OPTIMAL CONTROL
State Space Representation
Modeling and Analysis
2012/2013
José Sá da Costa
Chapter 1
OPTIMAL CONTROL
The aim of a control system is to force a given set of process
variables to behave in some desired and prescribed way by
either fulfilling some requirements of the time and frequency
domain or achieving the best performances as expressed by an
optimization index.
Classical control system design is generally a trial‐and‐error
process in which various methods of analysis are used iteratively
to determine the design parameters of an "acceptable" system.
Acceptable performance is generally defined in terms of time
and frequency domain criteria such as rise time, settling time,
peak overshoot, gain and phase margin, and bandwidth.
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Optimal Control
2
OPTIMAL CONTROL
A radically different performance criteria must be satisfied,
however, by the complex, multiple‐input, multiple‐output
systems required to meet the demands of modern technology.
For example, the design of a spacecraft attitude control system
that minimizes fuel expenditure is not amenable to solution by
classical methods.
A new and direct approach to the synthesis of these complex
systems, called optimal control theory, has been made feasible
by the development of the digital computer.
The objective of optimal control theory is to determine the
control signals that will cause a process to satisfy the physical
constraints and at the same time minimize (or maximize) some
performance criterion.
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Optimal Control
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State‐Space Representation: modeling and analysis
Summary
1. State‐Space Representation: modeling and analysis
1.1 State‐Space Definitions
1.2 State‐Space Representation of Dynamic Systems
1.3 From State‐Space to Transfer Functions
1.4 Solution of the Time‐invariant State‐Space Equations
1.5 Solution of the Time‐variant State‐Space Equations
1.6 Discrete Time State‐Space
1.7 Controllability and Observability
1.8 Stability Analysis in State‐Space
1.9 Relevant MATLAB Functions for State‐Space Modelling and Analysis
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State‐Space Representation: summary
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1. State‐Space Representation
A nontrivial part of any control problem is modeling the system.
The objective is to obtain the simplest mathematical description
that adequately predicts the response of the physical system to
all anticipated inputs.
The classical control design approach uses the external
mathematical representation of the process to synthetize the
controller: the transfer function for the Single Input Single
Output (SISO) case and the matrix transfer function for the
Multiple Input, Multiple Output (MIMO) case.
However, this external representation of the model does not give
information about the internal state of the system neither is
applicable to represent nonlinear systems.
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State‐Space Representation
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1. State‐Space Representation
The internal representation of the system dynamics can be made
by means of the state space representation.
The state‐space approach is a generalized time‐domain method
for modeling, analyzing and designing a wide range of control
systems and is particularly well suited to digital computational
techniques.
The approach can deal with
• Multiple Input, Multiple Output (MIMO) systems, or multivariable
systems
• Non‐linear and time‐variant systems
• Alternative controller design approaches, namely optimal control.
Let us first introduce the basic definitions associated to state
space representation.
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State‐Space Representation
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1. State‐Space Representation
,
Modeling and Analysis
1.1 State‐Space Definitions
1.1.1 State
1.1.2 State variables
1.1.3 State vector
1.1.4 State‐space
1.1.5 State‐space equations
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State
The state of a dynamical system is the smallest set of variables (called
state variables) such that the knowledge of these variables at t  t0 ,
together with the knowledge of the input for t  t0 , completely
determines the behavior of the system for any time t  t0 .
State variables
The state variables of a dynamic system are the variables making up
the smallest set of variables that determine the state of the dynamic
system.
If at least n variables x1 , x 2 , , xn , are needed to completely describe
the behavior of a dynamical system, then such n variables are a set of
state variables.
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State variables (cont.)
State variables need not be physically measurable or observable
quantities.
However, it is convenient to choose meaningful or easily measurable
quantities for the state variables, if this is possible at all, because state
space and optimal control laws will require the feedback of all state
variables.
State vector
If n state variables are needed to completely describe the behavior of
a given system, then these n state variables can be considered the n
components of a vector x. Such a vector is called a state vector.
A state vector is thus a vector that determines uniquely the system
state x(t) for any time t  t0 , once the state at t  t0 is given and the
input u(t) for t  t0 is specified.
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space
The n‐dimensional space whose coordinate axes consist of the x1 axis,
x2 axis, . . . , xn, axis, where x1 , x2,. . . , xn, are state variables is called
state‐space.
Any state can be represented by a point in the state space.
The manner in which the state variables change as a function of time
may be thought of as a trajectory in n dimensional space, called the
state‐space trajectory.
Two‐dimensional state‐space is sometimes referred to as the phase‐
plane when one state is the derivative of the other.
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations
The dynamic system must involve elements that memorize the values
of the input for t  t0 .
Since integrators (poles) in a continuous‐time system serve as memory
devices, the outputs of such integrators can be considered as the
variables that define the internal state of the dynamic system.
The number of state variables to completely define the dynamics of
the system is equal to the number of poles involved in the system.
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations
(cont.)
Assume that a multiple‐input‐multiple‐output (MIMO) system
involves: n states x1(t), x2(t), . . . , xn(t), r inputs u1(t), u2(t), . . . , ur(t)
and m outputs y1(t), y2(t), .. . , ym(t).
Then the system may be described by
(1)
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations (cont.)
and
(2)
If we define
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations (cont.)
and
and
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations (cont.)
Then eq.s (1) and (2) become
x (t )  f (x, u, t )
y (t )  g (x, u, t )
state equation
output equation
(3)
(4)
Since f and/or g involve time t explicitly, then the system is a time‐
varying system.
If eq.s (3) and (4) are linearized about the operating state, then we
have the following linearized state equation and output equation:
(5)
x (t )  A(t )x(t )  B(t )u(t )
(6)
y (t )  C(t )x(t )  D(t )u(t )
where A(t) is the state matrix, B(t) the input matrix, C(t) the output
matrix, and D(t) the direct transmission matrix.
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations (cont.)
The block diagram representation of eq.s (5) and (6) is
If vector functions f and g do not involve time t explicitly (time‐
invariant case) then the system is called a time‐invariant system. In
this case, eq.s (5) and (6) can be simplified to
x (t )  Ax(t )  Bu(t )
y (t )  Cx(t )  Du(t )
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State‐Space Representation Definitions
(7)
(8)
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1.1 State‐Space Definitions
State‐space equations (cont.)
where
The coefficients aij, bij, cij, and dij are constants, some of which may be
zero.
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations (cont.)
That corresponds to
and
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
State‐space equations (cont.)
The block diagram representation for the time‐invariant case, eq.s (7)
and (8), is
Matrices A, B, C, and D are called the state matrix, input matrix (or
control matrix), output matrix, and direct transmission matrix,
respectively.
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State‐Space Representation Definitions
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1.1 State‐Space Definitions
In conclusion in a state‐space representation :
 A system is represented by a state equation and an output
equation.
 For the Linear Time‐Invariant (LTI) case the internal structure of
the system is described by a first‐order vector‐matrix differen‐
tial equation.
 This fact indicates that the state‐space representation is funda‐
mentally different from the transfer function representation, in
which the dynamics of the system are described by the input
and the output, but the internal structure is put in a black box.
 The nonlinear case can be linearized most of the times by
Taylor's expansion around an equilibrium point
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State‐Space Representation Definitions
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1. State‐Space Representation
,
Modeling and Analysis
1.2 State‐Space Representation of Dynamic Systems
1.2.1 State‐space modelling
1.2.2 Systems without derivative terms in the input
1.2.3 Systems with derivative terms in the input
1.2.4 State‐space realizations
1.2.4.1 Controllable canonical form
1.2.4.2 Observable canonical form
1.2.4.3 Observable canonical form
1.2.4.4 Diagonal canonical form
1.2.4.5 Jordan canonical form
1.2.4.6 Transformation of state‐space representation
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State‐Space Representation of Dynamic Systems
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1.2 State Space Representation of Systems
As stated before the dynamics of a system have elements that
memorize the values of the input for t  t0 .
Mathematically speaking, integrators (poles) in a continuous‐time
system serve as memory devices. The outputs of such integrators can
be considered as the variables that define the internal state of the
dynamic system.
For a physical system energy storage elements retain the memory of
the system. A possible choice of state variables is often those variables
that represent energy storage.
The number of state variables to completely define the dynamics of
the system is equal to the number of poles (energy storages) involved
in the system.
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State‐Space Representation of Dynamic Systems
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1.2.1 State Space Modeling
Choosing state variables of physical systems
Linear Lumped Component
Variable
Electrical capacitor
Voltage
Electrical coil
Current
Spring
Position (displacement)
Mass (kinetic)
Velocity
Mass (potential)
Position (elevation)
Rotational inertia
Angular velocity
Rotational spring
Rotational angle
Thermal capacity
Temperature
Pressure vessel
Pressure
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example
Consider the mechanical system we assume
linear. The external force u(t) is the input to the
system and the displacement y(t) is the output.
From the figure the system equation is
It is a second order SISO system which means the
system involves two states variables. Let us define
Then we obtain
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example
(cont.)
or
The output equation is
In a vector‐matrix form, the state equations can be written as
The output equation can be written as
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example (cont.)
The previous equations are in the standard form
where
The corresponding block diagram is given by
Note that the state varia‐
bles are the outputs of
the integrators (storages)
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example
Consider the mechanical system shown below. The displacements z1
and z2 are measured from the respective equilibrium positions of the
carts before f is applied. The force is f = αu where u is a forcing func‐
tion (N).
Assuming that m1 = 10 kg, m2 = 20 kg, b = 20 N‐s/m, k1 = 30 N/m, k2 =
60 N/m, and α = 10, and null initial conditions derive a state space
representation of the system.
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example (cont.)
From the second Newton’s law the equations of motion for this system
are
If we choose z1, ż1, z2, and ż2 as state variables (outputs of the energy
storages) for the system and thus
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example (cont.)
The previous dynamic equations can be rewritten as
The state equation now becomes
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example (cont.)
Assuming that z1 and z2 are the outputs of the system; hence, the
output equations are
In terms of vector‐matrix equations, we have the state space repre‐
sentation of the system, i.e.
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example
(cont.)
Next, we substitute the given numerical values of the parameters into
the state equation. The result is
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State‐Space Representation of Dynamic Systems
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1.2.1 State‐Space Modeling
Example (cont.)
Thus the state space matrices are
Note that this system is a Single Input Multiple Output (SIMO) system
since we have one input ( f ) and two outputs (z1 and z2).
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State‐Space Representation of Dynamic Systems
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1.2.2 Systems without derivative terms in the input
The two examples of the modeling of dynamic systems in state space
form presented before are limited to the case where derivatives of the
input functions do not appear explicitly in the equations of motion, i.e.
no zeros in the model.
However, in real cases sometimes we have systems with zeros and
poles and the modeling of these systems in state space representation
should be addressed in a different way. In the next section we deal
with this case.
The methodology followed before for the case that the system
dynamics does not depend of derivative terms of the input (no zeros)
can be generalize for a n order system as follows:
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1.2.2 Systems without derivative terms in the input
Consider the following nth order system
(9)
Noting that the knowledge of
, together with
the input u(t) for t ³ 0 , determines completely the future behaviour
of the system, we may take
as a natural set of n
state variables.
Mathematically, this natural choice of state variables is quite
convenient. Practically, however, because higher‐order derivative
terms are inaccurate, due to the noise effects, such a choice of the
state variables may not be desirable.
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State‐Space Representation of Dynamic Systems
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1.2.2 Systems without derivative terms in the input
Let us define
Then eq. (9) can be written as
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State‐Space Representation of Dynamic Systems
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1.2.2 Systems without derivative terms in the input
or
where
x  Ax  Bu
(10)
The output can be given by
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State‐Space Representation of Dynamic Systems
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1.2.2 Systems without derivative terms in the input
or
y  Cx
where
(11)
C = [1 0  0 ]
Note that D is zero since the system is strictly proper.
Note that the state‐space representation for the transfer function
system
is given also by eq.s (10) and (11).
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1.2.3 Systems with derivative terms in the input
In this section, we take up the case where the equations of motion of a
system involve one or more derivatives of the input function.
In such a case, the variables that specify the initial conditions do not
qualify as state variables.
The main problem in defining the state variables is that they must be
chosen such that they will eliminate the derivatives of the input
function u in the state equation.
Example
Consider the mechanical system.
The displacements y and u are
measured from their respective
positions.
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1.2.3 Systems with derivative terms in the input
Example (cont.)
The equation of motion for this system is
or
If we choose the state variables
then we get
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1.2.3 Systems with derivative terms in the input
Example (cont.)
The right‐hand side of last equation involves the derivative term .
Note that, in formulating state‐space representations of dynamic
systems, we constrain the input function to be any function of time of
order up to the impulse function, but not any higher order impulse
functions, such as dδ(t)/dt, d2δ(t)/dt2, etc.
To explain why the right‐hand side of the state equation should not
involve the derivative of the input function u, suppose that u is the
unit‐impulse function δ(t).
Then the previous equation of ẋ2 by integration becomes
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1.2.3 Systems with derivative terms in the input
Example (cont.)
Notice that x2 (velocity) includes the term (k/m) δ(t).
This means that x2(0) = ∞, which is not acceptable as a state variable.
We should choose the state variables such that the state equation will
not include the derivative of u.
Suppose that we try to eliminate the term involving from the second
state equation. One possible way to accomplish this is to define
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State‐Space Representation of Dynamic Systems
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1.2.3 Systems with derivative terms in the input
Example (cont.)
Then
Thus, the acceptable state equations can now be given by
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1.2.3 Systems with derivative terms in the input
For the case where the dynamic equations involve , , , etc., the
choice of state variables becomes more complicated. Fortunately,
there are systematic methods for choosing state variables for a general
case of equations of motion that involve derivatives of the input
function u.
In what follows we shall present one systematic method for elimi‐
nating derivatives of the input function from the state equations.
Method
Consider the differential equation system
(12)
Having in mind that the state variables must be such that they will
eliminate the derivatives of u in the state equation.
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1.2.3 Systems with derivative terms in the input
Method (cont.)
One way to obtain a state equation and output equation is to define
the following n variables as a set of n state variables:
(13)
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1.2.3 Systems with derivative terms in the input
Method (cont.)
where b0 , b1 , b2 , , bn are determined from
(14)
With this choice of state variables the existence and uniqueness of
the solution of the state equation is guaranteed.
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1.2.3 Systems with derivative terms in the input
Method (cont.)
Thus, the state equation and output equation can be given in terms
of vector‐matrix equations by
(15)
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1.2.3 Systems with derivative terms in the input
Method (cont.)
(16)
Note that if β0 = bo = 0, then the state variable x1 is the output signal y,
which can be measured, and, in this case, the state variable x2 is the
output velocity ẏ minus b1u.
Note that this is not the only choice of a set of state variables when
you have input derivative terms. There are other methods to achieved
adequate state space representation.
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1.2.4 State Space Realization
Many techniques are available for obtaining state‐space representa‐
tions of transfer function (TF) of systems.
In the previous section we present a few methods. However, there
are some canonical forms that are convenient for control design.
State‐space Representation in canonical forms
Consider a system defined by
(17)
where u is the input and y is the output. This equation can also be
written as
(18)
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1.2.4 State Space Realization
The following state‐space representation is called a controllable
canonical form:
(19)
A in controllable companion form (ill conditioned)
(20)
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1.2.4 State Space Realization
The controllable canonical form is important in discussing the pole‐
placement approach to the control systems design.
The following state space representation is called an observable
canonical form
(21)
A in observable companion form (ill conditioned)
Note that the n x n state matrix of the state equation given by eq. (21)
is the transpose of that of the state equation defined by eq.(19).
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1.2.4 State Space Realization
and
(22)
Diagonal canonical form
Consider the transfer function system defined by eq. (18). For the
distinct roots case, eq. (18) can be written as
(23)
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1.2.4 State Space Realization
The diagonal canonical form of the state‐space representation of this
system is given by
(24)
(25)
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1.2.4 State Space Realization
Jordan canonical form
Next we shall consider the case where the denominator polynomial of
eq. (18) involves multiple roots. For this case, the preceding diagonal
canonical form must be modified into the Jordan canonical form.
Suppose, for example, that the pi's are different from one another,
except that the first three pi's are equal, or p1 = p2 = p3.
Then the factored form of Y(s)/U(s) becomes
The partial‐fraction expansion of this system in the Jordan canonical
form is given by
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1.2.4 State Space Realization
where
Jordan block
(26)
(27)
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1.2.4 State Space Realization
Example
Consider the system given by
Obtain state‐space representations in the controllable canonical form,
observable canonical form, and diagonal canonical form.
Resolution
Controllable Canonical Form from eq.s (19) and (20)
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1.2.4 State Space Realization
Example
(cont.)
Observable Canonical Form from eq.s (21) and (22)
Diagonal Canonical Form from eq.s (24) and (25)
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1.2.4 State Space Realization
Transformation of State Space Representation
It has been stated that a set of state variables is not unique for a given
system.
Suppose that x1(t), x2(t), . . . , xn(t) are a set of state variables. Then we
may take as another set of state variables any set of linear functions
provided that, for every set of values 1(t), 2(t), . . . , n(t), there
corresponds a unique set of values x1(t), x2(t), . . . , xn(t), and vice versa.
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
Thus, if x is a state vector, then , where
is also a state vector, provided the matrix P is nonsingular.
The above result allow us to conclude that different state vectors
convey the same information about the system behavior.
Also, we conclude that is always possible by simple linear trans‐
formation to transform the state matrix A to a convenient form since
the state vector transformation leads to the transformation of the
state matrix to
Note that the eigenvalues of A do not change by linear transforma‐
tion, thus the dynamics of the model remains the same after a
similarity transformation.
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
That is, a state space model
(27)
(28)
can be transformed into another state space model by transforming
the state vector x into state vector by means of the transformation
where P is nonsingular. Then eq.s (27) and (28) can be written as
or
(29)
(30)
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
Equations (29) and (30) represent another state space model of the
same system.
Since infinitely many n x n nonsingular matrices can be used as a
transformation matrix P, there are infinitely many state space models
for a given system.
Note that any state transformation leads to a different meaning for the
transformed state.
These results can be used to transform the system state space repre‐
sentation from one form to another, e.g. from a natural form to a
controllable canonical form, or to a diagonal canonical form.
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
Diagnolization of matrix A (n x n)
Consider an n x n state matrix
(31)
with distinct eigenvalues λ1, λ2, . . ., λn.
If the state vector x is transformed into another state vector z with the
use of a transformation matrix P, or
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
where
(32)
is known as the Vandermonde matrix.
Then P-1AP becomes
a diagonal matrix, or
(33)
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
If the matrix A defined by eq. (31) involves multiple eigenvalues, then
diagonalization is impossible.
For example, if the 3 X 3 matrix A
has the eigenvalues λ1, λ1 and λ3, where λ1 ≠ λ3.
In this case the transformation x = Sz, to diagonalize the matrix A is
given by
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
will yield
which corresponds to the Jordan canonical form.
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
Example
Consider the following state‐space representation of a system
where the states have a physical meaning (displacement, velocity and
acceleration).
a) Obtain the state‐space diagonal canonical form of the system.
b) Conclude about the pros and cons of this representation regarding
the initial one.
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
Rewriting the model in a standard form
(34)
where
The eigenvalues of A are the roots of the characteristic equation
and the eigenvalues of A are λ1 =‒1, λ2 = ‒2, and λ3 =‒3.
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
Thus, three eigenvalues are distinct. If we define a set of new state
variables z1 , z2 , and z3 obtained by the Vandermonde transformation
or
(35)
then, by substituting eq. (35) into eq. (34), we obtain
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
By premultiplying both sides of this last equation by P-1, we get
or
Simplifying gives
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1.2.4 State Space Realization
Transformation of State Space Representation (cont.)
The output equation, eq. (34), is modified to y = CPz
or
b) The new states z do not have physical meaning since they result
from the transformation x = Pz.
This makes difficult the physical reasoning about this diagonal
canonical form.
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State‐Space Representation of Dynamic Systems
69
1.2.4 State Space Realization
,
From Transfer Function to State Space in Matlab
MATLAB is quite useful in transforming a system model from single
input transfer function to state space and vice versa.
Let us write the closed‐loop transfer function as
Once we have this transfer function expression, the MATLAB com‐
mand allows to obtain the state space representation
will give a state‐space representation.
The MATLAB command gives one possible state‐space representation
from infinity of possibilities.
This command can be used when the system equation involves one or
more derivatives of the input function.
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State‐Space Representation of Dynamic Systems
70
1.2.4 State Space Realization
,
From transfer function to state space in Matlab
Example
Consider the transfer‐function system
From MATLAB command tf2ss results
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State‐Space Representation of Dynamic Systems
71
1.2.4 State Space Realization
,
From transfer function to state space in Matlab
Example
Consider the SIMO system
 2s  3 
 s 2  2 s  1

G (s)   2
s  0.4  1
From MATLAB command tf2ss results
 x1   0.4 1.0   x1  1 
  u
 x    1.0



0   x2  0 
 2 
 2.0 3.0   x1  0 
  u
y



1.6 0   x2  1 
José Sá da Costa
b = [0 2 3; 1 2 1];
a = [1 0.4 1];
[A,B,C,D] = tf2ss(b,a)
A=
‐0.4000 ‐1.0000
1.0000
0
B=
1
0
C=
2.0000 3.0000
1.6000
0
D=
0
1
State‐Space Representation of Dynamic Systems
72
1. State Space Representation
,
Modeling and Analysis
1.3 From State‐Space to Transfer Functions
1.3.1 SISO systems
1.3.2 MIMO systems
1.3.3 From state‐space to transfer function in Matlab
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From State Space to Transfer Function
73
1.3.1 SISO systems
,
Derivation of transfer function of SISO system from state‐
space equations
Let us consider the system whose transfer function is given by
(36)
This system may be represented in state space by the following
equations:
x (t )  Ax(t )  Bu(t )
y (t )  Cx(t )  Du(t )
(37)
(38)
where x is the state vector, u is the input, and y is the output.
The Laplace transforms of eq.s (37) and (38) are given by
(39)
(40)
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From State Space to Transfer Function
74
1.3.1 SISO systems
,
Since the transfer function was previously defined as the ratio of the
Laplace transform of the output to the Laplace transform of the input
when the initial conditions were zero, we set x(0) in eq. (39) to be
zero. Then we have
or
By pre‐multiplying
obtain
to both sides of this last equation, we
(41)
By substituting eq. (41) into eq. (40), we get
(42)
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From State Space to Transfer Function
75
1.3.1 SISO systems
,
Upon comparing eq. (42) with eq. (36), we see that
(43)
This is the transfer function expression of the system in terms of A,
B, C, and D.
Note that the right‐hand side of eq. (43) involves (sI – A)-1. Hence
G(s) can be written as
where Q(s) is a polynomial in s. Therefore, |sI – A|-1 is equal to the
characteristic polynomial of G(s). In other words, the eigenvalues of
A are identical to the poles of G(s).
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From State Space to Transfer Function
76
1.3.1 SISO systems
,
Example
Consider again the previous mechanical system. We saw
that state‐space equations for the system are given by
We shall obtain the transfer function for the system from
these state‐space equations by using eq. (43).
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From State Space to Transfer Function
77
1.3.1 SISO systems
,
resulting
Since
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From State Space to Transfer Function
78
1.3.1 SISO systems
,
we have
which is the transfer function of the mechanical system.
The same transfer function can be obtained applying directly the
Laplace transform to the differential equation representing the model
of the system, assuming initial conditions equal zero.
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From State Space to Transfer Function
79
1.3.2 MIMO systems
,
Derivation of transfer function of MIMO system from state‐
space equations
Next, consider the multiple‐input‐multiple‐output (MIMO) system.
Assume there are r inputs u1(t), u2(t), . . . , ur(t) and m outputs y1(t),
y2(t), .. . , ym(t). Define
The transfer matrix G(s) relates the output Y(s) to the input U(s), or
where G(s) is given by
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From State Space to Transfer Function
80
1.3.3 From SS to TF in Matlab
,
To obtain the transfer function from state space equations, use the
following MATLAB command
iu must be specified for systems with more than one input. For exam‐
ple, if the system has three inputs (u1, u2, u3), then iu must be either
1,2, or 3, where 1 implies u1, 2 implies u2, and 3 implies u3.
If the system has only one input, then either
or
may be used.
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From State Space to Transfer Function
81
1.3.3 From SS to TF in Matlab
,
Example
Obtain the transfer function of the system defined by the following
state‐space equations
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From State Space to Transfer Function
82
1.3.3 From SS to TF in Matlab
,
Example
(cont.)
From MATLAB program using command ss2tf
the transfer function obtained is given by
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From State Space to Transfer Function
83
1.3.3 From SS to TF in Matlab
,
Example
Consider a system with multiple inputs and multiple outputs
This system involves two inputs and two outputs. Four transfer
functions are involved:
and
.
When considering input u1, we assume that input u2 is zero, and vice
versa.
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From State Space to Transfer Function
84
1.3.3 From SS to TF in Matlab
,
Example
José Sá da Costa
(cont.)
From State Space to Transfer Function
85
1.3.3 From SS to TF in Matlab
,
Example
(cont.)
Resulting the following four transfer functions:
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From State Space to Transfer Function
86
1.3.3 From SS to TF in Matlab
,
Note on MIMO systems
To deal with the MIMO systems in Matlab it is easier to work with
state space descriptors, since it is difficult to represent transfer
function matrices as this requires 3D structures.
To this end, to facilitate this dual representation, the following
notation for a transfer function matrix of a system based on its state
space matrices has been introduced
A B 
G ( s )  C( sI  A) B  D  

C
D


1
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From State Space to Transfer Function
87
State Space Representation Problems
Problem 1
Consider the spring‐mass‐dashpot system of
the Figure.
Assuming angle θ to be the output of the system,
obtain the non‐linear and linear state‐space
representation of the system.
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State Space Representation
88
State Space Representation Problems
Problem 2
Consider the spring‐mass‐dashpot system
mounted on a massless cart as shown in
Figure.
In this system, u(t) is the displacement of
the cart and is the input to the system.
At t = 0, the cart is moved at a constant speed.
The displacement y(t) of the mass is the output. In this system, m
denotes the mass, b denotes the viscous friction coefficient, and k
denotes the spring constant. Assume the dashpot and spring have
linear behavior.
Obtain the following mathematical models of this system by assuming
that the cart is standing still for t < 0 and the spring‐mass‐dashpot
system on the cart is also standing still for t < 0:
a) The natural state‐space representation of this system.
b) The controllable canonical form of this system.
c) The observable canonical form of this system.
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State Space Representation
89
1.2 State Space Representation of Systems
,
,
Problem
3
Consider the transfer function system
Y ( s)
s

U ( s ) ( s  10)( s 2  4 s  16)
a) Verify if the state space representation below is a possible state‐space
representation of the above transfer function.
1
0   x1   0 
 x1   0
 x    0
 x    1  u
0
1
2
  
 2 

 x3   160 56 14   x3   14 
 x1 
y  1 0 0  x2    0 u
 x3 
b) Verify if the state‐space representation below is a possible state‐space
representation of the above transfer function.
 x1   14 56 160   x1  1 
 x    1
  x   0  u
0
0
2
  
 2  
 x3   0
1
0   x3  0 
 x1 
y  1 0 0  x2    0 u
 x3 
c) Obtain the state‐space representation of the above transfer function
using the MATLAB command tf2ss. Comment the result.
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State Space Representation
90
,
State Space Representation Problems
Problem 4
Consider the following state‐space representation of a system
1
0   x1  0 
 x1   0
 x    0
  x   0  u
0
1
 2 
 2  
 x3   6 11 6   x3  6 
 x1 
y  1 0 0  x2 
 x3 
where the state variables have a physical meaning (displacement,
velocity a
and acceleration).
a) Obtain the state‐space diagonal canonical form of the system.
b) Conclude about the pros and cons of this representation regarding
the initial one.
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State Space Representation
91
,
State Space Representation Problems
Problem 5
Consider the system given by
 9
x   26
 24
y  1 2
1 0
2
0 1  x   5  u
 0 
0 0 
1 x
a) Obtain the modal matrix
b) Obtain the diagonal (modal) state‐space representation.
Note: Modal matrix is the matrix that transforms a matrix A in to a diagonal matrix by
similarity transformation. This matrix is made of the eigenvector associated with the
eigenvectors of matrix A.
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State Space Representation
92
,
State Space Representation Problems
Problem 6
Consider the system given by
2
2


x  2 x  (  d ) x  u (t )
a) Obtain the state‐space representation of the system.
b) Obtain the modal matrix.
c) Obtain the modal state‐space representation
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State Space Representation
93
,
State Space Representation Problems
Problem 7
Consider the modal state‐space representation obtained in c) of
Problem 6.
a) Obtain the state‐space modal representation equivalent with only
real parameters.
b) Find the transformation matrix that directly transforms the state‐
space representation obtained in a) of Problem 6 to the
representation obtained in a) of Problem 7.
c) Generalize the transform obtained in b) for both complex and real
eigenvalues.
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State Space Representation
94
,
State Space Representation Problems
Problem 8
a) Determine the transfer functions and draw a block diagram for the
two‐input two –output system represented by
0 1
1 1 
0 2 
x  
x
u, y  
x



 2 3
0 2 
1 0 
b) Confirm the result using MATLAB.
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State Space Representation
95
,
State Space Representation Problems
Problem 9
a) Determine the state‐space modal representation of
4 s 2  15s  13
G(s)  2
s  3s  2
b) Confirm the result using MATLAB.
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State Space Representation
96
,
State Space Representation Problems
Problem 10
Consider the system represented in the block diagram.
a) Calculate the state‐space equations.
b) Determine the eigenvalues from the state equation.
c) Find the equivalent transfer function.
d) Verify if the system is controllable and / or observable.
e) Calculate the modal matrix and the modal state‐space canonical
form.
f) Draw the block diagram of the modal canonical form obtained in e).
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State Space Representation
97
1. State Space Representation
,
Modeling and Analysis
1.4 Solution of the Time‐Invariant State‐Space Equations
1.4.1 Solution of homogeneous state equations
1.4.2 Solution of nonhomogeneous state equations
1.4.3 Useful results in vector‐matrix analysis
1.4.4 Response of Systems in State Space Form in Matlab
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1.4 Solution of the Time‐Invariant State‐Space Equations
,
It is well known that the response of a linear time‐invariant
system is given by the convolution integral independently of the
adopted form of representation – transfer function or state‐
space.
For the general case of a arbitrary input and initial conditions
different of zero the system response is given by
t
x(t )  Φ(t )x(0)   Φ( )Bu(t   )d
0
(44)
where Φ(t ) corresponds to a generalization of the impulse
response (the unforced solution or the solution of the autono‐
mous system). For the SISO case we have the impulse response
function and for the MIMO case we have the matrix of impulse
response functions.
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Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.1 Solution of Homogeneous State Equations
Before we solve vector‐matrix differential equations, let us review the
solution of the scalar differential equation
(45)
In solving this equation, we may assume a solution x(t) of the form
(46)
By substituting this assumed solution into eq. (45), we obtain
(47)
If the assumed solution is to be the true solution, eq. (47) must hold
for any t.
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Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.1 Solution of Homogeneous State Equations
Hence, equating the coefficients of the equal powers of t, we obtain
The value of b0 is determined by substituting t = 0 into eq. (46), or
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Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.1 Solution of Homogeneous State Equations
Hence, the solution x(t) can be written as
We shall now solve the vector‐matrix differential equation
(48)
By analogy with the scalar case, we assume that the solution is in the
form of a vector power series in t, or
(49)
By substituting this assumed solution into eq. (48), we obtain
(50)
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Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.1 Solution of Homogeneous State Equations
Thus by equating the coefficients of like powers of t on both sides of
eq. (50), we obtain
By substituting t = 0 into eq. (49), we obtain
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Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.1 Solution of Homogeneous State Equations
Thus, the solution x(t) can be written as
The similarity of this equation to the infinite power series for a scalar
exponential, we call it the matrix exponential and write
In terms of the matrix exponential, the solution of eq. (48) can be
written as
(51)
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Solution of the Time‐Invariant State ‐Space Equation
104
,
1.4.1 Solution of Homogeneous State Equations
Laplace transform approach
Let us first consider the scalar case
(52)
Taking the Laplace transform of eq. (52), we obtain
(53)
Solving eq. (53) gives
The inverse Laplace transform of this last equation gives the solution
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Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.1 Solution of Homogeneous State Equations
Laplace transform approach (cont.)
Extending this procedure to the homogeneous state equation case
(54)
Taking the Laplace transform of eq. (54), we obtain
(55)
Solving eq. (55) gives
or premultiplying both sides of this last equation by (sI - A)-1, we
obtain
The inverse Laplace transform of this last equation gives the solution
(56)
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Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.1 Solution of Homogeneous State Equations
Laplace transform approach (cont.)
Note that
Hence, the inverse Laplace transform of (sI - A)-1 gives
(57)
Note: this is the closed solution for the matrix exponential.
From eq.s (56) and (57), the solution of eq. (54) is obtained as
or
where the state‐transition
matrix is
(58)
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Solution of the Time‐Invariant State ‐Space Equation
107
,
1.4.1 Solution of Homogeneous State Equations
Example
Obtain the state‐transition matrix Φ(t) of the following system
For this system,
The state‐transition matrix Φ(t) is given by
Since
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Solution of the Time‐Invariant State ‐Space Equation
108
,
1.4.1 Solution of Homogeneous State Equations
Example (cont.)
the inverse of (sI - A) is given by
Hence
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Solution of the Time‐Invariant State ‐Space Equation
109
,
1.4.2 Solution of Nonhomogeneous State Equations
Having address the homogeneous state equation solution case, for
both the time domain approach and the Laplace transform approach,
now we will concentrate in the Laplace transform to solve the
nonhomogeneous state equation. To solve it in the time domain the
procedure is analogous to the homogeneous case.
Laplace transform approach
Let us now consider the nonhomogeneous state equation described by
(59)
x = Ax + Bu
where x = n‐vector
u = r‐vector
A = n x n constant matrix
B = n x r constant matrix
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1.4.2 Solution of Nonhomogeneous State Equations
Laplace transform approach (cont.)
The Laplace transform of this last equation (59) yields
or
Premultiplying both sides of this last equation by (sI - A)-1, we obtain
Using the relationship given by eq. (53) gives
The inverse Laplace transform of this last equation can be obtained by
use of the convolution integral as follows:
(60)
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Solution of the Time‐Invariant State ‐Space Equation
111
,
1.4.2 Solution of Nonhomogeneous State Equations
Example
Obtain the time response of the following system
where u(t) is the unit‐step function occurring at t = 0, or u(t) = 1(t)
For this system
The state‐transition matrix Φ(t) = eAt was obtained in previous
Example as
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Solution of the Time‐Invariant State ‐Space Equation
112
,
1.4.2 Solution of Nonhomogeneous State Equations
Example (cont.)
The response to the unit‐step input is then obtained as
or
If the initial state is zero, or x(0) = 0, then x(t) can be simplified to
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,
1.4.2 Solution of Nonhomogeneous State Equations
Solution in terms of x(t0)
So far we have assumed the initial time to be zero. If, however, the
initial time is given by t0 instead of t = 0, then the solution to eq. (54)
must be modified to
(61)
**************************************************************************************
Next subsection will introduce several useful results in vector‐matrix
analysis that will be useful to calculate the state‐transition matrix and
the structural properties introduced in next section.
**************************************************************************************
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Solution of the Time‐Invariant State ‐Space Equation
114
,
1.4.3 Useful results in vector‐matrix analysis
Matrix Exponential
It can be proved that the matrix exponential of an n x n matrix A,
k k
¥
1
1
A
t
e At = I + At + A 2t 2 +  + A k t k = å
(63)
2!
k!
k =0 k !
converges absolutely for all finite t. Hence, computer calculations for
evaluating the elements of e At by using the series expansion can be
easily carried out.
¥
Ak t k
Because of the convergence of the infinite series å
, the series
k =0 k !
can be differentiated term by term to give
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Solution of the Time‐Invariant State ‐Space Equation
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1.4.3 Useful results in vector‐matrix analysis
Matrix Exponential
(cont.)
The matrix exponential has the property that
In particular, if s = t, then
Thus, the inverse of e At is e-At .
Since the inverse of e At always exists, e At is nonsingular.
Note that
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1.4.3 Useful results in vector‐matrix analysis
State‐Transition Matrix
Previously, we defined the state‐transition matrix as
(64)
For a given system this matrix is unique.
Also, this matrix holds the properties of the matrix exponential
Other relevant properties for LTI systems are
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Solution of the Time‐Invariant State ‐Space Equation
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1.4.3 Useful results in vector‐matrix analysis
State‐Transition Matrix
(cont.)
Has referred before the state‐transition matrix is unique and contains
all information about the free motion of the system.
If the eigenvalues λ1, λ2, . . ., λn of the matrix A are distinct, then Φ(t)
will contain the n exponentials
In particular, if the matrix A is diagonal, then
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Solution of the Time‐Invariant State ‐Space Equation
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1.4.3 Useful results in vector‐matrix analysis
State‐Transition Matrix
(cont.)
If there is a multiplicity in the eigenvalues, for example, if the
eigenvalues of A are
λ1, λ1, λ1, λ4, λ5, . . ., λn ,
then Φ(t) will contain, in addition to the exponentials
terms like
and
.
Cayley‐Hamilton Theorem
This theorem is very useful in proving theorems involving matrix
equations or solving problems involving matrix equations.
Consider an n x n matrix A and its characteristic equation:
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Solution of the Time‐Invariant State ‐Space Equation
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1.4.3 Useful results in vector‐matrix analysis
Cayley‐Hamilton Theorem
(cont.)
The Cayley‐Hamilton theorem states that the matrix A satisfies its
own characteristic equation, or that
(65)
The proof can be found in any algebra text book.
Minimal Polynomial
The characteristic equation is not, however, necessarily the scalar
equation of least degree that A satisfies.
The least‐degree polynomial having A as a root is called the minimal
polynomial, i.e. the minimal polynomial of an n x n matrix A is
defined as the polynomial ϕ(λ) of least degree,
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,
1.4.3 Useful results in vector‐matrix analysis
Minimal Polynomial
(cont.)
Such that ϕ(A) = 0 , or
(66)
The minimal polynomial plays an important role in the computation of
polynomials in an n x n matrix.
The minimal polynomial ϕ(λ) of an n x n matrix A can be determined
by the following procedure:
1. Form adj(λI - A) and write the elements of adj(λI - A) as factored
polynomials in λ.
2. Determine d(λ) as the greatest common divisor of all the
elements of adj(λI - A). Choose the coefficient of the highest‐
degree term in λ of d(λ) to be 1. If there is no common divisor,
d(λ) = 1.
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,
1.4.3 Useful results in vector‐matrix analysis
Minimal Polynomial
(cont.)
3. The minimal polynomial ϕ(λ) is then given as |λI - A| divided by
d(λ).
Calculation of the Matrix exponential
In solving control engineering problems, it often becomes necessary
At
to compute e .
If matrix A is given with all elements in numerical values, MATLAB
AT
provides a simple way to compute e , where T is a constant, with
the command “expm”.
In other cases we need to calculate e At by analytical methods.
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Solution of the Time‐Invariant State ‐Space Equation
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1.4.3 Useful results in vector‐matrix analysis
Computation of e At : Method 1
If matrix A can be transformed into a diagonal form, then e At can
be given by
(67)
where P is a diagonalizing matrix for A.
If matrix A can be transformed into a Jordan canonical form, then e At
can be given by
(68)
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Solution of the Time‐Invariant State ‐Space Equation
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1.4.3 Useful results in vector‐matrix analysis
Computation of e At : Method 1. (cont.)
Example
Consider the following matrix A
The characteristic equation is
Thus, matrix A has a multiple eigenvalue of order 3 at λ = 1. It can be
shown that matrix A has a multiple eigenvector of order 3. The
transformation matrix that will transform matrix A into a Jordan
canonical form can be given by
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1.4.3 Useful results in vector‐matrix analysis
Computation of e At : Method 1. (cont.)
Example (cont.)
The inverse of matrix S is
Then it can be seen that
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1.4.3 Useful results in vector‐matrix analysis
Computation of e At : Method 1. (cont.)
Example (cont.)
¥
Jkt k
Jt
e =å
Noting that
k =0 k !
(69)
results
Thus, we find
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126
,
1.4.3 Useful results in vector‐matrix analysis
Computation of e At : Method 2.
The second method of computing e At uses the Laplace transform
approach. Previously we had
At
Thus, to obtain e , first invert the matrix (sI ‒ A). This results in a
matrix whose elements are rational functions of s. Then take the
inverse Laplace transform of each element of the matrix.
Example
Consider the following matrix A
José Sá da Costa
Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.3 Useful results in vector‐matrix analysis
Computation of e At : Method 2. (cont.)
Example (cont.)
The eigenvalues of A are 0 and 2. Since
We obtain
Hence
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1.4.4 Response of Systems in SS Form in Matlab
MATLAB can be used to obtaining response curves of systems
that are written in state‐space form.
To do so we first define the system with
Step response
For a unit‐step input, the MATLAB command
or
will generate plots of unit‐step responses.
The time vector is automatically determined when t is not
explicitly included in the step commands.
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1.4.4 Response of Systems in SS Form in Matlab
Note that when step commands have left‐hand arguments, such
as
no plot is shown on the screen. Hence, it is necessary to use a
plot command to see the response curves.
The matrices y and x contain the output and state response of
the system, respectively, evaluated at the computation time
points t.
Matrix y has as many columns as outputs and one row for each
element in t.
Matrix x has as many columns as states and one row for each
element in t.
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1.4.4 Response of Systems in SS Form in Matlab
Note also that:
• the scalar iu is an index into the inputs of the system and
specifies which input is to be used for the response;
• t is the user‐specified time.
• if the system involves multiple inputs and multiple outputs,
the step commands produces a series of step response plots,
one for each input and output combination of
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1.4.4 Response of Systems in SS Form in Matlab
Example
Obtain the unit‐step response curves for the following system:
Since this system involves two inputs and
two outputs, we have four individual step‐
response curves.
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1.4.4 Response of Systems in SS Form in Matlab
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1.4.4 Response of Systems in SS Form in Matlab
To plot two step‐response curves for the input u1 in one diagram
and two step‐response curves for the input u2 in another
diagram, we may use the commands
and
.
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1.4.4 Response of Systems in SS Form in Matlab
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1.4.4 Response of Systems in SS Form in Matlab
Impulse response
The unit‐impulse response of a dynamic system defined in a state
space may be obtained with the use of one of the following MATLAB
commands:
Example
Obtain the impulse response of the system given by the state‐space
matrices
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1.4.4 Response of Systems in SS Form in Matlab
Program
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Impulse‐response curves
Solution of the Time‐Invariant State ‐Space Equation
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,
1.4.4 Response of Systems in SS Form in Matlab
Response to arbitrary input
The command lsim produces the response of linear time‐invariant
systems to arbitrary inputs.
If the initial conditions of the system in state‐space form are zero, then
produces the response of the system to an arbitrary input u with user‐
specified time t.
If the initial conditions are nonzero in a state‐space model, the
command
where x0 is the initial state, produces the response of the system,
subject to the input u and the initial condition x0.
To find the response to the initial condition x0 the previous command
may be used where u is a vector consisting of zeros having length size(t),
or the command
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,
1.4.4 Response of Systems in SS Form in Matlab
Example
Plot the unit‐ramp response curve of the following system:
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1.4.4 Response of Systems in SS Form in Matlab
Example
Obtain the response curves to initial condition of the following system:
,
José Sá da Costa
with
Solution of the Time‐Invariant State ‐Space Equation
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,
Response of Systems in State‐Space Form
Problem 11
Consider the system represented by
0 6
0 
x  
x   u

 1 5
1 
a) Find the state transition matrix Φ(t)=exp [At] in the time domain
space.
b) Find the state transition matrix Φ (t)=exp [At] by using the
frequency domain space.
c) Confirm in MATLAB the result obtained for the transition matrix in
a) and b).
d) Solve the state equation for a unit step function scalar input
u(t)=u‐1(t).
a) Using MATLAB compute and plot the time response of d).
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Response of Systems in State‐Space Form
Problem 12
Obtain the response y(t) of the following system
where u(t) is the unit‐step input occurring at t = 0.
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,
Response of Systems in State‐Space Form
Problem 13
Compute e At , where
Problem 14
Consider the following matrix A
Compute e At by three methods.
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Response of Systems in State‐Space Form
Problem 15
Given the system equation
find the solution in terms of the initial conditions x1(0), x2(0) and x3(0).
Problem 16
Obtain a state‐space representation of the following system with
MATLAB
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1. State Space Representation
,
Modeling and Analysis
1.5 Solution of the Time‐variant State‐Space Equations
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Solution of the Time‐Variant State ‐Space Equation
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1.5 Solution of Time‐Variant State‐Space Equation
The state‐space form of a time‐variant system is given
x (t )  A(t )x(t )  B(t )u(t )
x(0)  x0
with
y (t )  C(t )x(t )  D(t )u(t )
(70)
Assume that the coefficient matrices in the above model are
sufficiently well behaved for there to exist a unique solution to the
state‐space model for any specified initial condition x(t0) and any
integrable input u(t). For instance, if these coefficient matrices are
piecewise continuous, with a finite number of discontinuities in any
finite interval, then the desired existence and uniqueness properties
hold.
In this case for the general case of a arbitrary input and initial condi‐
tions different of zero the system response is unique given by
t
x(t )  Φ(t , t0 )x(t0 )   Φ(t , )B( )u(t   )d
0
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Solution of the Time‐Variant State ‐Space Equation
(71)
146
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1.5 Solution of Time‐Variant State‐Space Equation
where the state‐transition matrix Φ(t , ) is unique.
Unlike the time‐invariant case, we cannot define Φ as a simple
exponential matrix. In fact, Φ can't be defined in general, because it
will actually be a different function for every system. However, the
state‐transition matrix does follow some basic properties that we can
use to determine the state‐transition matrix.
Φ holds the following proprieties
 (t , )  A(t )Φ(t , )
Φ
(72)
Φ( , )  I
(73)
and also
Φ(t2 , t1 )Φ(t1 , t0 )  Φ(t2 , t0 ),
Φ (t , )Φ(t , )  I,
1
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Φ 1 (t , )  Φ( , t ),
dΦ(t0 , t0 )
 A(t )
dt
Solution of the Time‐Variant State ‐Space Equation
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1.5 Solution of Time‐Variant State‐Space Equation
However, in the time‐variant case, there are many different functions
that may satisfy these requirements, and the solution is dependent on
the structure of the system since Φ has the general form
(74)
and the homogeneous solution is given by
x(t )  Φ(t , t0 )x0
(75)
There are several ways to calculate the state‐transition matrix. Here
we will refer some of the must used approaches.
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1.5 Solution of Time‐Variant State‐Space Equation
Method based on the Fundamental Matrix
The solutions of the zero input case (homogeneous equation)
(76)
x (t )  A(t )x(t )
form an n‐dimensional vector space in the interval T = [t0, +∞[.
Any set of n linearly‐independent solutions {x1, x2, ..., xn} to the
homogeneous equation is called a fundamental set of solutions.
The fundamental matrix is formed by creating a matrix out of the n
fundamental vectors, i.e. Ψ(t )   x1 x 2 x3  x n 
This matrix will satisfy the state equation
 (t )  A(t )Ψ(t )
Ψ
(77)
Also, any matrix that solves this equation can be a fundamental matrix
if and only if the determinant of the matrix is non‐zero for all time t in
the interval T.
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1.5 Solution of Time‐Variant State‐Space Equation
Once we have the fundamental matrix of a system, we can use it to
find the state transition matrix of the system:
Φ(t , t0 )  Ψ(t )Ψ 1 (t0 )
(78)
Example
Given the following fundamental matrix, find the state‐transition
matrix.
 t 1 t 
e
e
Ψ(t )  
2

t 
0 e 
From (77) results
 t
e
1

Φ(t , t0 )  Ψ(t )Ψ (t0 ) 

0
José Sá da Costa
1 t   t0
e  e
2
t  
e  0
1 3 t0    t  t0
e  e

2


t0
e   0
1 t  t0

( e  e  t  3 t0 ) 
2

t  t0
e

Solution of the Time‐Variant State ‐Space Equation
150
,
1.5 Solution of Time‐Variant State‐Space Equation
Method 2
If A(t) is triangular (upper or lower triangular),
(79)
then
(80)
The state transition matrix can be determined by sequentially integra‐
ting the individual rows of the state equation, starting from the third
one
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1.5 Solution of Time‐Variant State‐Space Equation
Method 3
If A(t) is diagonal
(81)
Then the state‐transition matrix is given by
(82)
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1.5 Solution of Time‐Variant State‐Space Equation
Method 4
If for every τ and t, the state matrix commutes as follows:
t
t



A(t )  A( )d   A( )d  A(t )
 t0
  t0

Then the state‐transition matrix is given by
(83)
t
Φ(t , t0 )  e
t0 A ( ) d
(84)
Method 5
If A(t) can be decomposed as the following sum
n
A(t )   α i fi (t )
i 1
(85)
where αi are constant matrices such that αiαj = αjαi and fi is a single‐
value function.
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1.5 Solution of Time‐Variant State‐Space Equation
Method 5 (cont.)
then the state‐transition matrix is given by
n
Φ(t , t0 )   e
αi
t
t0 fi ( ) d
i 1
(86)
Example
Calculate the state‐transition matrix for the system with the state
matrix given by
 t 1
A(t )  


1
t


We can decompose this matrix as
1 0   0 1 
A (t )  
t
1


0 1   1 0 
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1.5 Solution of Time‐Variant State‐Space Equation
Example (cont.)
Then the state‐transition matrix will be given by
Φ(t , t0 )  e
α1
t
t0
f1 ( ) d α 2
e
t
t0 f2 ( ) d
Solving the two integrations we obtain
Φ(t , t0 )  e
José Sá da Costa
2
2
0   0
1  ( t  t0 )

 
2
2
2  0
( t t0 )    t  t0
e
t  t0 
0 
Solution of the Time‐Variant State ‐Space Equation
155