Introduction to Control Theory
2.2 Linear State-Variable Systems
Many physical systems such as the robots considered in this book are described by differential or difference equations. These describing equations, which are usually obtained from fundamental physical laws, provide the starting point for the analysis and control of systems. There are, of course, some systems which are so complicated that describing differential (or difference) equations are not available. We do not consider those systems in this book.
In this section we study the state-space model of physical systems that are linear. We limit ourselves to systems described by ordinary differential equations which will lead to a finite-dimensional state space. Partial differential equations, leading to infinite-dimensional systems, are needed to study flexible robotic manipulators, but those are not considered in this textbook. We stress that the material of this chapter is intended as a quick introduction to these topics and will not be comprehensive. The readers are referred to [Kailath 1980], [Antsaklis and Michel 1997] for more rigorous presentations of linear control systems.
Continuous-Time Systems
A continuous-time system is said to be linear if it obeys the principle of superposition] that is, if the output y1(t) results from the input u1(t) and the output y2(t) results from the input u2(t), then the output resulting from
2.2 Linear State-Variable Systems 23 u(t)=α1u1(t)+α2u2(t) is given by y(t)=α1y1(t)+α2y2(t), where α1 and α2 are scalar constants. Linear, single-input/single-output (SISO), continuous-time, time-invariant systems are described by linear, scalar, constant-coefficient ordinary differential equations such as
(2.2.1)
where ai, bi, i=0,…,n are scalar constants, y(t) is a scalar output and u(t) is a scalar input. Moreover, we are given for some time t0 the initial conditions,
Note that the input u(t) is differentiated at most as many times as the output y(t). Otherwise, the system is said to be non-dynamic.
State-Space Realization
The state of the system is defined as a sufficient set of variables, which when specified at time t0 along with the input u(t), t≥t0, is sufficient to completely determine the behavior of the system for all t≥t0 [Kailath 1980]. The state vector then contains all necessary variables needed to determine the future behavior of any signal in the system. By definition, such a state vector x(t) is not unique, a feature that will be exploited later. In fact, if x is a state vector then so is any (t)=Tx(t), where T is any n×n invertible matrix. For the continuous-time system described in (2.2.1), the following choice of a state
vector is possible:
where , i=1,2,…, n. The input-output equation then reduces to (2.2.3) (2.2.2)
This particular state-space representation is known as the controllable canonical form [Kailath 1980], [Antsaklis and Michel 1997]. In general, a linear, time-invariant, continuous-time system will have more than one input and one output. In fact, u(t) is an m×1 vector and y(t) is a p×1 vector. The differential equations relating u(t) to y(t) will not be presented here, but the state-space representation of the multi-input/multi-output (MIMO) system becomes
(2.2.6)
where A is n×n, B is n×m, C is p×n, and D is p×m. For the specific forms of A, B, C, and D, the reader is again referred to [Kailath 1980], [Antsaklis and Michel 1997]. A block diagram of (2.2.6) is shown in Figure 2.2.1a. Note that the minimal number of states is equal to the required number of initial conditions in order to find a unique solution to the set of differential equations.
(2.2.4)
where
(2.2.5)
2.2 Linear State-Variable Systems 25
EXAMPLE 2.2–1: Double Integrator Consider a SISO system described by
ÿ(t)=u(t)
This system is known as the double integrator and represents a wide variety of physical systems described by Newton’s Law. In order to obtain a state-space description, let
so
Figure 2.2.1: (a) State-space block diagram of (2.2.6); (b) Transfer function block diagram of (2.2.6)
䊏
There are 2 inputs to the system given by u2 which causes the ground to move and u1 which causes the platform m1 to move. The system also has 2 outputs, namely the motion y1 of platform m1 and the motion y2 of platform m2. The experiments will be conducted on top of platform m1 and therefore, one would like to minimize the size of y1. The differential equations describing this system are obtained using Newton’s second law:
A state-space formulation of this system can be obtained by choosing
Transfer Functions
Another equivalent representation of linear, time-invariant, continuous-time systems is given by their transfer function, which relates the input of the system u(t) to its output y(t) in the Laplace variable s or in the frequency domain. It is important to note that the transfer function description has no 䊏
2.2 Linear State-Variable Systems 27
information about the initial conditions of the system and, as such, will not provide a unique output to a particular input unless all initial conditions are zero [Antsaklis and Michel 1997], [Kailath 1980]. The transfer function formalism, however, is important in practice, since many engineers are familiar with frequency-domain specifications. In addition, the identification of many systems may be effectively performed in the frequency domain [Ljung 1999].
It is therefore imperative that one should be able to move between the state-space (or modern) description and the transfer function (or classical) description.
Let us consider the system described by (2.2.6) and take its Laplace transform,
(2.2.7)
where X(s), U(s), and Y(s) are the Laplace transforms of x(t), u(t), and y(t) respectively. The initial state vector is x(0). By eliminating X(s) between the two equations in (2.2.7), we find the following relation:
Figure 2.2.2: Two-platform system
between the input U(s) and the output Y(s) when x(0)=0, that is,
Y(s)=[C(sI-A)-1B+D] U(s). (2.2.9) The transfer function of this particular linear, time-invariant system is given by
P(s)=C(sI-A)-1B+D (2.2.10)
Y(s)=P(s)U(s) (2.2.11)
EXAMPLE 2.2–3: Transfer Function of Double Integrator
Consider the system of Example 2.2.1. It is easy to see that the transfer function is
Discrete-Time Systems
In the discrete-time case, a difference equation is used to described the system as follows:
(2.2.12)
where ai, bi, i=0,…,n are scalar constants, y(k) is the output, and u(k) is the input at time k. Note that the output at time k+n depends on the input at time k+n but not on later inputs; otherwise, the system would be non-causal.
䊏 such that (see Fig.2.2.1)
2.2 Linear State-Variable Systems 29
The input-output equation then reduces to
y(k)=b0x1(k)+b1x2(k)+…+ bn-1xn(k)+u(k) (2.2.14) A more compact formulation of (2.2.7) and (2.2.8) is given by
(2.2.15)
where
State-Space Representation
In a similar fashion to the continuous-time case, the following state-vector is defined:
(2.2.13)
(2.2.16)
(2.2.17)
where A is n×n, B is n×m, C is p×n, and D is p×m.
In many practical cases, such as in the control of robots, the system is a continuous-time system, but the controller is implemented using digital hardware. This will require the designer to translate between continuous-and discrete-time systems. There are many different approaches to
“discretizing” a continuous-time system, some of which are discussed in problem is referred to [Åström and Wittenmark 1996], [Franklin et al.
1997].
EXAMPLE 2.2–4: Double Integrator in Discrete Time
Recall Example 2.2.1 which presented a model of the double integrator or Newton’s system. One discrete-time version of the differential equation is given by the following difference equation
where T is the sampling period in seconds. If we choose x1(k)=y(k) and x2(k)=x1(k+1), we obtain the state-space description
Transfer Function Representation
In a similar fashion to the continuous-time case, a linear, time-invariant, discrete-time system given by (2.2.17) may be described in the Z-transform domain, from input U(z) to output Y(z) by its transfer function P(z) such that
Y(z)=P(z)U(z)
䊏 Chapter 3. The interested reader in this very important aspect of the control
31
where
P(z)=C(zI-A)-1B+D
Note that the Z transform is used in the discrete-time case versus the Laplace transform in the continuous-time case.
EXAMPLE 2.2–5: Tranfer Function of Discrete-Tiem Double Integrator The transfer function of the Example 2.2.4 is given by