-H (s) ?
6
-r ?d y
v n
x1 u x2
x3
Figure 3.3. Linear feedback control structure with disturbances.
Definition 3.1. The feedback system shown in Figure 3.3 is said to be well-posed if all nine closed-loop transfer functions from the input signals (r, d, n) to the output signals (u, y, v) exist.
Well-posedness can be determined by the following theorem.
Theorem 3.1. The system is well-posed if and only if the 3× 3 matrix
1 0 H (s)
−Gc(s) 1 0
0 −G(s) 1
(3.1)
is nonsingular, i.e., the determinant of the matrix 1+ G(s)Gc(s)H (s)does not equal zero.
Definition 3.2. The system shown in Figure 3.3 is said to be internally stable if all nine closed-loop transfer functions from the inputs (r, d, n) to the internal signals (x1, x2, x3) are stable.
The nine transfer functions can be described by
x1 x2
x3
= 1
1+G(s)Gc(s)H (s)
1 −G(s)H (s) −H (s) Gc(s) 1 −Gc(s)H (s) G(s)Gc(s) G(s) 1
r d n
. (3.2)
Theorem 3.2. The system is internally stable if and only if the following two conditions are satisfied:
(1) The transfer function 1+ H (s)G(s)Gc(s)has no zeros on Re[s] ≥ 0.
(2) H (s)G(s)Gc(s)has no pole-zero cancellation on Re[s] ≥ 0.
The first condition is equivalent to saying that the closed-loop system is input-output stable. So, one can concentrate on the second condition, which is also very easy to check by using the MATLAB functionintstable():
1 function key=intstable(G,Gc,H)
2 GG=minreal(feedback(G*Gc,H)); Go=H*G*Gc; Go1=minreal(Go); p=eig(GG);
3 z0=eig(Go); z1=eig(Go1); zz=setdiff(z0,z1); % find the cancellations 4 if (G.Ts>1), % discrete-time system
5 key=any(abs(p)>1); if key==0, key=2*any(abs(zz)>1); end 6 else, % continuous system
7 key=any(real(p)>0); if key==0, key=2*any(real(zz)>0); end
8 end
Its syntax is [V,c]=intstable(G,Gc,H). In the function call, the returned vari-ables are defined as follows:
(1) If the system is internally stable, V = 0 is returned and c is empty.
(2) If V = 1, the system is input-output unstable and the unstable closed-loop poles are returned in the vector c.
(3) If V = 2, the system is input-output stable, but not internally stable, and the internally canceled unstable poles are returned in c.
Example 3.3. Consider the typical feedback system with G(s)= 5(s− 1)(s + 2)
s3+ 4s2+ 3s + 4, Gc(s)= s2+ 3s + 4
(s− 1)(s2+ 3s + 2), H (s)= 1.
The stability of the system can be tested by the following MATLAB statements:
>> s=tf(’s’); G=5*(s-1)*(s+2)/(sˆ3+4*sˆ2+3*s+4);
Gc=(sˆ2+3*s+4)/((s-1)*(sˆ2+3*s+2)); H=1;
G_a=minreal(ss(feedback(Gc*G,H))); eig(G_a)
The closed-loop poles of the system, after the two pairs of pole-zero cancellation, are located at−0.2328 ± j2.0546, −2.2672 ± j0.6879, which are all on the left-hand side of the s-plane. It can be seen that the closed-loop system is stable. However, by checking the internal stability from the statement
>> [V,cc]=intstable(G,Gc,H)
we conclude that the system is not internally stable with the canceled pole-zero having positive real parts zi = pj = 1 returned incc.
3.1.2 Controllability and Observability Analysis
Controllability of linear systems
Controllability is an important property of a control system and plays a crucial role in many control problems, such as stabilization of unstable systems by feedback control.
Definition 3.3. The state xi(t )is said to be controllable if there is an input that in finite time drives it to any specified xi(tf)from initial state xi(0). The system is said to be fully controllable if all its states are controllable.
Since the full controllability of the system depends only upon the A and B matrices of the state space model, it is simply said that (A, B) is controllable.
Construct a transformation matrix Tcin the form
Tc = [B, AB, . . . , An−1B], (3.3) where n is the order of the system or the number of states. The matrix Tcis referred to as the controllability matrix and it can be generated using the MATLAB functionctrb(),
provided in the Control Systems Toolbox, with the syntax Tc=ctrb(A,B). The rank of Tc, i.e., rank(Tc), is called the controllability index of the system and equals the number of controllable states in the system. If rank(Tc)= n, the system is fully controllable.
Under a suitably chosen transformation matrix Tc, the state space model can be trans-formed into the following canonical form, through the staircase transformation;
Ac= A¯c 0
The above transformed state space representation is known as the controllability staircase form. The eigenvalues of A¯care called the uncontrollable modes. If the system is control-lable, the uncontrollable subspace A¯cwill be empty. The simplified transfer function of the system can be obtained from the controllable subspace
G(s)= Cc(sI− Ac)−1Bc+ D. (3.5) A MATLAB functionctrbf(), provided in the Control Systems Toolbox, can be used to perform the controllable staircase form transformation. The syntax of this function is [Ac,Bc,Cc,Tc]=ctrbf(A,B,C), where (A,B,C) is the given state space model and the returned state space model (Ac,Bc,Cc) has a staircase format which separates the controllable and uncontrollable subspaces. The returned matrix Tc is the transformation matrix.
Example 3.4. Consider a state space model given by
˙x =
One can use the following statements to check the controllability of the system:
>> A=[0,1,0,0; 0,0,-1,0; 0,0,0,1; 0,0,5,0]; B=[0; 1; 0; -2];
C=[1,0,0,0]; D=0; Tc=[B, A*B, Aˆ2*B, Aˆ3*B]; rank(Tc)
Since the rank of Tc is 4, which equals the order of the system, the system is fully controllable.
Example 3.5. Let us consider another system model given by
˙x =
The controllability can be analyzed using the following MATLAB statements:
>> A=[0,1,0,0; 3,0,0,2; 0,0,0,1; 0,-2,0,0]; B=[0;1;0;0];
C=[1,0,0,0]; Tc=[B, A*B, Aˆ2*B, Aˆ3*B]; rank(Tc) [Ac,Bc,Cc,T]=ctrbf(A,B,C)
The controllable index of the system is 3 since the rank of Tcis 3. The controllable staircase form can be written, in partitioned form, as
Ac=
0 0 0 0
−0.4472 0 −0.8944 0
0 0 0 −2.236
3.5777 0 0.4472 0
, Bc=
0 0 0
−1
, Cc=
−0.8944 0 0.4472 0
and it can be observed that the uncontrollable mode is at s = 0, which is the eigenvalue of the A¯cmatrix.
Observability of linear systems
Observability is a measure of how well internal states of a system can be inferred from knowledge of its external inputs and outputs. The observability and controllability of a system are mathematical duals.
Definition 3.4. A state xi(t )is said to be observable if for any tf > 0, the initial state xi(0) can be determined from the time history of the input u(t) and the output y(t) in the interval of[0, tf]. The system is said to be fully observable if all the states in the system are observable.
Since the observability of the system depends only upon the A and C matrices of the state space model, one can simply say that (A, C) is observable.
Construct a transformation matrix Toin the following form:
To=
C CA
... CAn−1
, (3.6)
where n is the order of the system. To is referred to as the observability matrix, which can be generated using the MATLAB functionobsv(), provided in the Control Systems Toolbox by To=obsv(A,C). rank(To)is called the observability index of the system or the number of observable states. If rank(To)= n, the system is then fully observable.
With a suitable transformation matrix To, the state space model can be transformed into the following canonical form:
Ao= A¯o A12 0 Ao
, Bo=B¯o Bo
, Co= [0, Co], (3.7)
known as the observability staircase form. The eigenvalues of A¯oare called the unobservable modes. If the system is fully observable, the unobservable subspace A¯owill be empty. The transfer function of the system can be simply expressed by
G(s)= Co(sI− Ao)−1Bo+ D. (3.8) By comparing the controllability problem to the observability problem, it is not dif-ficult to see that these problems are dual. That is, the observability problem of the sys-tem (AT, CT, BT, DT)is exactly the same as the controllability problem of the system (A, B, C, D).
A MATLAB function obsvf() in the Control Systems Toolbox can be used to perform the staircase form transformation. The syntax is
[Ao,Bo,Co,To]=obsvf(A,B,C)
and the arguments are similar to those of thectrbf()function.
Example 3.6. Consider again the uncontrollable system shown in Example 3.5. The ob-servability index and the staircase form can be obtained by
>> rank([C; C*A; C*Aˆ2; C*Aˆ3]) % or rank(obsv(A,C)) [Ao,Bo,Co,T,K]=obsvf(A,B,C); Ao,Bo,Co
In the partitioned matrix form, the staircase form can be written as
Ao=
0 1 0 0
0 0 −2 0
0 2 0 3
0 0 1 0
, Bo=
0 0 1 0
, Co=
0 0 0 1 .
The unobservable mode is at s= 0. In fact, it is not hard to verify that there are two eigenvalues at s= 0, with one uncontrollable and the other unobservable.
Controllability and observability Gramians
The controllability and observability Gramians are very important in the balanced realization of a transfer function model and the related model reduction method [30]. In terms of controllability and observability, one may ask how controllable and observable the system is. The controllability and observability Gramians can be used to address this concern.
Definition 3.5. The controllability and observability Gramians of the system (A, B, C, D)