Sometimes, the control impacts the state evolution in an affine manner.
Definition 52
A control affine system is given by the differential equation
When designing controllers, there is a wide variety of techniques we can use. Some simple techniques involve canceling out various types of nonlinearities in the system using the input. Here are some examples.
While these techniques can work, there are also more principled ways of designing controllers to satisfy different criteria, particularly for the case of control affine systems.
Control Lyapunov Functions
This result motivates the following definition.
Definition 53
Once we have a control lyapunov function, we can prove that it is possible to find a state feedback law that will make the origin globally asymptotically stable.
Theorem 27
Suppose that we have a control affine system, and we want to construct a control lyapunov function for the system.
Theorem 28
if
Definition 54
The small control property means that CLF will lead to a controller which has a small value that does not get too large when close to the equilibrium.
Hence, let
When the plant dynamics are naturally stabilizing, this controller exerts no control effort. When the plant dynamics are not naturally stabilizing, then the controller applies some control to stabilize the system. We can show that this is a minimum norm controller as it solves the optimization problem
Another type of controller is known as the Sontag controller.
Theorem 29
Feedback Linearization
SISO Case
Suppose we have a SISO control-affine system
Definition 55
If we differentiate again, then
Therefore, the relative degree of the system is essentially telling us which derivative of the output that we can control. By sequentially taking derivatives, we are essentially looking at the system
This change of coordinates allows us to put the system into a canonical form.
Definition 56
The normal form of a SISO control affine system is given by
With this parameterization, it is quite easy to see how we can make our system behave linearly. In particular, choose
Theorem 30
MIMO Case
Suppose instead we have a MIMO control affine system where
Definition 57
\label{thm:vector-relative-degree}
Definition 58
The normal form of a square MIMO system is given by
Theorem 31
Dynamic Extension
Sliding Mode Control
In sliding mode control, we design a controller
Reaching Mode
Sliding Mode
Backstepping
Definition 59
A system expressed in strict feedback form is given by
When systems are expressed in this way, we have a convenient method of designing controllers.
Theorem 32 (Backstepping Lemma)
the function
is a valid CLF and the control input
is a stabilizing controller.
If we apply Theorem 32 to a system expressed in strict feedback form, then we can recursively define controllers until we arrive at a controller for the full system.
x is the state of the system, and u is the input to the system. Note that for simplicity, the system is time-invariant. Further assume, without loss of generality, that f(0,0)=0. The goal of nonlinear feedback control is to find a state feedback law α(x) such that the equilibrium point xe=0 is globally asymptotically stable for the closed loop system
dtdx=f(x,α(x)).
dtdx=f(x)+G(x)u
where G(x) is a matrix dependent on the state vector x.
Set u such that it cancels out nonlinear terms and adds a stable linear term, effectively making the nonlinear system behave linear in the closed loop.
Set u to cancel destabilitizing nonlinear terms and add a stable linear term, so the stable nonlinearities help the input drive the system to equilibrium.
Set u to cancel destabilizing nonlinear terms, so the nonlinear system dynamics drive the system to equilibrium.
Set u to dominate destabilizing terms so they have a minimal impact on the overall system behavior.
If we can find an α(x) that makes the origin globally asymptotically stable, then the converse Lyapunov theorem says that we can find a corresponding Lyapunov function for the system.
A continuously differentiable, PDF, radially unbounded V:Rn→R is a Control Lyapunov Function for the system dtdx=f(x,u) if
∀x=0,infu(∇xV)⊤f(x,u)<0
Suppose f is Lipschitz and V is a control Lyapunov function, then there exists a smooth function α such that the origin is a globally asympototically stable equilibrium point of dtdx=f(x,α(x)).
infu∂x∂Vf(x,u)=infuLfV+∑iLgiVui<0
Here, each gi(x) is a column of G(x). If ∀i,LgiV=0, then Definition 53 is satisfied so long as LfV<0.
A function V is a control lyapunov function for a control affine system
dtdx=f(x)+∑igi(x)ui
LgiV∀i⟹LfV≤0
Notice that the condition in Theorem 28 is essentially saying that the l2-norm of the vector composed of the LgiV is equal to 0. The choice of CLF is important because different CLFs have different properties when used to derive controllers.
A CLF V(x) satisfies the small control property if ∀ϵ>0, ∃δ>0 such that x∈Bδ(0), then if x=0,∃u∈Bϵ(0) satisfying
dtdV=LfV+LGVTu<0.
Given a control lyapunov function for a control affine system V(x,u), we can devise a controller which stabilizes the system. In particular, we need
Suppose V:Rn→R is a CLF for SISO control affine system dtdx=f(x)+g(x)u where f(0)=0 where f and g are Lipschitz. Then the Sontag feedback control law is given by
makes the origin globally asymptotically stable. Moreover, αS(x) is continuous everywhere except x=0, is continuous at x=0 if V satisfies the small control property, and if V(x) is K+1 times continuously differentiable and f(x),g(x) are K times continously differentiable ∀x=0, then αS is Ktimes continuously differentiable.
Since we have a large number of tools which allow us to control linear systems, it would be ideal if we could somehow leverage those tools for nonlinear control. Feedback linearization is the process of finding a feedback control law u=α(x) such that under a nonlinear change of coordinates z=Φ(x), the system dtdx=f(x,α(x)) behaves like a linear system dtdz=Az. When the system is control-affine, there are well-established results which help us do this.
dtdxy=f(x)+g(x)u=h(x)
A SISO control affine system with an equilibrium point xe has strict relative degree γ if in a neighborhood U around the equilibirum point, LgLfγ−1h(x) is bounded away from 0 and
∀i=0,⋯,γ−2,LgLfγ−1h(x)=0
To understand relative degree, suppose we differentiate y once
dtdy=Lfh+Lghu.
If ∀x∈U,Lgh(x)=0 where U is some region around the equilibrium, then
∀x∈U,dtdy=Lfh(x).
∀x∈U,dt2d2y=Lf2h(x)+LgLfh(x).
Suppose that ∀x∈U,LgLfh(x)=0, then we can differentiate again. At some point, after γ differentiations, we will get
Suppose ∀i=0,⋯,γ−1, we let ξi(x)=dtidiy. These are γ linearly independent coordinates. Since the distribution
Δ(x)=span{g(x)}
is involutive, it is integrable, and so there must be n−1 functions ηi such that
∀x∈U,(∇xηi)⊤g(x)=0.
We can now choose n−γ of them which are linearly independent of the ξi and linearly independent with each other, and this forms a change of coordinates
, which is a linear system. Therefore, we can design a linear feedback controller v=α(x) where we have all of the tools of linear control at our disposal. However, notice that the ηi cannot be impacted by the control effort. These are known as the internal dynamics of the system. When ξ=0, then
dtdη=q(0,η)
are known as the Zero Dynamics of the system. Zero dynamics for a system can be dangerous because if they are unstable, then the system could be blowing up. When γ=n, then there are no zero dynamics. When this happens, we say the system is Full State Linearizable. Fortunately, there are necessary and sufficient conditions which guarantee full state linearization.
There exists a function h such that a control affine system dtdx=f(x)+g(x)u has relative degree n at x0 if and only if
[g(x)⋯adfn−2g(x)adfn−1g(x)]
has rank n and
[g(x)⋯adfn−3g(x)adfn−2g(x)]
has rank n−1 and is involutive in the neighborhood of x0. The h is chosen to satisfy
(∇xh)⊤[g(x)⋯adfn−3g(x)adfn−2g(x)]=0
dtdxy=f(x)+G(x)u=h(x)
We will assume that the number of outputs is equal to the number of inputs (i.e y,u∈Rm). To linearize the system, we can take the same idea of relative degree from the SISO case and apply it to the MIMO case. Define γj to be the lowest derivative of yj which is impacted by at least one input.
As before the dtdη represent the internal dynamics of the system that are not impacted by the control. As with the linear case, we can design a controller
u=A−1(x)Lfγ1h1(x)⋮Lfγmhm(x)+v
which renders the system linear. We can now choose v where each entry of v controls a different output. For this reason, we call A(x) the decoupling matrix. As in the SISO case, unless ∑jγj=n, there are zero dynamics to the system.
A control affine square system dtdx=f(x)+G(x)u has vector relative degree ∑jγj=n if and only if Δi is involutive for all i≤n−2, Δi has constant rank for all 1≤i≤n−1 and Δn−1 has rank n where
Sometimes, we can use full-state linearization even if h does not satisfy the conditions in Theorem 30. We do this by adding additional states to the system and corresponding pseudo-control inputs which help control these states. Sometimes, this can be done in a way which makes the extended system full-state linearizable.
u=β(x)sgn(s)
where s(x) describes a manifold called the “sliding manifold”. Sliding mode controllers have two states
During the reaching mode, the controller drives the state towards the sliding manifold s(x)=0. We choose s such that on the manifold, when the system is in sliding mode, the system naturally converges asymptotically to the equilibrium. If s(x)=0 is an invariant manifold, then the system will smoothly travel along the manifold to equilibrium. If the sliding manifold is not invariant, then the state will chatter around the manifold towards equilibrium as the controller continuously drives it back to the manifold once it leaves. To choose s, we need to find a CLF V(s) which converges to 0 in finite time when applying the sliding mode controller.