Equipos de mejora Kaizen

1. y(t)2≤ G(s)∞u(t)2. 2. y(t)∞≤ G(s)2u(t)∞.

3. G1(s)G₂(s)_∞≤ G1(s)_∞G2(s)_∞. H₂-norm and H_∞-norm evaluations

If the system model is given by an LTI object G, theG(s)2andG(s)_∞norms of the system can be evaluated, respectively, from the MATLAB function calls norm(G) and norm(G,inf). The norms of discrete-time systems can also be obtained with the same functions.

Example 3.11. Consider the discrete-time system

The H2- andH_∞- norms of the discrete-time system can be evaluated directly with the following statements:

3.2 Time Domain Analysis of Linear Systems

We remark that the time domain analytical solution to a linear system is always possible given a typical input signal. For general input signals, however, the time domain analysis has to be performed numerically.

3.2.1 Analytical Solutions to Continuous Time Responses

State space method

Consider an LTI system with its n-dimensional state space model

! ˙x(t) = Ax(t) + Bu(t)

y(t)= Cx(t) + Du(t) with initial condition x(0)= x0. (3.27) It has been stated in Chapter 2 that the time domain solution of (3.27) is

x(t)= e^Atx₀+

 _t

0

e^A(t−τ)Bu(τ )dτ, (3.28)

where e^Atis called the state transition matrix.

It can be seen from (3.28) that the most difficult part in the solution is the evaluation of the integral. If a certain transformation is introduced to remove the B term, the solution to the original problem can be significantly simplified.

Assume that the input signal is a unit step signal; then an extra state x_n+1(t )= u(t) can be introduced. Clearly,˙xn+1(t )= 0. Thus, the state space equation can be rewritten as

 ˙x(t)

So, the original state space equation can be converted into an autonomous system A class of commonly used input signals, which can be converted into an autonomous system, is defined as

u(t )= u1(t )+ u2(t )=

One may introduce some extra states, called augmented states, such that xn+1 = e^d1tcos(d4t ), xn+2 = e^d1tsin(d4t ), xn+3 = u1(t ), . . . , xn+m+3 = u^(m−1)₁ (t ). It can be shown that the augmented state space equations under such an input signal can be written as

A=

whose analytical solution is



x(t)= e^Atx(0). (3.34) A MATLAB functionss_augment()is written to establish the augmented state space model for the typical input signal:

1 function [Ga,Xa]=ss_augment(G,cc,dd,X) 2 G=ss(G); Aa=G.a; Ca=G.c; Xa=X; Ba=G.b; D=G.d;

3 if (length(dd)>0 & sum(abs(dd))>1e-5), 4 if (abs(dd(4))>1e-5),

5 Aa=[Aa dd(2)*Ba, dd(3)*Ba; ...

6 zeros(2,length(Aa)), [dd(1),-dd(4); dd(4),dd(1)]];

7 Ca=[Ca dd(2)*D dd(3)*D]; Xa=[Xa; 1; 0]; Ba=[Ba; 0; 0];

8 else,

9 Aa=[Aa dd(2)*B; zeros(1,length(Aa)) dd(1)];

10 Ca=[Ca dd(2)*D]; Xa=[Xa; 1]; Ba=[B;0];

11 end, end

12 if (length(cc)>0 & sum(abs(cc))>1e-5), M=length(cc);

13 Aa=[Aa Ba zeros(length(Aa),M-1); zeros(M-1,length(Aa)+1) ...

14 eye(M-1); zeros(1,length(Aa)+M)];

15 Ca=[Ca D zeros(1,M-1)]; Xa=[Xa; cc(1)]; ii=1;

16 for i=2:M, ii=ii*i; Xa(length(Aa)+i)=cc(i)*ii;

17 end, end

18 Ga=ss(Aa,zeros(size(Ca’)),Ca,D);

The syntax of the function is [G,x₀]=ss_augment(G,c,d,x₀), where the vectors c= [c0, c₁, . . . , c_m] and d = [d1, d₂, d₃, d₄] are used to describe the input function u(t ) in (3.32). The arguments G and x₀ are the model object and initial state vector, respectively, while the returned variables Gandx₀are, respectively, the augmented state-space model and its initial vector. Once the augmented system is established, the analytical solutions to the system can be easily obtained using the Symbolic Toolbox.

Example 3.12. Assume that a state space model is given by

 2e^−3tsin(2t), the functionss_augment()can be used to construct the augmented state space model:

and the augmented model is

˙x(t) =

The following statements can be used to find the analytical solution of the system:

>> syms t; y=Ga.c*expm(Ga.a*t)*xx0;

The output signal of the system is obtained as y(t )= −54+127

Let us consider the equivalent transfer function model:

G(s)= b₁s^m+ b2s^m−1+ · · · + bms+ bm+1

sⁿ+ a1sⁿ⁻¹+ a2sⁿ⁻²+ · · · + an−1s+ an

. (3.35)

For any input signal u(t ) with U (s) as its Laplace transform, the output signal can be obtained from Y (s)= G(s)U(s). Thus, in order to find y(t), an inverse Laplace transformation is needed such that y(t )= L⁻¹[Y (s)]. The Symbolic Toolbox of MATLAB can be used to evaluate the Laplace transform of given input signals, and the inverse Laplace transform function can be used to evaluate the analytical solution of the system.

Example 3.13. Assume that

G(s)= s³+ 7s²+ 3s + 4 s⁴+ 7s³+ 17s²+ 17s + 6

is the transfer function to be analyzed, and the input signal is given by u(t)= 2+2e^−3tsin 2t . The analytical solution to the output signal can be evaluated using the statements

>> syms s t;

G=(sˆ3+7*sˆ2+3*s+4)/(sˆ4+7*sˆ3+17*sˆ2+17*s+6);

u=2+2*exp(-3*t)*sin(2*t); U=laplace(u);

y=ilaplace(G*U)

and the analytical solution can be written as y(t )=4

3.2.2 Analytical Solutions to Discrete-Time Responses

Similar to the s-domain approach to the analytical solution for continuous systems, the Z transform can be used for discrete systems to evaluate the response to an input signal U (z).

Then, the analytical solution of the system H (z) can be obtained by solving the inverse Z transform such that y(n)= Z⁻¹[H (z)U(z)].

Example 3.14. Assume that

G(z)= (z− 1/3)

(z− 1/2)(z − 1/4)(z + 1/5)

is a discrete-time transfer function of the system. Also assume that the input signal is a unit step signal. The analytical solution can be obtained using the statements

>> syms z; u=sym(1); U=ztrans(sym(u));

H=(z-1/3)/(z-1/2)/(z-1/4)/(z+1/5);

y=iztrans(H*U)

and the analytical solution can be written as y(n)= 40 If the sampling interval T is given, the analytical solution can be rewritten as

y(kT )=40

3.3 Numerical Simulation of Linear Systems

The analytical solutions to linear systems were studied in the previous section. In real applications, one may prefer to have numerical solutions, and based on the results, the time domain responses can be plotted. Graphical visualization of system responses is usually more straightforward and informative for control engineers.

In this section, the numerical solution techniques to linear systems are presented, with a focus on some common responses such as step responses, impulse responses, and more generally, the time domain responses to arbitrary input signals.

3.3.1 Step Responses of Linear Systems

Step input signals and their responses are commonly used in control systems analysis and design. The typical step response of a second-order system is studied and specifications are given. Then, MATLAB-based evaluation of step responses are given.

Second-order system analysis

In classical control courses, second-order systems are often used as an example, where many properties of the linear control systems are illustrated.

Theorem 3.5. The closed-loop unit step response of a second order system