Välkomna till TSRT15 ReglerteknikFöreläsning 10

Summary of lecture 9State-space modelsLinearizationsTransfer functions vs state-space modelsStabilityControllability and observability

2Summary of lecture 9

Σ

G(s)F(s)

-1

R(s) Σ Z(s)ΣU(s)

N(s)

V(s)

Y(s)

We want to have both S(s) and T(s) close to zero

Fundamental limitation: T(s)+S(s) = 1

Additionally, Bodes integral tells us that S(s) cannot be arbitrarily small

3Summary of lecture 9

We use models with relative model errors

Robustness criteria: Stable if

Robust performance:

(Y(s) is the output signal without model errors, Y0(s) output with model errors)

G(s)F(s)

-1

R(s) Σ ΣY0(s)

Δ(s)

4

Different transfer functions (filter) on reference and feedback signal gives us the possibility to form the closed-loop transfer function and the complementary sensitivity function more freely

All theory regarding T(s) and S(s) still holds

Summary of lecture 9

Σ

G(s)R(s) Σ Z(s)ΣU(s)

N(s)

V(s)

Y(s)

Fr(s)

-Fy(s)

5State-space models

y(t): Missile positionu(t): Forcem: Mass

Newton:

Solution with force u(t)=1

y(t)

u(t)

To compute C1 and C2 we must know the initial conditions

This vector thus contains all information needed to determine the future position of the missile (for a given input signal)

Information completely defining the future of a system is called a state

6State-space models

It turns out that everything is much easier if we limit everything to first-order differential equations

We can write our system using first-order differential equations by introducing a state vector x(t)

We have

In matrix form

7State-space models

The same strategy can be applied to arbitrary linear differential equations

The result is a first-order differential equation

x(t) is the state of the system (typically derivatives of y(t)), u(t) is the system input, and y(t) is the system output

A is a nxn matrix, B is a nx1 matrix, C is a 1xn matrix, D is a scalar

8State-space models

Nonlinear state-space models

Most systems are nonlinear in reality

Can not be written directly in ouyr state-space formLaplace transform not possibleNo transfer functionSinusoidal input sinusoidal output no longer holds

We can often approximate the nonlinear system by using a linearizationaround a working-point (x0,u0)

9State-space models

Linearization of nonlinear state-space models

Taylor expansion gives

Now assume we have a stationary point (equilibrium) where f(x0,u0)=0

Introduce new variables Δx(t)=x(t)-x0, Δu(t)=u(t)-u0 och Δy(t)=y(t)-h(x0,u0)

10State-space models

y(t): Missile altitudeu(t): Forcem: Masscd: aerodynamic drag coefficient

Newton:

Introduce statesy(t)

u(t)

Example: Missile flying straight up with a nonlinear aerodynamic model

mg

11State-space models

Nonlinear state-space model

Linearizations (Jacobians)

Possible stationary points (arbitrary altitude, zero speed)

12State-space models

G(s)U(s) y(t)u(t)Y(s)

We now have two alternative (but equivalent) representations of a system described by a linear differential equation

How can we tell if the state-space model is stable?

How do we convert from state-space models to transfer function?

How do we convert from transfer function to state-space model?

13State-space models

State-space model to transfer function

Laplace transform the state-space model!

Eliminate X(s)

14State-space models

Example: The first missile model

State-space model

Start with the inverse

Final result

15State-space modelsStability: How can we determine stability (poles) from the state-space model?

Most easily seen using Cramers rule

The pole polynomial is given by det(sI-A). The poles are thus the solutions to

This is the definition of eigenvalues of A!

Poles = eigenvalues of A

16State-space models

Transfer function to state-space model

This is the trickier direction (called realization)…

1 If b0= b1=…= bm-1=0 you can use derivatives of y(t) as states and proceed as in our missile example

2 Controllable canonical form

3 Observable canonical form

Note: this means that the realization not is unique. There are infinitely many ways to realize a transfer function, corresponding to different coordinate systems for the state-space

17Controllability and observability

x1(t), x2(t), x3(t): Water levelu(t): Water fed to tank 2f1(t): Flow from tank 1 to 2f2(t): Flow from tank 2 to 3

Input: u(t)Output: y(t)=x2(t)

State-space model(linear, all coefficients = 1)x2(t)

x1(t)

x3(t)

f1(t)

f2(t)

u(t)

18Controllability and observability

We have a state-space model with the following matrices

Transfer function from u(t) to y(t)

The system has three states, so A has three eigenvalues, but the transfer function has only one pole! Two states have disappeared?!

The transfer function says we have a system with one state

19Controllability and observability

The state x1(t) can not be controlled and is called a uncontrollable state

The state x3(t) can not be measured (or derived from y(t) and its derivatives) and is called a unobservable state

A state-space model that contains no uncontrollable or unobservable states is said to be controllable and observable

A model which not is controllable and observable seems, in some sense, to have unnecessarily many states

x2(t)

x1(t)

x3(t)

f1(t)

f2(t)

u(t)

20State-space models

Definition: A vector x* is controllable is there exist an input u(t) drivingthe state from the initial condition x(0)=0 to x* in finite time. If all elements in Rn are controllable, the system is said to be controllable

Theorem: A state-space model is controllable if det([B AB A2B…An-1B]) ≠0

21State-space models

Definition: If x(0)=x* ≠0, u(t)=0 leads to y(t)=0 then x* is unobservable. The system is observable if there exist no such x*

Theorem: A state-space model is observable if det([C ;CA;CA2;CAn-1]) ≠ 0

(alternative interpretation: The state x(t) can be computed from y(t) and its derivatives)

22Controllability and observability

Definition: A state-space model is said to be a minimal realization ifthere does not exist any other state-space model representing the same transfer function, but with a lower number of states

Theorem: A controllable and observable state-space model is minimal

23State-space models

x2(t)

x1(t)

x3(t)

f1(t)

f2(t)

u(t)

Not controllable

Not observable

Our model is thus not minimal (which we saw when we computed the transfer function, and also realized physically)

24SummarySummary of todays lecture

All information needed to compute the future output from a system, given an input, is available in the state

Stability for a state-space model is checked using the eigenvalues of the A-matrix

The transfer function is obtained by Laplace transforming the state-space model

A state-space model is obtained from a transfer function by using stadnard forms (controllable and observable form)

If the state-space model is minimal, the transfer function has as many poles as there are statesA system is minimal if it is controllable and observable

25Summary

State: A collection of variables defining the future of the system

Minimal realization: A state-space model using as few states as possible to describe an input-output relation

Controllable: There are no states that we cannot control arbitrarily

Observable: There are no states that can be non-zero without us noticing it

Important concepts