2 ẋ 2 = ( + mlx 2 4 sin x 3 mg sin x 3 cos x 3 )/(M + m sin 2 x 3 ) ẋ 3 = x 4 ẋ 4 = ((M + m)g sin x 3 mlx 2 4 sin x 3 cosx 3 cosx 3 )/(l(m + m sin 2 x 3 )) ( ) x1 y = x 3 (2.48a) (2.48b) (2.48c) (2.48d) (2.48e) Eqilibrim Set Since the eqilibrim state x e and eqilibrim inpt e mst satisfy f(x e, e ) = 0 we have immediately x 2e = 0, x 4e = 0 from (2.48). As well, we immediately conclde that x 1e = const. (i.e., x 1e can be any constant vale) given that f is independent of x 1. Finally, from (2.48b) we have and from (2.48d) we have Sbbing (2.49) into (2.50) we have e = mg sin x 3e cosx 3e (2.49) (M + m)g sin x 3e = e cosx 3e (2.50) M + m m sin x 3e = sin x 3e cos 2 x 3e (2.51) which is satisfied iff x 3e = kπ, k Z. This is becase if x 3e kπ then we can cancel sin x 3 from both sides of (2.51) and we obtain a relation which is never satisfied: M + m m = cos2 x 3e

28 48 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS since (M + m)/m > 1. Hence, from (2.49) we have e = 0. Hence the eqilibrim set is given by {(x e, e ) : x 2e = x 4e = 0, x 1e = const., x 3e = kπ, k Z} (2.52) The set (2.52) has physical significance as it indicates that the only way x = const is if the pendlm is the pendant or inverted position, and that these two configrations reqire a zero force inpt to remain there. A similar sitation arises in the lab with the ROTPEN. Linearized system abot eqilibrim point We compte the linearization of the system abot x e = (x 1e 0 0 0) T, e = 0 (x 1e is any constant) by applying (2.37). With little comptation we have f 1 x = ( ), Straightforward bt tedios calclation shows f 3 x = ( ) (2.53) f ( ) 2 x = 0 0 m(lx2 4 cos x 3+g(sin 2 x 3 cos 2 x 3 )) M+msin 2 x 3 2msin x 3 cos x 3 (+mlx 2 4 sinx 3 mg sin x 3 cos x 3 ) 2mlx 4 sinx 3 (M+m sin 2 x 3 ) 2 M+msin 2 x 3 Evalated at (x e, e ) gives Frther we have f 4 x = (2.54) f 2 x (x e, e ) = ( 0 0 mg/m 0 ) (2.55) 0 0 (M+m)g cos x 3 +mlx 2 4 (sin2 x 3 cos 2 x 3 )+ sin x 3 l(m+m sin 2 x 3 2msin x 3 cos x 3 ((M+m)g sin x 3 mlx 2 4 sin x 3 cos x 3 cos x 3 ) ) l(m+msin 2 x 3 ) 2 2mx 4 sinx 3 cos x 3 M+msin 2 x 3 Evalated at (x e, e ) gives (2.56) f 4 x (x e, e ) = ( 0 0 (M + m)g/(lm) 0 ) (2.57) Hence we have the linearized system ẋ = 0 0 mg/m x + 1/M 0 (2.58) 0 0 (M + m)g/(ml) 0 1/(Ml) A B ( ) y = x (2.59) }{{} C We remark that in this case x = (x 1 x 1e, x 2, x 3, x 4 ) T, y = (x 1 x 1e, x 3 ) T, and =. T

29 2.5. LINEARIZATION OF NONLINEAR STATE SPACE SYSTEMS 49 MATLAB code to perform linearization Compting the Jacobian matrices can be very tedios, fortnately MATLAB provides an easy soltion with the Symbolic Math Toolbox. The code to generate the linearization can be downloaded at and is given here % EE Linearization of Cart Pendlm sing Symbolic Math Toolbox syms m M L g positive; syms x1e; syms x1 x2 x3 x4 x=[x1;x2;x3;x4]; x_=[x1,x2,x3,x4,]; % Eqilibrim state (inverted configration) and corresponding constant inpt xe_e=[x1e,0,0,0,0]; % Nonlinear state space form f=[x2;... (+m*l*x4^2*sin(x3)-m*g*sin(x3)*cos(x3))/(m+m*sin(x3)^2);... x4;... ((M+m)*g*sin(x3)-m*L*x4^2*sin(x3)*cos(x3)-*cos(x3))/(L*(M+m*sin(x3)^2))]; h=[x1;x3]; % Jacobian matrices dfdx=jacobian(f,x); dfd=jacobian(f,); dhdx=jacobian(h,x); dhd=jacobian(h,); % Evalate Jacobian matrices A=sbs(dfdx,x_,xe_e); B=sbs(dfd,x_,xe_e); C=sbs(dhdx,x_,xe_e); D=sbs(dhd,x_,xe_e); disp( Linearized system matrices ); A B C D % Compte transfer matrix of linearized system syms s; % Otpt is y = [x1;x3] C2=[ ; ];

30 50 CHAPTER 2. STATE SPACE MODELS BASIC CONCEPTS TM=simple(C2*inv(s*eye(4)-A)*B); pretty(tm) [n,d]=nmden(tm(2)); pretty(simple(solve(d))) This program also comptes the transfer matrix of the linearized system.

31

32 Bibliography [Bay99] John S. Bay. Fndamentals of linear state space systems. McGraw-Hill, [Che99] Chi-Tsong Chen. Linear System Theory and Design. Oxford, New York, NY, 3rd edition, [Max68] J.C. Maxwell. On governors. Proceedings of the Royal Society of London, 16: ,