Lesson 5: State Space Systems

Systems and Control Theory Lecture Notes

Laura Giarr´ e

Lesson 5: State Space Systems

 State

 Dimension

 Infinite-Dimensional systems

 State-space model (nonlinear)

 LTI State Space model

 Linearization

State of a System

 If a system is causal, in order to compute its output at a given time t ₀ , we need to know only the input signal over (−∞, t ₀ ].

(Similarly for DT systems.) A lot of information!

 Can we summarize it with something more manageable?

(memory!) Definition ( State)

The state x (t 1 ) of a causal system at time t 1 is the

information needed, together with the input u between times t ₁ and t ₂ , to uniquely predict the output at time t ₂ , for all t ₂ ≥ t 1 .

 The state of the system at a given time summarizes the whole history of the past inputs to predict the output at future times.

 The state of a system is a vector in the Euclidean space R ⁿ .

State Property of DT state-space models

Given the initial state x (t0) and input u (t) for t 0 ≤ t ≤ t _f

(with t ₀ and t _f arbitrary),

we can compute the output y (t) for t 0 ≤ t ≤ t _f and the state x (t) for t ₀ ≤ t ≤ t _f

 Thus, the state at any time t ₀ summarizes everything about the past that is relevant to the future.

 Keeping in mind this fact that the state variables are the

memory variables (or, in more physical situations, the energy

storage variables) of a system, often guides us quickly to good

choices of state variables in any given context.

Dimension of a System

 The choice of a state for a system is not unique ( there are infinite choices, or realizations).

 However, there are come choices of state which are preferable to others; in particular, we can look at minimal realizations.

Definition ( Dimension of a system )

We define the dimension of a causal system as the minimal number of variables sufficient to describe the system’s state (i.e., the dimension of the smallest state vector).

 We will deal mostly with finite-dimensional systems, i.e.,

systems which can be described with a finite number of

variables.

Infinite-Dimensional Systems

 Even though we will not address infinite-dimensional systems in detail, some examples are very useful:



(CT) Time-delay systems: Consider the very simple time delay S _T , defined as a continuous-time system such that its input and outputs are related by

y (t) = u(t − T ).

In order to predict the output at times after t, the knowledge of the input for times in (t − T , t] is necessary.



PDE-driven systems: Many systems in engineering, arising,

e.g., in structural control and flow control applications, can

only be described exactly using a continuum of state variables

(stress, displacement, pressure, temperature, etc.). These are

infinite-dimensional systems.

State-Space model

 Finite-dimensional linear systems can always be modeled using a set of differential (or difference) equations as follows:

Definition (Continuous-time State-Space Models)

d

dt x (t) = ˙x(t) =f (x(t), u(t), t) y (t) =g(x(t), u(t), t);

Definition (Discrete-time State-Space Models) x (k + 1) =f (x(k), u(k), k)

y (k) =g(x(k), u(k), k);

State-Space model (linear)

 If the system satisfies Linearity:

Definition (Continuous-time State-Space Models)

d

dt x (t) = ˙x(t) =A(t)x(t) + B(t)u(t) y (t) =C(t)x(t) + D(t)u(t);

Definition (Discrete-time State-Space Models) x (k + 1) =A(k)x(k) + B(k)u(k)

y (k) =C(k)x(k) + D(k)u(k);

 The matrices appearing in the above formulas are in general

functions of time, and have the correct dimensions.

LTI State-Space model

 If the system satisfies Linearity and Time Invariance (LTI systems) an n is the dimension of the state vector:

LTI TC systems:

d

dt x (t) = ˙x(t) =Ax(t) + Bu(t) y (t) =Cx(t) + Du(t);

LTI TD systems:

x (k + 1) =Ax(k) + Bu(k) y (k) =Cx(k) + Du(k);

 A ∈ R ^n×n , B ∈ R ^n×m , C ∈ R ^p×n , D ∈ R ^m×p , x ∈ R ^n×1 ,

y ∈ R ^p×1 , u ∈ R ^m×1 .

Finite-Dimensional linear Systems

 Recall the definition of a linear system.

 Essentially, a system is linear if the linear combination of two inputs generates an output that is the linear combination of the outputs generated by the two individual inputs.

 The definition of a state allows us to summarize the past inputs into the state, i.e.,

u (t), −∞ ≤ t ≤ ∞ ⇔

 x (t ₀ ),

u (t), t ≥ t 0 ,

 (similar formulas hold for the DT case.)

 We can extend the definition of linear systems

Finite-Dimensional linear Systems

Definition (Linear system (again))

A system is said a Linear System if, for any u

, u

, t

, x

, x

, and any two real numbers α, β, the following are satisfied:

 x(t

) = x

,

u(t) = u

(t), t ≥ t

, → y

 x(t

) = x

,

u(t) = u

(t), t ≥ t

, → y

 x(t

) = αx

+ βx

,

u(t) = αu

(t) + βu

(t), t ≥ t

, → αy

+ βy

 Similar formulas hold for the discrete-time case.

Linearization 1

 Linear models frequently arise as description of small

perturbations away from a nominal solution of the system

 Suppose x _o (t), u _o (t) and y _o (t) constitute a nominal solution of the system, i.e. a collection of CT signals that jointly

satisfy the state space equations.

 Let the input and the initial condition be perturbed from their nominal values x (0) = x o (0) + δx(0), u(t) = u o (t) + δu(t)

 Let the state trajectory accordingly be perturbed x (t) = x _o (t) + δx(t)

 Substituting them and performing a Taylor expansion to

first-order terms, we find

Linearization 2

δ ˙x(t) ≈

 ∂f

∂x



o δx(t) +

 ∂f

∂u



o δu(t) δy(t) ≈

 ∂g

∂x



o δx(t) +

 ∂g

∂u



o δu(t)

 Where the n × n matrix [ _∂x ^∂f ] _o denotes the Jacobian of f (. . .) with respect to x and the other matrices are defined

accordingly.

 The ij-th entry is the partial derivative of f _i with respect to x _j .

 The subscript _o indicates that the Jacobian are evaluated

along the nominal trajectories.

Linearization 3

 The linearized model is evidently linear:

δ ˙x(t) =A _l (t)δx(t) + B _l (t)δu(t) δy(t) =C _l (t)δx(t) + D _l (t)δu(t)

 where

A _l (t) =

 ∂f

∂x



o

B _l (t) =

 ∂f

∂u



o

C _l (t) =

 ∂g

∂x



o

D _l (t) =

 ∂g

∂u



o

 Similar formulas hold for the discrete-time case.

Examples

System Type

˙x(t) = tx ² (t) NLTV

˙x(t) = x ² (t) NLTI

˙x(t) = tx(t) LTV

˙x(t) = (cost)x(t) LPV

˙x(t) = x(t) LTI

Thanks

DIEF- [email protected] Tel: 059 2056322

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